implied volatility

The Concepts and Practice of Mathematical Finance

by Mark S. Joshi  · 24 Dec 2003

say. Actually he will quote two vols, the price to buy and the price to sell. The vol used to price is often called the implied volatility as it is the volatility implied by 73 74 Practicalities the price. A market maker is expected to quote two prices or two vols, which

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50 60 70 80 90 100 110 120 130 140 150 160 170 Strike Fig. 4.1. Some possible smiles. Indeed, if one plots the implied volatility as a function of the strike of an option, one obtains a curve which is roughly smile-shaped. The qualitative nature of the curve will

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as 6 - 0+ will become the digital (see Section 6.3). The price this gives will be different from the Black-Scholes formula with the implied volatility of the call at K plugged in. To see this, we write a, the volatility, as a function of strike. Our approximating portfolio is then

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that the price of the digital call is ac M (K, aQ (K, Q(K)) au (K). If we had just plugged the Black-Scholes implied volatility into the digital call option formula, we would have only got the first term and not the second. The error is therefore the Vega of

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; (iii) numeric integration; (iv) Monte Carlo; (v) replication. Replication is the safest method of pricing as it automatically takes smile effects into account. Using the implied volatility to price options other than vanillas can lead to pricing errors. Monte Carlo pricing relies on the fact that the price of an option is

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pays -5 otherwise. Decompose it into vanilla options. 200 The practical pricing of a European option Exercise 7.7 Suppose two smiles have the same implied volatility at 100. One smile is downwards sloping and the other one is upwards sloping. How will the prices of digital calls struck at 100 compare

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that when spot is on the barrier B the smile should be symmetric - a call option of strike K/B must trade with the same implied volatility as a put (or call) option of strike B/K. This symmetry condition is certainly satisfied by a Black-Scholes model with time-dependent volatility

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we now outline. (i) Write the instrument's payoff in terms of the behaviour of forward rates. (ii) Observe the current forward rates and their implied volatility in the market. (iii) Choose a numeraire which makes the rates as driftless as possible. (iv) Compute the drifts of the forward rates in the

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Fig. 14.6. The smiles produced by displaced-diffusion models by varying the displacement but rescaling the volatility to keep the at-the-money (6%) implied volatility constant. The displacements vary from zero (flat line) to 5% (most sloped line). involves moving rate and strike by a we have a similar

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against the markets' fickleness. Note that the fact that the market's choice of measure has changed will mean that the implied volatilities of options in the market have changed. Thus implied volatilities can and do change from day to day without the real-world volatility and jump-intensity changing, simply because the market

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for f and write down an expression for it. 16 Stochastic volatility 16.1 Introduction It is an observed fact in the market that the implied volatilities of traded options vary from day to day. We have seen that in an incomplete model, such as jumpdiffusion, this can be caused by the

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added uncertainty, and one then needs to plug the instantaneous volatility into the new formula and invert the Black-Scholes formula to get the new implied volatility. Having decided to make the instantaneous volatility stochastic, it is necessary to decide what sort of process it follows. Volatility is generally chosen to follow

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the-money option might be used to hedge an out-of-the-money option with the same maturity. This is because it is really the implied volatilities that are moving around rather than the instantaneous. They are being driven as much by changing risk-preferences and expectations of future volatility as by

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of large movements in the underlying stock, stochastic-volatility models lead to fatter tails for the distribution of the final stock price. This leads to implied-volatility smiles which pick up out-of-the-money; that is, smile-shaped smiles! If we allow correlation between the underlying and the volatility then a

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auxiliary variable which increases the dimensionality again. 16.8 Key points Stochastic-volatility models are currently quite popular. They provide a simple mechanism for allowing implied volatilities of options in the market to vary from day to day. A rapid pricing formula can be developed. They have the appealing property that it

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volatility in the risk-neutral measure but in practice a mean-reverting volatility is used. In a stochastic-volatility model, the instantaneous volatility and the implied volatility are quite different things. Prices can be developed by Monte Carlo, transform methods and PDE solutions. If volatility and spot are uncorrelated then the spot

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instantaneous volatilities stochastic but let the implied volatility drive the process. Such an approach has been developed by Schonbucher, [1321. One approximation to stochastic volatility that has recently become very popular is the

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choice of density for a single time horizon is equivalent to specifying the call option prices, and that in turn is equivalent to specifying the implied volatility smiles. Our choice of is therefore a choice of smile dynamics. In other words, how we choose c is a statement about how we

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we need to think about: how the smile changes with spot, and how the smile changes with time. 18.2.1 Sticky or floating The implied volatility smile is a function of strike. The crucial question is how does that function change when spot is moved. Two fairly obvious functional forms are

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Gamma model with constant parameters, then everything is defined relative to the current value of spot and the current time. Once again we obtain an implied volatility function of the form &(S, K, t, T) = &(K/S, T - t). (18.5) Smile dynamics and the pricing of exotic options 418 0.

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will be driven up, and increase skewness in the short-dated smiles. A second problem with jump-diffusion models is that the at-the-money implied volatility is much greater than the diffusive volatility since a large component of the price comes from jump risk. However, if one measures the diffusive volatility

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of a major index such as the S&P and compares it with the implied volatility of at-the-money options they are not particularly different. This is an argument against using jump-diffusion models. (See Figure 18.12.) A

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P is the discount factor. This means that the value of our cliquet at time T1 is 0.4Q T2 - T1, where o- is the implied volatility observable in the market at time T1 for options expiring at time T2. Thus when we buy and sell cliquets, we are really trading the

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We will call this option an optional cliquet. Suppose we use a deterministic smile model such as Black-Scholes, jumpdiffusion or Variance Gamma. Then the implied volatility prevailing at time T1 is already known at time To and a unique price, C(1), at time T1 for the cliquet is known. The

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obvious candidate for such a model is a stochastic-volatility model. Of course, by this we mean a stochastic instantaneous volatility model not a stochastic implied volatility model, so the connection is not direct as the name suggests. Another possibility is to use a jump-diffusion model with stochastic parameters. For example

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to be drawn from this product is that when compound optionality is involved it is very important to take account of the stochastic nature of implied volatility. 18.7 Key points 427 Consider another related cliquet product. Suppose we trade two cliquets on the same underlying with different strikes. Suppose one has

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we consider the portfolio consisting of the difference of the two options, what are we trading? If the two options are trading at the same implied volatility at time T1, then their values will cancel as they have the same moneyness, and the trade will be of zero value. Our value at

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principal components analysis of the volatility surface for index options in S&P and FTSE. They have many interesting results including that relative movements of implied volatilities have little correlation with the underlying. Emanuel Derman carried out an analysis of how changes in spot related to changes in skew for the S

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vanilla options for a jump-diffusion model with log-normal jumps. Implement a Monte Carlo pricer also and check they give the same answers. Implied volatility Implement an implied volatility function - this is a function which inverts the Black-Scholes price function to get the unique volatility which gives the correct price for the

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Asian call option with monthly resets. Comparison with Black-Scholes Fit a Black-Scholes model with time-dependent volatility so that it gives the same implied volatility to at-the-money call options at the monthly reset times. Reprice the Asian option with this model. Now do a discrete barrier call option

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Asian call option with monthly resets. Comparison with Black-Scholes Fit a Black-Scholes model with time-dependent volatility so that it gives the same implied volatility to at-the-money call options at the monthly reset times. Reprice the Asian option with this model. Now do a discrete barrier call option

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of strike for options with maturity 0.25, 0.5, 1, 2, 4, and 8 years. (See project 12 for discussion of how to do implied volatilities.) How does the shape change with maturity? Compare with jump-diffusion smiles and stochastic volatility smiles. Repeat trying varying values of 6. B.16 Project

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Asian call option with monthly resets. Comparison with Black-Scholes Fit a Black-Scholes model with time-dependent volatility so that it gives the same implied volatility to at-the-money call options at the monthly reset times. Reprice the Asian option with this model. Now do a discrete barrier call option

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techniques that do not depend on the stock price being continuous will work. Exercise 17.2 2N. References [1] C. Alexander, Principal component analysis of implied volatility and skews, ISMA Centre Discussion Paper in Finance, 2000-10. [2] C. Alexander, Market Models: a Guide to Financial Data Analysis, Wiley, 2001. [3] L

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of the Art, Thompson International Press, 1997. [39] J.H. Cochrane, Asset Pricing, Princeton University Press, 2001. [40] R. Cont, Jose da Fonseca, Dynamics of implied volatility surfaces, Quantitative Finance 2, 2002, 45-60. [41] R. Cont, P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall, 2003. [42] J.C. Cox, S

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fundamental theorem of asset pricing, Mathematische Annalen 300, 1997, 463-520. [46] E. Derman, Regimes of volatility: some observations on the variation of S&P implied volatilities, Goldman Sachs Quantitative Strategy Research Note, January 1999. [47] E. Derman, I. Kani, Riding on a smile, Risk 7, 1994, 32-9. [48] D. Duffle

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Probability 32(4), 1995, 1077-88. [131] W. Rudin, Principles of Mathematical Analysis, McGraw Hill, 1976. [132] P.J. SchSnbucher, A market model for stochastic implied volatility, Philosophical Transactions of the Royal Society A 357(1758), 1999, 2071-92. [133] W. Schoutens, Levy Processes in Finance: Pricing Financial Derivatives, Wiley, 2003. [134

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jump-diffusion world, 372 Vega, 79 hedging strategy, 17-18, 44, 76 stop-loss, 143 hedging, discrete, 76 HJM model, 322 homogeneity, 274, 281, 383 implied volatility, see volatility, implied importance sampling, 193 in-the-money, 30 incomplete, 431 incomplete market, 50, 361, 367-375, 389, 390 incomplete model, 89 incremental path

Expected Returns: An Investor's Guide to Harvesting Market Rewards

by Antti Ilmanen  · 4 Apr 2011  · 1,088pp  · 228,743 words

Treasury Inflation-Protected Securities, real bonds UIP Uncovered Interest Parity (hypothesis) VaR Value at Risk VC Venture Capital VIX A popular measure of the implied volatility of S&P 500 index options VMG Value Minus Growth, equity value premium proxy WDRA Wealth-Dependent Risk Aversion X Cash flow Y Yield YC

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. Realized time series volatility of inflation and bond yields likewise show a peak around 1980 and later fall sharply. More relevant measures include option-based implied volatility, dispersion among many forecasters, and self-reported inflation uncertainty (individual forecasters’ wide probability density). • Ex ante real yields and BRPs. The mountain shape in

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predictors and subsequent excess returns of the IG corporate bond index for the period 1990–2009. Wide credit spreads are the strongest bullish indicators. High implied volatilities—both equity market volatility and interest rate volatility—are also bullish, partly reflecting the contemporaneous correlation between volatility and spread levels. Weak real activity and

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currency index—would have raised the SR from 0.61 to 0.70. Dynamic portfolio construction. Scaling position sizes inversely by recent historical or option-implied volatility tends to improve Sharpe ratios. Using mean variance optimization in sizing carry positions is another potential approach, but its chief benefit is to take

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variance exposure (i.e., exposure to vega risk) can be achieved via trading delta-hedged straddles or variance swaps. Both approaches capture the difference between implied volatility/variance (IV) and realized volatility/variance (RV) over the life of the contract. I assume that the reader is familiar with basic option-pricing

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positions benefit from lower realized volatility. Under idealized BSM conditions, a delta-hedged short at-the-money straddle’s return reflects the difference between contracted implied volatility and realized volatility over the option’s life. In reality, market price moves are not continuous, hedging is costly and done discretely, and volatility

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The cross-sectional implication of BSM assumptions (constant volatilities, no market frictions, etc.) is that all options across all strike prices should have the same implied volatility. Thus, real-world deviations from BSM assumptions should explain observed smile effects and other apparent violations of no-arbitrage pricing. (The smile refers to the

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premia in addition to volatility expectations. Thus, analogous to the way a term spread reflects the market’s rate expectations plus required bond risk premia, implied volatilities also reflect the market’s volatility expectations plus some volatility-related risk premia. Any decomposition requires a specified model and empirical estimation, and any resulting

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estimates are noisy. As a convenient simplification, it is common to interpret excess implied volatility over realized volatility as a volatility risk premium. Empirically, this approximation works well, at least for assessing historical average premia. If we recognize the time

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5), roughly half of the strategy’s long-run return is related to directional risk (the equity risk premium) and half to the fact that implied volatility is typically higher than realized volatility (the volatility risk premium). Between 1994 and 2005, each component earned 24 bp per month, the latter component

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one-month or 3-month variance swaps on the S&P 500. Excess return for such a product is proportional to the difference between squared implied volatility and squared realized volatility over the life of the contract. Backtested results were extremely impressive until 2007 but the losses in autumn 2008 were

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in less than two months. (All index volatility-selling strategies plummeted in autumn 2008, but leverage made the losses of this index exceptionally high.) Although implied volatilities were high, realized volatilities exceeded them, reaching levels only seen during the 1987 crash. Indeed, volatility-selling strategies have suffered serious losses mainly on these

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–short strategies. Timing overlays on volatility selling could include simple rules that have been effective in the past—say, do not sell volatility if current implied volatility is well below the “normal” (past average) implied, recent realized, or predicted volatility, where predicted volatility is based on a GARCH-type model or

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as profitable (or risky) as with equity index options. Trolle–Schwartz (2008) study variance risk premia for energy commodities. Between 1996 and 2006 average implied volatility exceeded average realized volatility by 4% for crude oil and by 3% for natural gas. The SRs from shorting estimated variance swaps were 0.59

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returns. • One explanation is that correlation risk is more consistently priced than volatility risk. This could explain the fact that for equity index options, implied volatility exceeds realized volatility most of the time, but the same is not true for single-stock options. In the same vein, and again contrary to

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times may conceal growing imbalances, excessive leverage, stretched market valuations, and vulnerability to regime change. Critics noted that many measured risks—including realized volatilities and implied volatilities—were at their lowest in 2007 when the true risk of loss was at its highest. • Some researchers distinguish measurable volatility from ambiguity, unquantifiable (“Knightian

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in one day), that is an “unknown unknown” and represents a qualitatively greater degree of uncertainty. 19.2 FACTOR HISTORY Chapter 15 presented evidence on implied volatilities. Figure 19.1 gives a long perspective on realized market volatilities. More frequent data sampling makes volatility estimates sharper, so here I use daily return

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can artificially create a positive contemporaneous relation between volatility and returns as well as a negative lead–lag relation. Finally, stocks whose options have high implied volatilities appear to have higher near-term returns. In another line of study, Ang–Bali–Cakici (2010) document evidence that information moves slowly between option and

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. As Table 19.1 shows, the contemporaneous relation between realized volatility—both level and change—and realized return is clearly negative, and even stronger with implied volatility data (last row) [1]. (Realized one-month volatility is measured as the standard deviation of daily returns during the month.) However, the predictive relation

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strategies As a reminder, the most interesting results in Chapter 15 about profitable option-trading strategies are that• Equity index options tend to be expensive (implied volatilities higher than realized volatilities), especially OTM puts (with a more pronounced skew than for single-stock options):• index volatility selling can be profitable, with high

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on an index while buying volatility on its constituent assets) can give more consistent profits as some volatility risk is hedged. • For single-stock options, implied volatilities do not consistently exceed realized volatilities, but OTM calls have especially large negative average returns (perhaps because skewness-seeking investors prefer them as speculative vehicles

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-allocated from loss-making, insurance-providing, or liquidity-providing strategies. I do not claim such timing is easy. Many speculators who began to buy implied volatility in 2005, already then at historically low levels, bled money for two years and may have gone out of business before volatility finally rose in

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for equity indices, perhaps a sign of “crash-o-phobia”, was highlighted in Chapter 15. Jump risk has also attracted much academic attention for explaining implied volatility skew, time-varying volatility, and fat-tailed return distributions. Some studies estimate jump risk premia, but I leave the topic aside. Source notes. Tail

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abnormally high reward for volatility selling. I find a similar split between PEH-like and risk premium hypotheses in the FX market when I study implied volatilities for the EUR/USD exchange rate. 22.4 CONCLUSIONS Empirical evidence clearly shows that carry measures are better at predicting near-term asset returns

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returns are based on the simulated Merrill Lynch Equity Volatility Arbitrage Index since 1989, which tries to gain from the typically positive gap between market-implied volatility and subsequent realized volatility of the S&P 500 index’s Bloomberg ticker MLHFEV1 index. Chapter 15 also analyzes the simulated covered-call-writing

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premium), Greenwood–Vayanos (debt maturity share), the Chicago Fed (CFNAI or Chicago Fed National Activity Index), Consensus Economics (expected fiscal balance), Bloomberg (VIX and MOVE implied volatility indices for stocks and Treasuries), and other macro-data from Haver Analytics. The default rates in Chapter 10 are from Moody’s (2010). Bibliography Acharya

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lottery features,” working paper, available at SSRN: http://ssrn.com/abstract=1089562 Doran, James S.; and Kevin Krieger (2010), “Implications for asset returns in the implied volatility skew,” Financial Analysts Journal 66(1), 65–76. Driessen, Joost; Pascal J. Maenhout; and Grigory Vilkov (2009), “The price of correlation risk: Evidence from

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hedge funds long-horizon investors market frictions premium reward—risk ratios see also liquidity illiquidity premium definition liquidity risk premium stocks time-varying implied correlation implied volatility tail risks volatility selling incubation bias index OTM puts indices CCW strategies equity fundamental market cap weighted real estate vulnerability see also individual indices industry

Frequently Asked Questions in Quantitative Finance

by Paul Wilmott  · 3 Jan 2007  · 345pp  · 86,394 words

We must have This is put-call parity. Another way of interpreting put-call parity is in terms of implied volatility. Calls and puts with the same strike and expiration must have the same implied volatility. The beauty of put-call parity is that it is a model-independent relationship. To value a call

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it’s much easier to spot violations of put-call parity. You must look for non-overlapping implied volatility ranges. For example, suppose that the bid/offer on a call is 22%/25% in implied volatility terms and that on a put (same strike and expiration) is 21%/23%. There is an overlap

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With an option this may not happen until expiration. When you hedge options you have to choose whether to use a delta based on the implied volatility or your own estimate of volatility. If you want to avoid fluctuations in your mark-to-market P&L you will hedge using the

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implied volatility, even though you may believe this volatility to be incorrect. Example A stock is trading at $47, but you think it is seriously undervalued. You

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, it is very easy to rack up an imaginary profit this way. Whatever volatility is used it cannot be too far from the market’s implied volatilities on liquid options with the same underlying. • Use prices obtained from brokers. This has the advantage of being real, tradeable prices, and unprejudiced. The main

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hedging of derivatives. Take the simple case of a vanilla equity option bought because it is considered cheap. There are potentially three different volatilities here: implied volatility; forecast volatility; hedging volatility. In this situation the option, being exchanged traded, would probably be marked to market using the

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will depend on the realized volatility (let’s be optimistic and assume it is as forecast) and also how the option is hedged. Hedging using implied volatility in the delta formula theoretically eliminates the otherwise random fluctuations in the mark-to-market value of the hedged option portfolio, but at the cost

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will be in the future, things are not that simple. Example Actual volatility is the σ that goes into the Black-Scholes partial differential equation. Implied volatility is the number in the Black-Scholes formula that makes a theoretical price match a market price. Long Answer Actual volatility is a measure of

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days is the same as over the previous sixty days. This will give us an idea of what a sixty-day option might be worth. Implied volatility is the number you have to put into the Black-Scholes option-pricing equation to get the theoretical price to match the market price. Often

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us what volatility will be in the future. There are several models for measuring and forecasting volatility and we will come back to them shortly. Implied volatility is the number you have to put into the Black-Scholes option-pricing formula to get the theoretical price to match the market price. This

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of-the-money puts as insurance against a crash. For example, the market falls, people panic, they buy puts, the price of puts and hence implied volatility goes up. Where the price stops depends on supply and demand, not on anyone’s estimate of future volatility, within reason

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. Implied volatility levels the playing field so you can compare and contrast option prices across strikes and expirations. There is also forward volatility. The adjective ‘forward’ is

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similar to CAPM for returns. Deterministic models: The simple Black-Scholes formulæ assume that volatility is constant or time dependent. But market data suggests that implied volatility varies with strike price. Such market behaviour cannot be consistent with a volatility that is a deterministic function of time. One way in which the

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Black-Scholes world can be modified to accommodate strike-dependent implied volatility is to assume that actual volatility is a function of both time and the price of the underlying. This is the deterministic volatility (surface) model

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the Black-Scholes closed-form formulæ are no longer correct. This is a simple and popular model, but it does not capture the dynamics of implied volatility very well. Stochastic volatility: Since volatility is difficult to measure, and seems to be forever changing, it is natural to model it as stochastic. The

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differently. Part of this is because of the way traders look at option prices. Equity traders look at implied volatility versus strike, FX traders look at implied volatility versus delta. It is therefore natural for implied volatility curves to behave differently in these two markets. Because of this there have grown up the sticky strike, sticky

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delta, etc., models, which model how the implied volatility curve changes as the underlying moves. Poisson processes: There are times of low volatility and times of high volatility. This can be modelled by volatility

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On Quantitative Finance, second edition. John Wiley & Sons What is the Volatility Smile? Short Answer Volatility smile is the phrase used to describe how the implied volatilities of options vary with their strikes. A smile means that out-of-the-money puts and out-of-the-money calls both have higher

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be a download sloping graph of implied volatility versus strike. Example Figure 2-9: The volatility ‘smile’ for one-month SP500 options, February 2004. Long Answer Let us begin with how to calculate

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price are the same? Although we have the Black-Scholes formula for option values as a function of volatility, there is no formula for the implied volatility as a function of option value, it must be calculated using some bisection, Newton- Raphson, or other numerical technique for finding zeros of a function

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. Now plot these implied volatilities against strike, one curve per expiration. That is the implied volatility smile. If you plot implied volatility against both strike and expiration, as a three-dimensional plot, that is the implied volatility surface. Often you will find that the smile is quite flat for

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for short-dated options. Since the Black-Scholes formulæ assume constant volatility (or with a minor change, time-dependent volatility) you might expect a flat implied volatility plot. This appears not to be the case from real option-price data. How can we explain this? Here are some questions to ask. • Is

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with confidence that actual volatility is not constant it is altogether much harder to estimate the future behaviour of volatility. So that might explain why implied volatility is not constant, people believe that volatility is constant. If volatility is not constant then the Black-Scholes formulæ are not correct. (Again, there is

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dramatically we often see a temporary increase in its volatility. How can that be squeezed into the Black-Scholes framework? Easy, just bump up the implied volatilities for option with lower strikes. A low strike put option will be out of the money until the stock falls, at which point it may

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density function has fat tails then you would expect option prices to be higher than Black-Scholes for very low and high strikes. Hence higher implied volatilities, and the smile. Another school of thought is that the volatility smile and skew exist because of supply and demand. Option prices come less from

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after all those selling the insurance also want to make a profit. Thus out-of-the-money puts are relatively over priced. This explains high implied volatility for low strikes. At the other end, many people owning stock will write out-of-the-money call options (so-called covered call writing) to

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typically negative so that a fall in the stock price is often accompanied by a rise in volatility. This results in a negative skew for implied volatility. Unfortunately, this negative skew is not usually as pronounced as the real market skew. These models can also explain the smile. As a rule

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, J 2006 The Volatility Surface. John Wiley & Sons Javaheri, A 2005 Inside Volatility Arbitrage. John Wiley & Sons Taylor, SJ & Xu, X 1994 The magnitude of implied volatility smiles: theory and empirical evidence for exchange rates. The Review of Futures Markets 13 Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second edition. John

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of wanting to hedge as often as possible to reduce risk, but as little as possible to reduce any costs associated with hedging. Example The implied volatility of a call option is 20% but you think that is cheap, volatility is nearer 40%. Do you put 20% or 40% into the

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my risk? Third, when I do rehedge how big are my transaction costs? What is the correct delta? Let’s continue with the above example, implied volatility 20% but you believe volatility will be 40%. Does 0.2 or 0.4 go into the Black-Scholes delta calculation, or perhaps something else

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S,t;σ̃), where V(S,t;σ) is the Black-Scholes formula for the call option and σ denotes actual volatility and σ̃ is implied volatility. That profit is realized in a stochastic manner, so that on a marked-to-market basis your profit will be random each day. This is

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outset. Most traders dislike the potentially large P&L swings that you get by hedging using the forecast volatility that they hedge using implied volatility. When you hedge with implied volatility, even though it is wrong compared with your forecast, you will still make money. But in this case the profit each day is

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make a lot of money. If the stock goes quickly far into or out of the money then your profit will be small. Hedging using implied volatility gives you a nice, smooth, monotonically increasing P&L but at the cost of not knowing how much money you will make. The profit each

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time step is where Γi is the Black-Scholes gamma using implied volatility. You can see from this expression that as long as actual volatility is greater than implied you will make money from this hedging strategy. This

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implied volatility, and would be parabolic in the stock move. The index would hardly move and the profit there would also be related to the index option

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calibration. A vanilla trades at $10, say. That is the price. The quant then backs out from a Black-Scholes valuation formula the market’s implied volatility. By so doing he is assuming that price and value are identical. Related to this topic is the question of whether a mathematical model explains

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over the option’s lifetime, conditional upon its current value, into the formulæ. There are no arbitrage opportunities: Even if there are arbitrage opportunities because implied volatility is different from actual volatility you can still use the Black-Scholes formulæ to tell you how much profit you can expect to make, and

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The asymptotic solution is then a power series in ∈1/2. Schönbucher’s stochastic implied volatility Schönbucher begins with a stochastic model for implied volatility and then finds the actual volatility consistent, in a no-arbitrage sense, with these implied volatilities. This model calibrates to market prices by definition. Jump diffusion Given the jump-diffusion modeldS

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stochastic volatility models. In New Directions in Mathematical Finance, Ed. Wilmott, P & Rasmussen, H, John Wiley & Sons Schönbucher, PJ 1999 A market model for stochastic implied volatility. Phil. Trans. A 357 2071-2092 Schönbucher, PJ 2003 Credit Derivatives Pricing Models. John Wiley & Sons Vasicek, OA 1977 An equilibrium characterization of the term

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interpreted via Jensen’s Inequality as a convexity adjustment because of volatility of volatility. The VIX volatility index is a representation of SP500 30-day implied volatility inspired by the one-over-strike-squared rule. Chapter 8 Popular Quant Books The following are the dozen most popular quant books in the wilmott

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of real-world option pricing that are not captured by the Black- Scholes model. These features include the ‘smile’ pattern and the term structure of implied volatility. The book includes Mathematica code for the most important formulæ and many illustrations. The Concepts and Practice of Mathematical Finance by Mark Joshi “Mark Joshi

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example, what volatility when put into the Black-Scholes formula gives a theoretical price that is the same as the market price? This is the implied volatility. Intimately related to calibration. Lévy A probability distribution, also known as a stable distribution. It has the property that sums of independent identically distributed random

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Woodward, that exploits asymptotic analysis to make an otherwise intractable problem relatively easy to manage. See page 292. Skew The slope of the graph of implied volatility versus strike. A negative skew, that is a downward slope going from left to right, is common in equity options. Smile The upward curving shape

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of the graph of implied volatility versus strike. A downward curving profile would be a frown. Sobol’ A Russian mathematician responsible for much of the important breakthroughs in low-discrepancy sequences

My Life as a Quant: Reflections on Physics and Finance

by Emanuel Derman  · 1 Jan 2004  · 313pp  · 101,403 words

-the-money put is more attractive than Y40 for a deep out-of-the-money put. A better measure of value is the option's implied volatility. The Black-Scholes model views a stock option as a kind of bet on the future volatility of a stock's returns. The more volatile

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price into the future volatility the stock must have in order for the option price to be fair. This measure is called the option's implied volatility. It is, so to speak, an option's view of the stock's future volatility. The Black-Scholes model was the market standard. When I

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traders sometimes use more complex models, the Black-Scholes model's implied volatilities are still the market convention for quoting prices. Options are generally less liquid than stocks, and implied volatility market data is consequently coarse and approximate. Nevertheless, Dave pointed out to me what I was already

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lopsided shape, though it's commonly called "the smile," is more of a smirk. With implied volatility as your measure of value, low-strike puts are the most expensive Nikkei options. Anyone who was around on October 19, 1987 could easily guess

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1987, in contrast, more light-heartedly naive options markets were happy to charge about the same implied volatility for all strikes, as illustrated by the dashed line in Figure 14.1. Figure 14.1 A typical implied volatility smile for three-month options on the Nikkei index in late 1994. The dashed line shows

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for options of any expiration, so that implied volatility varied not only with strike but also with expiration. We began to plot this double variation of

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implied volatility in both the time and strike dimension as a two-dimensional implied volatility surface. A picture of the surface for options on the Standard & Poor's (S&P) 500 index is illustrated in Figure 14.2

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. Like the yield curve, it changes continually from minute to minute and day to day. Figure 14.2 A typical implied volatility surface for the S&P 500 in mid-1995. This tentlike surface was a challenge to theorists everywhere. The Black-Scholes model couldn't account

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realism, to add complexity to the simple evolution of stock prices assumed by Black and Scholes, but complexity without calibration is pointless. Figure 14.3 Implied volatility surfaces. (a) In the standard Black-Scholes model. (b) In an enhanced Black-Scholes model where volatility varies with time to expiration. Figure 14.4

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parameter richer than Black-Scholes, which contained only a single volatility. We calibrated these parameters by matching the model's option prices to the two implied volatilities that defined the shape of the three-month smile, those of an at-the-money option and a put option struck 5 percent out of

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Mars around the sun. The right place to start was with a model that could match the market prices of all standard options, the entire implied volatility surface. Only then, when it was correctly calibrated, could you sensibly use it to calculate the value of an exotic. How could we find a

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each future instant of time and index level. This constancy of the index's volatility in the Black-Scholes model leads to the associated flat implied volatility surface that is inconsistent with actual options markets. Figure 14.6 At left, a binomial tree of future index moves. Each future percentage move is

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identical and represents a constant index volatility. At right, the shape of the corresponding implied volatility surface. Iraj and I developed an alternate view of the future tree of index levels. We pictured the usual constant-volatility binomial tree redrawn on

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the index at a particular future level and time. The constant or global volatility of Figure 14.6 was inconsistent with the market's tentlike implied volatility surface of Figure 14.2. There must be, we figured, an implied binomial tree whose local volatility could be chosen to match the market's

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implied volatility surface. We expected it to look like the tree in Figure 14.7, in which the local volatility of the index rises as the index

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percentage move has a local volatility that rises as the index level drops. Can one deduce the shape of the implied binomial tree from the implied volatility surface at right? It was easy to imagine such a tree. It was even easy to build such a tree by literally making up a

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tree and then constructing it. Given such a tree, you could use it to calculate the prices of many different options and then plot their implied volatility surface. We could see that it was possible to pick a local volatility whose variation produced a realistic-looking volatility surface. But the ultimate problem

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we faced was the inverse of what we were doing. We needed to start with the implied volatility surface the market presented and deduce from it the unique local volatilities that reflected it. The implied volatility surface was the primary object, and the whole procedure we envisaged would only constitute a true theory

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to define a scheme for uniquely extracting the local volatility at each future node of the implied binomial tree. We began with the market's implied volatility surface on a given day, as illustrated in Figure 14.2. We then constructed a binomial tree like the one in Figure 14.8. Each

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to lower (paler) volatilities; lower index levels correspond to higher (darker) volatilities. How pale or how dark must you choose them to match the initial implied volatility surface of Figure 14.2? That was the question. Figure 14.8 An implied binomial tree with varying local volatilities represented by the shaded triangles

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volatility in Figure 14.8 is a local quality of the tree, the microscopically viewed volatility within each single small internal triangle. In contrast, the implied volatility in Figure 14.2 is a global quality, a wide-angle overview of all the internal triangles seen from 30,000 feet. We viewed the

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implied volatility of an option as the average of all the local volatilities that the index will experience during the life of that option. Consider the option

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index level at the location of the small flashlight in the nextto-last row of the tree in Figure 14.9. The value of its implied volatility depends upon the values of the local volatilities in the shaded rightstriped triangles; those are the local volatility regions that the index can traverse in

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, sheds light on just one triangle in the tree, the single node whose volatility was the obscure object of our desire. Figure 14.9 The implied volatility of the option whose expiration and strike lie at the circle illuminates the local volatilities in the right-striped region of the tree. We kept

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internal node. But each scheme we tried failedthere seemed to be no recipe for the local volatility at a single node. Figure 14.10 The implied volatility of the option whose expiration and strike lie at the location of the lantern illuminates the local volatilities in the left-striped region of the

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Figure 14.11. It was astonishing: We had found an algorithm that determined the local volatility at a single node in terms of the market implied volatilities of the options with strikes at the three surrounding nodes. Now we knew how to find every local volatility, step by step. We could select

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any node on the implied tree, read off the implied volatilities of the three options adjacent to it from the market's implied volatility surface, and then extract the local volatility at that node via our algorithm. In this way, one node at

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history of the problem. His initial solution to the inverse scattering problem focused on matching the market's implied volatilities at a single expiration, thereby ignoring some of the additional information embedded in the implied volatility surface. Bruno's IAFE talk was the most tantalizing. As a Frenchman, he had a taste for formal

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mathematics, and his very brief report proposed an elegant formula for the local volatilities in terms of the slope and curvature of the implied volatility surface at the same strike and expiration. His paper wasn't easy to understand and people I spoke to were not sure it was correct

The Rise of Carry: The Dangerous Consequences of Volatility Suppression and the New Financial Order of Decaying Growth and Recurring Crisis

by Tim Lee, Jamie Lee and Kevin Coldiron  · 13 Dec 2019  · 241pp  · 81,805 words

Figure 4.1 Cumulative returns to simulated currency carry strategy . . 50 Figure 4.2 Median monthly return across carry trade deciles . . . . . . 53 Figure 4.3 Implied volatility changes and stock returns during currency carry drawdowns . . . . . . . . . . . . . . . . . 54 Figure 4.4 Correlation of currency carry trade returns with CBOE short volatility strategies . . . . . . . . . . . . . . . 58 Figure

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Decile 10 the worst. Figure 4.2 shows the median carry trade return in each decile along with the median change in US stock market implied volatility proxied by the 3. Specifically, the developed market–only strategy had an average daily return of 0.022 percent and a daily standard deviation of

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AUG 98 OCT 98 DEC 01 OCT 08 SEP 11 AUG 13 DEC 14 AUG 15 AUG 18 200% Drawdown End Date FIGURE 4.3 Implied volatility changes and stock returns during currency carry drawdowns Source of data: Datastream, Global Financial Data, CBOE, authors’ calculations Two interesting features emerge—the pattern in

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Turkish lira fell by almost 40 percent. The lira was the carry strategy portfolio’s largest long position at that time. The VXO measures of implied volatility did rise during this period—by about 5 percent—consistent with the carry trade drawing down when volatility increases. However, this was only a mild

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 THE RISE OF CARRY time left until expiration of the option, and, importantly, the expectation of future volatility of the stock price or market index—“implied volatility.” Other things being equal, the higher the expected volatility, the more the option will cost and the greater will be the income to the writer

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distribution of tomorrow’s price change, even though it might not be useful for thinking about daily price changes in a year’s time. Therefore implied volatility, the market’s expectation for future realized volatility, tends to move in line with realized volatility—albeit that

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implied volatility is normally higher than realized volatility by some margin. In modern financial markets implied volatility can be sold directly. The simplest and most popular way to do this is through VIX futures or through exchange

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-traded notes (ETNs) that correspond to simple VIX futures strategies. The VIX is an index representing the implied volatility for the S&P 500 over the next 30 days, derived from the prices of options on the S&P 500 index.2 VIX futures

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VIX represents the volatility corresponding to the implied variance for the S&P 500 over the next 30 days. 3. The less widely followed VXO implied volatility index is based on prices of options on the S&P 100, which have a longer history; because of the longer data history, the VXO

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the carry trader as the price of the VIX future “rolls down” the curve toward the spot price. The trade is even more profitable if implied volatility falls and the spot VIX is lower at expiration than when the short futures trade was initiated. Thus, like the currency carry trade, the VIX

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certain amount and preparing to pay an uncertain amount. For instance, in the VIX roll-down trade described earlier, the carry trader sells long-dated implied volatility at a known price and will close out the trade by purchasing the spot VIX in the future at an uncertain price. Because the VIX

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as “Volmageddon,” during which the VIX more than doubled in one day). During these spikes all forms of S&P 500 volatility selling fail simultaneously. Implied volatility soars, especially at the spot and nearest forwards, causing the VIX curve to invert; realized volatility explodes higher than even current implied; and the market

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into 2015. At the beginning of the financial crisis in the summer of 2007, credit spreads had begun to widen and the VIX measure of implied volatility of the S&P 500 began to rise, as the carry trade began to come under pressure. However, the currency carry trade bubble—and also

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the strike but there is 1. The option replication strategy outlined here is equivalent to the Black-Scholes replication of an option with arbitrarily low implied volatility. 148 THE RISE OF CARRY still plenty of time before the option expires and the price is volatile enough to have a good chance of

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market itself and by extension into other asset markets. One very common, and very important, method is to construct a trade that profits directly if implied volatility in a certain period exceeds realized volatility. A way to do this is to sell options delta hedged. Delta hedging is a crucial idea. Any

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for selling the option. The fee, which equals the expected cost of replicating the option over its remaining life, is driven by the option’s implied volatility. Set against this income is the actual cost of managing the replicating portfolio. Since replicating the option means buying optionality—trading with the market—this

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the realized volatility of the asset. Thus, the profit from selling options delta hedged is determined by the difference between implied and realized volatility. Since implied volatilities systematically exceed realized volatilities, as shown in Figure 9.1, it is systematically profitable to sell options delta hedged. What if a trader sells options

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measures of volatility. Different measures of volatility correspond to different tenors of implied or realized volatility; the second VIX future corresponds to two-month forward implied volatility, while the activity of delta-hedging an option position weekly corresponds to five-day realized volatility, and so forth. Selling volatility is using leverage because

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the structure of volatility premiums at equilibrium on a single asset. This chart depicts three critical features of volatility. First, the top line shows that implied volatilities are greater further forward. This seems intuitive—it expresses the requirement that shorting VIX futures be profitable at all points along the VIX term structure

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. (The horizontal axis for implied volatility is at the top of the chart and goes from spot VIX out to VIX futures five months forward.) Realized 28 VIX 1 Month 2

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-Term Value 12 10 Instantaneous 1 Day 1 Week 1 Month 1 Year 7 Years FIGURE 9.4 Equilibrium volatility premium term structure Chart shows implied volatility spot and forward on the upper gray line, read off the top horizontal axis, and realized volatility across measuring horizons on the lower black line

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, read off the bottom horizontal axis. Data for implied volatility is a stylized version of the average VIX curve since 2009. Data for realized volatility is a stylized version of realized volatilities by horizon for

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is plotted over intervals from instantaneous to seven years, shown on the lower horizontal axis.) The third feature is that implied volatilities are never below realized volatilities: even the lowest point on the implied volatility curve, the spot VIX, is not below the highest point on the realized volatility curve, which is instantaneous realized

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tenor, and can be delta-hedged at some shorter frequency by replicating a long volatility position in the underlying. This exploits the last feature, of implied volatility being systematically higher than realized. The trader running this strategy is compensated for providing instantaneous liquidity to the market—in effect, behaving as a market

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maker. There are a few other kinds of volatility premiums not described so far. There is skew, which is the fact that implied volatilities are excessively high for further out-of-the-money put options. That is, it is “expected” that volatility will rise if the market falls. Selling

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to the market maker. Empirically, for the S&P 500 front e-mini future, realized volatility over horizons of less than three minutes somewhat exceeds implied volatility. Finally, all the above refer to single assets. The volatility of portfolios of assets depends on both the volatilities of the assets and the correlations

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price, shorter-term realized volatilities need to exceed longer-term realized volatilities. For implied volatility the curve is of implied volatilities at different forward points; for liquidity to have a positive price, further forward implied volatilities need to exceed nearer forward implied volatilities—and all implied volatilities must exceed all realized volatilities. The slope of realized volatility means that the

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to have liquidity and would have to provide that liquidity to the market. Many other apparent market inefficiencies can be viewed analogously. The slope of implied volatility superficially appears to mean that the market expects price volatility to rise. Since volatility does not on average rise—it is empirically mean reverting to

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a stable level of long-run volatility—this cannot really be understood as an expectation. Instead this curvature is a risk premium paid to forward implied volatility sellers, that is, forward liquidity providers. In the same way, the gap between implied and realized volatilities makes selling spot

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implied volatility—selling contracts that pay out depending on realized volatility of the underlying starting immediately—profitable over time, as a risk premium to providers of instantaneous

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to become identified with each other. So in equity markets, liquidity short squeezes are to the downside. They form and cause the skew of both implied volatility and realized returns, this skew being another way in which the liquidity price manifests. The high return-to-risk ratio of liquidity provision strategies is

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at characterizing this counterfactual world would be just to invert the features of the world as we know it. In this counterfactual world, far forward implied volatility would be oversupplied. People would “want” to sell it—that is, they would on average be willing to pay for the privilege of selling it

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. Plausibly, as a result, options markets would “expect” volatility to fall into the future, with long-dated implied volatility below short-dated implied volatility. Furthermore, it would cost more to trade with the market on longer horizons than on shorter horizons. Volatility would be greater over longer than

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and volatile inflation. Does the Carry Regime Have to Exist? 167 In this world, long-run average realized volatility would likely exceed long-run average implied volatility, so that buyers of spot options would on average earn money. At first glance, a negative implied-realized gap would seem to suggest a negative

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14 12 VIX 1 Month 2 Months 3 Months 4 Months 5 Months FIGURE 10.1 Hypothetical “mirror” equilibrium volatility premiums term structure Chart shows implied volatility spot and forward on the lower gray line, read off the bottom horizontal axis, and realized volatility across measuring horizons on the upper black line

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regime is to consider one curious feature of the current regime: that the most risk is closest to the present. Nearer forward implied volatilities are more volatile than far forward implied volatilities. While spot volatility whips around, far forward volatility barely moves. The distant future is considered to be relatively certain; the idea that

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“volatility is mean reverting to a stable long-run level” is fully accepted by the market. For the S&P 500, implied volatility five months forward has been barely a quarter as volatile as the spot VIX (Figure 10.2). Standard Deviation of Daily Change in VIX Spot

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into the future, and the point of greatest uncertainty would be the point at infinity rather than the current instant. It seems highly unlikely that implied volatility could be in backwardation in such a world; distant future volatility must be both more volatile and higher than near future volatility. And the

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implied volatility curve would be flattest near the spot and steepest in the distant future, in order to keep ex-ante risk-reward ratios for trading volatility

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is to fail and volatility buyers are to be paid, then volatility must simply be rising over time faster than the upward slope of the implied volatility curve. This is the true anti-carry regime, where market “expectations” systematically underestimate future change. All of this would be consistent with a completely uncontrolled

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and Reality VIX 1 Month 2 Months 3 Months 4 Months 5 Months FIGURE 10.3 Hypothetical anti-carry volatility premiums term structure Chart shows implied volatility spot and forward on the lower gray line, read off the bottom horizontal axis. Realized volatility across measuring horizons is plotted on the upper black

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, can a carry crash be detected ahead of time and a hedge initiated just prior to the event? There is some evidence that once the implied volatility curve inverts— that is, once the spot VIX starts to increase above further forward levels— short-term volatility continues to trend higher. However, the timing

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from short to long; it will become profitable to buy, rather than sell, volatility. There will be three major visible parts to this transition. First, implied volatility forward curves will settle into consistent backwardation. Second, prices will begin to behave with momentum over all time horizons: realized volatility measured weekly or monthly

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liability categories for, 72 HFR (Hedge Fund Research), 73 high-frequency trading firms (HFTs), 84 Hungarian forint, 34 IMF (International Monetary Fund), 14, 16, 198 implied volatility, 57, 90, 164, 167–168 anti-carry regime and, 171–172 realized volatility relationship to, 158 income streams, from carry trades, 2 India, 19 industrial

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activity, measures of, 56 real estate booms, currency carry trades contributing to, 13 228 realized volatility, 90, 164, 167–168 anti-carry regime and, 172 implied volatility relationship to, 158 recessions, carry and consequences of, 6 recipient currencies, 10–11, 13, 65 crashes in, 23 volatility in, 215 regulatory capture, 176 rent

Mathematical Finance: Theory, Modeling, Implementation

by Christian Fries  · 9 Sep 2007

0.50 0.75 1.25 2.00 Price V(Ki ) 0.5277 0.3237 0.1739 0.0563 Implied Volatility σ(Ki ) 0.4 0.4 0.6 0.6 The given implied volatility is the σ of a Black-Scholes Modells with assumed interest rate (short rate) of r = 0.05. Note

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One could hope that a local smoothing would solve this problem. Instead we will give an example that a linear grow of implied volatility may be inadmissible alone, although the interpolated prices are admissible. We consider two prices V(K2 ) > V(K3 ). The monotony ensures that these two prices

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Implied Volatility σ(Ki ) 0.9 0.1 The implied volatility decreases with the strike K. Figure 6.5 shows the linear interpolation of the implied volatilities (center). The density (see Figure 6.5, right) shows a large region with negative

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values, thus it is not a probability density. The reason is a too fast decay of the implied volatility. 95 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd/2.5/deed.en Comments welcome. ©2004, 2005

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strike 1,00 1,25 underlying value Figure 6.5.: Linear interpolation for decreasing implied volatility. Conclusion: An arbitrarily fast decrease of implied volatility is not possible. 6.2.2.2. Lineare Interpolation for increasing Implied Volatilities For the example of increasing implied volatility we consider the prices i 2 3 Strike Ki 0.75 1.25 Price

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V(Ki ) 0.2897 0.2532 Implied Volatility σ(Ki ) 0.2 0.8 Figure 6.5 shows the linear increasing interpolation of

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the implied volatilities. Interpolated prices Interpolated volatlities Probability density 0,50 1,5E-2 volatility price 0,30 0,20

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,00 strike 1,25 0,75 1,00 1,25 underlying value Figure 6.6.: Linear interpolation for increasing implied volatility. At first sight the density implied by the linear interpolation of the implied volatilities exhibits no flaw (positive, no point measure). However, it is not a probability density. The integral below the segment

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region with negative values beyond the interpolation region (this however will make the integral of the density to 1). Conclusion: An arbitrarily fast increase of implied volatility is not possible. 6.3. Arbitrage Free Interpolation of European Option Prices The examples of the previous sections bring up the question for arbitrage free

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a continuum (T, K) 7→ V(T, K) of prices of European options with maturity T and strike K. Let σ(T, K) denote the implied volatility of V(T, K), i.e. the volatility 97 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd

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Black volatility. This is of course just another unit of the price, since the Black model is a one-to-one map from price to implied volatility. 257 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd/2.5/deed.en Comments welcome. ©2004, 2005

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condition is one reason for it initial popularity. However, since the model restricted to a single LIBOR is a Black model, we have that the implied volatility does not depend on the strike of an option. Thus, in this form, the model may calibrate to a single caplet per maturity only. It

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consider only a single swap rate S i,k , then (17.27) is a Black modell for this swap rate and under this model the implied volatility is defined by inverting the pricing formula. The inversion of the pricing formula is what a calibration should achieve. Remark 199 (LIBOR Market Model versus

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options, all having strike K. This is our calibration product. 23.2.3.1. Market Price Let σ̄BS (T, K) denote the Black-Scholes implied volatility surface given from market prices. Then we have that the market price of V is V market (T, K; 0) √ ∂σ̄BS (T, K) = S

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requirement on the functionals (which is fulfilled by the BlackScholes model). Such models may calibrate only to a one dimensional sub-manifold of a given implied volatility surface, see [?]. For the Markov functional model this follows directly from (23.3). Assuming that the Markovian driver x is given and that the interest

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to a given interest rate dynamics by adding a drift to the Markovian driver, see Section 23.2.5.2. • Forward volatility: This is the implied volatility of an option with maturity T and strike K, given we are in state (t, ξ), i.e. S (t, ξ) · EQ
max(S (T

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Section 23.2.5.3. • Auto correlation / Forward spread volatility: The auto correlation of the process S impacts the forward spread volatility. This is the implied volatility of an option on S (T 2 ) − S (T 1 ) with maturity T 2 , given we are in state (t, ξ), i.e. S (t

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driver is given by x(ti+1 ) = x(ti ) + σi ∆W(ti ). (23.7) Consider a term structure of Black-Scholes implied volatilities, i.e. let σ̄BS (ti ) denote the implied volatility of an option with maturity ti . Assuming the simple Markovian driver (23.7) the corresponding functionals that calibrate to these options

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σ̄BS (ti )
S (ti , ξ) = S (0) · exp r · ti + σ̄BS (ti )2 t + ·ξ 2 σ̄i calibrating to European options with implied volatility σ̄BS (ti ).4 the standard deviation of the increment x(tk ) − x(ti ) is σ̄ti ,tk · √ Then we have that 1 Pk−1

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2 (tk − ti ), where tk −ti j=i σ j ∆t j . It follows that the implied volatility of an option with maturity tk , given we are in state (ti , ξ), is σ̄ti ,tk . σ̄tk σ̄BS (tk ) · Thus a decay

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. 2.8. 2.9. 2.10. Arbitrage-free option prices . . . . . . . . . . . . . . . . Linear interpolation of option prices . . . . . . . . . . . . Linear interpolation of implied volatilities . . . . . . . . . Spline interpolation of option prices and implied volatilities Linear interpolation for decreasing implied volatility . . . Linear interpolation for increasing implied volatility . . . 415 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd/2

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Market Model . . . . . . . . . . . . . . . . . . 357 – Equity-Hybrid LIBOR Market Model . . . . . . . . . . . . . . . . . . . . . . . . . 354 – LIBOR Market Model . . . . . . 263, 275 Implementierung . . . . . . . . . . . . . . . . . 362 Implied Black volatility . . . . . . . . . . . 150 Implied Black-Scholes volatility . . . . . 88 implied volatility . . . . . . . . . . . . . . . . . . 97 import (Java™ Schlüsselwort) . . . . . 380 Importance sampling – using a proxy scheme . . . . . . . . . . . 251 independence – of events . . . . . . . . . . . . . . . . . . . . . . . . 28 – of random variables . . . . . . . . . . . . . . 33 information (filtration) . . . . . . . . . . . . . 37 Instantaneous

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Model . . . . . . . . . . . . . . . . . . . 304 Vasicek model – Extended Vasicek model . . . . . . . . 304 Vega . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Vega Hedge . . . . . . . . . . . . . . . . . . . . . . 117 Vererbung . . . . . . . . . . . . . . . . . . . . . . . 378 void (Java™ Schlüsselwort) . . . . . . 365 Volatility – implied volatility . . . . . . . . . . . . . . . . 88 – time structure . . . . . . . . . . . . . . . . . . 309 volatility – forward volatility . . . . . . . . . . . . . . . 131 – implied volatility . . . . . . . . . . . . . . . . 97 volatility bootstrapping . . . . . . . . . . . . 268 volatility surface . . . . . . . . . . . . . . . . . . . 97 – definition . . . . . . . . . . . . . . . . . . . . . . . 97 W weak convergence . . . . . . . . . . . . . . . . . 39 weather derivative . . . . . . . . . . . . . . . . . 61 weighted Monte-Carlo . . . . . . . . . . . . 185 Wiener

Trading Risk: Enhanced Profitability Through Risk Control

by Kenneth L. Grant  · 1 Sep 2004

68 69 v vi CONTENTS Drawdown Correlations 70 73 Putting It All Together CHAPTER 4 The Risk Components of an Individual Portfolio Historical Volatility Options Implied Volatility Correlation Value at Risk (VaR) Justification for VaR Calculations Types of VaR Calculations Testing VaR Accuracy Setting VaR Parameters Use of VaR Calculation in Portfolio

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92 94 98 99 102 104 106 109 110 111 114 129 130 133 134 135 141 142 144 146 148 149 149 vii Contents Implied Volatility Asymmetric Payoff Functions Leverage Characteristics Summary CHAPTER 7 150 150 151 154 The Risk Components of an Individual Trade Your Transaction Performance Key Components of

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), 60-day (one-quarter), and 250-day (one-year) series as a means of obtaining a multidimensional view of the security’s price dynamics. OPTIONS IMPLIED VOLATILITY The critical drawback of using historical data to measure the volatility of a given security is that history is not always the best predictor of

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future volatility patterns that transcend what can be inferred directly by historical volatility analysis. One such methodology derives from the concept of implied volatility embedded in the price of every option. Implied volatility is a measure of the amount of price dispersion expected in a given security on a prospective The Risk Components of

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over a given period would theoretically be worth an infinite amount, as there is a theoretical opportunity to liquidate them at infinite intrinsic value. The implied volatility pricing for all options with economic meaning resides somewhere in between these extremes, and it is not an exaggeration to say that a preferred method

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of its volatility. Options traders will often speak in terms of purchasing/selling this or that option at, say, 10%, meaning that this is the implied volatility at which they were able to execute an options transaction. To elaborate further on the associated pricing implications, the execution of an options trade at

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of 10% over the life of the option. How does all of this pertain to the applicability of options implied volatility to our efforts to estimate portfolio risk? As it turns out, because implied volatility is quoted in the exact manner that historical volatility is expressed (i.e., in annualized terms), for the purposes

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of 88 TRADING RISK estimating price dispersion, the terms can be used interchangeably. In fact, implied volatility can be viewed very explicitly as the options market’s attempt to predict what historical volatility will be in the future. We can therefore substitute

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option. For example, just as is the case with historical volatility, if we take a $50,000 bet in a given security with a 10% implied volatility, we would expect the position in question to fluctuate at an annualized, one-standard-deviation rate of $5,000. Again, we don’t have to

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options to arrive at this estimate but can simply derive data from the options markets and apply it against our position in cash securities. Options implied volatility thus has the advantage over historical volatility of encompassing not only historical time series information but also “qualitative” data and prospective economic inputs into its

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estimates of price dispersion. The measure gains further credibility through the fact that options traders are actually risking financial capital on the basis of implied volatility valuations. Take my word for it—this type of reality dose does wonders for the accuracy of financial models. From these perspectives

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, implied volatility arguably is the superior measure. However, like everything else in our statistical tool kit, it has shortcomings. First, because the implied volatility statistic is derived entirely from the manner in which options are priced, it is subject

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for all types of options are often bid up to levels beyond what would be justified by the underlying economic data; and the use of implied volatility as an exposure input might lead to inaccuracies in these instances. In addition, there is a well-known concept within the universe of option trading

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, referred to as the volatility skew, or smile, that describes the tendency for out-of-the-money options to trade at higher implied volatilities than those that are at, near, or in the money. It is therefore necessary to understand the relationship between the strike price and the underlying

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market price (i.e., the “moneyness”) for the option on whose implied volatility you are relying. As a practical matter, I recommend the volatility for the at-the-money strike as the best approximation of future price dispersion

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to capture the attention of the market, while small enough to operate within its liquidity constraints). Wherever and whenever this occurs, it will lead to implied volatility outcomes that differ, to varying degrees, from what can be best described as the rational equilibrium. In addition, options for the same underlier may have

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, for instance, one option expires before a key information event (e.g., corporate earnings), and the other expires after this occasion. For these reasons, while implied volatility offers a unique and critical insight into the likely price dispersion characteristics of a given financial instrument, it must be viewed, as with other elements

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, to paraphrase that righteous man of letters George Santayana, should not be relinquished too readily). I believe that it makes sense to utilize both the implied volatility statistic and historical volatility figures, with both statistics calculated over multiple time spans. Line these up side by side, and examine the extent to which

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may also help you sharpen any hypothesis you may be forming about what’s happening in the markets). One final note: You can find extensive implied volatility data sets through news services such as Reuters and Bloomberg or in periodicals such as Barron’s. You can also calculate them (if you have

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out, converging toward the equilibrium levels that in most cases can be ascertained through close observation of such factors as historical pricing patterns and options implied volatility levels. Under these circumstances, traders and investors are likely to capture the full measure of the volatility inherent in each of the securities they hold

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types of movements for options at or near the money as underlying markets change in value and display shifting price dispersion characteristics. Implied Volatility We already covered the topic of implied volatility, which, as you may recall from Chapter 3, is a useful means of characterizing future volatility, both for the options themselves and

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for their associated underlying instruments. Because implied volatility is as dynamic a concept as underlying price, it can shift abruptly, often causing discontinuous movements in the price of options that may or may

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, 162–165 time horizon and, 142–144 trading with an edge, 222 “House,” being the, 223 Humor, importance of, 243–244 Impact ratio, 186–188 Implied volatility, 86–87, 89 Incremental risk, 126, 141–142 Individual portfolio, risk components: correlation, 90–91 historical volatility, 84–88, 96–97 options

…

implied volatility, 86–89 scenario analysis, 104–106 technical analysis, 106–108 value at risk (VaR), 91–104 Individual trades, risk components of: core transactions-level statistics,

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–175 256 OGHET. See Scientific method Optimal f, 245–251 Optimism, importance of, 4 Options: asymmetric payoff functions, 150–151 implications of, generally, 148–149 implied volatility, 86–89, 150 leverage, 151–153 nonlinear pricing dynamics, 149 pricing, 88–89, 106 strike price/underlying price, relationship between, 149–150 volatility arbitrage, 106

Market Risk Analysis, Quantitative Methods in Finance

by Carol Alexander  · 2 Jan 2007  · 320pp  · 33,385 words

White’s heteroscedasticity test I.5.1 Excel’s Goal Seek I.5.2 Using Solver to find a bond yield I.5.3 Interpolating implied volatility I.5.4 Bilinear interpolation I.5.5 120 I.5.6 123 I.5.7 127 131 I.5.8 I.5.9 132

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the Black–Scholes–Merton price of an in-the-money or an out-of-the money option is a convex monotonic increasing function of the implied volatility. I.1.3.4 Stationary Points and Optimization When f x = 0 x is called a stationary point of f . Thus the tangent to the

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set of zero coupon returns of different maturities, a set of commodity Essential Linear Algebra for Finance 65 futures returns of different maturities, or an implied volatility surface. In short, PCA works best for term structures. PCA can achieve many things and, as a result, has many purposes. Its statistical applications include

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data, and orthogonal regression to cope with the problem of collinear explanatory variables. Financial applications include multi-factor option pricing models, predicting the movements in implied volatility surfaces and quantitative fund management strategies. In this book we shall focus on its many applications to market risk management. PCA is commonly used to

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should be negative, β2 is the coefficient of the equity return R and should also be negative, and β3 is the coefficient of the equity implied volatility  and should be positive. We now test this theory using daily data from 21 June 2004 to 19 September 2007 on the iTraxx Europe credit

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-year euro swap rate, an equity index constructed from the share prices of the 125 firms that are included in the iTraxx index and the implied volatility index for the Eurostoxx index, which is called the Vstoxx.19 The data are shown in Figure I.4.7, with each series set to

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. But the regression R2 is only 0.1995 and there is a high degree of multicollinearity between the equity index return and the change in implied volatility. In fact in our sample their correlation is −0829, and the square of this is 0.687, which far exceeds the regression R2 . To

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can simply drop one of the collinear variables from the regression, using either the equity return or the change in implied volatility in the model, but not both. If we drop the implied volatility from the regression the estimated model becomes ŝ = 002618 − 21181 r − 05312 R 00610 −10047

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return from the regression, the estimated model becomes ŝ = −00265 − 21643 r + 05814  −06077 −10086 126154 Hence, the equity implied volatility alone is a very significant determinant of the credit spread. Its effect was being masked by its collinearity with the equity return in the original

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model. In fact, the effect of equity implied volatility on the credit spread is almost equal and opposite to the effect of the equity index. However, we do not require both variables in the

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prices. However, it is not possible to invert the Black–Scholes–Merton formula so that we obtain an analytic solution for the implied volatility of the option. In other words, the implied volatility is an implicit function, not an explicit function of the option price (and the other variables that go into the Black

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–Scholes–Merton formula such as the strike and the maturity of the option). So we use a numerical method to find the implied volatility of an option. The allocations to risky assets that give portfolios with the minimum possible risk (as measured by the portfolio volatility) can only be

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of problems, using Excel’s Goal Seek or, more frequently, the Solver optimizer. For instance, in this chapter we use Goal Seek to find the implied volatility of an option and the yield on a bond. But the Excel Solver is a more powerful and more flexible optimizer that is used in

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interpolation and extrapolation. These are methods that ‘fill in the gaps’ where data are missing. They are required in many situations: fitting yield curves and implied volatility surfaces being two common examples. Section I.5.4 covers the optimization problems that arise in three broad areas of financial analysis: the efficient allocation

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y − PMarket = 0, where PVy and PMarket denote the theoretical and market prices of the bond; • in Chapter III.4 we find the option implied volatility  as a root of fBSM  − fMarket = 0, where fBSM and fMarket denote the Black–Scholes–Merton model price and the observed market price of an

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very basic algorithm that does not allow you to change convergence criteria or levels of tolerance. The following example uses Goal Seek to find the implied volatility of an option. This is the volatility that is implicit in the observed market price – the higher the market price of the option, the higher

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price of 4 when the underlying price is 100. The strike of the option is 98 and the maturity is 30 days. What is the implied volatility of this option? Solution We use the Goal Seek function from the Tools menu with the settings shown in Figure I.5.2. The objective

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+ 25 × 12 = 11 613 31 Similarly, linear interpolation can be applied to construct constant maturity implied volatility series from the implied volatilities of options of different maturities. The next example illustrates this application. Example I.5.3: Interpolating implied volatility Suppose we have two options with the same strike but different maturities: option 1 has maturity

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10 days and option 2 has maturity 40 days. If the implied volatility of option 1 is 15% and the implied volatility of option 2 is 10%, what is the

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linearly interpolated implied volatility of an option with the same strike as options 1 and 2 but with maturity 30 days

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a long out-of-the-money (OTM) put and a long OTM call. Strangles are usually quoted as the spread between the average of the implied volatilities of the put and call components and the ATM forward volatility, and we denote this ST . • A risk reversal is a long OTM call and

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the spreadsheet for this example we find the solution a = −0 320 b = −0 340 c = 0 270 Figure I.5.8 plots the fitted implied volatility smile and from this we can read off the interpolated values of the 10-delta and the 90-delta volatilities as 23.9% and 22

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in Castagna and Mercurio (2007). If further data on 10-delta strangles and risk reversals are available, two more points can be added to the implied volatility smile: 10 = 50 + ST10 + 21 RR10  90 = 50 + ST10 − 21 RR10 (I.5.13) A more precise interpolation and extrapolation method

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splines to the discount rates associated with each interest rate.8 Another common application of cubic splines is to the interpolation and extrapolation of an implied volatility surface. Liquid market prices are often available only for options that are within 10% of at-the-money and with a maturity no more than

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assess these costs one has to consider many possible future values of delta. But delta depends on the implied volatility. Hence, from the relatively sparse market data available, one must interpolate and extrapolate the implied volatility surface at extreme strikes. Instead of natural cubic splines we could use basis splines or Hermite polynomials.9

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‘calibrate’ the parameters of the Heston (1993) option pricing model by finding the best fit to a current ‘snapshot’ of market data, such as an implied volatility smile. Let y =  y1      yn  denote the data that we want to fit;10  denote the parameters that we want to estimate/calibrate to the

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an option are the partial derivatives of an option price g S      with respect to its risk factors – i.e. the underlying price S, the implied volatility , and so forth. The Greeks of an option depend on the formula used to price the option. The Black– Scholes–Merton Greeks have a simple

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to model financial risk. These include the following: • • • • • • Iterative methods for solving implicit equations, with applications that include the yield on a bond and the implied volatility of an option. Polynomial and spline techniques for interpolation and extrapolation, with applications to term structures of interest rates and

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implied volatility surfaces. Algorithms for finding the maximum or minimum value of a multivariate function, subject to certain constraints on the parameters. These constrained optimization techniques have

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error process 148 financial modelling 186 GEV distribution 101 regression 148, 157, 175 stable distribution 106 stochastic process 134–5 Implicit function 185 Implied volatility 194, 196, 200–1 Implied volatility surface 200–1 Incremental change 31 Indefinite integral 15 Independent events 74 Independent and identically distributed (i.i.d.) variables central limit theorem

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Vector notation, functions of several variables 28 Vectors 28, 37–9, 48–54, 59–61, 70 Venn diagram 74–5 Volatility equity 3, 172–3 implied volatility 194, 196–7, 200–1 interpolation 194, 196–7 linear portfolio 57–8 long-only portfolio 238–40 minimum variance portfolio 240–4 portfolio variance

Python for Finance

by Yuxing Yan  · 24 Apr 2014  · 408pp  · 85,118 words

binomial tree method for American options Hedging strategies Summary Exercises Chapter 10: Python Loops and Implied Volatility Definition of an implied volatility Understanding a for loop Estimating the implied volatility by using a for loop Implied volatility function based on a European call Implied volatility based on a put option model The enumerate() function Estimation of IRR via a for

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loop Estimation of multiple IRRs Understanding a while loop Using keyboard commands to stop an infinitive loop Estimating implied volatility by using a while loop Nested (multiple) for loops Estimating implied volatility by using an American call Measuring efficiency by time spent in finishing a program The mechanism of a binary search

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another computer language could go through the first few chapters relatively quickly and move to more advanced topics (chapters). They should focus on option theory, implied volatility and measures of volatility, and GARCH models. One feature of this book is that most chapters after Chapter 3, Using Python as a Financial Calculator

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+=v is equivalent to x=x+v. We will discuss the for loop and other loops in more detail in Chapter 10, Python Loops and Implied Volatility. If there is no error message, we could use the npv_f() function easily. To find information about the enumerate() function, we could use help

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put option. In addition, we will explain how to download real-world option data from several public available sources. Using that data, we will estimate implied volatility, volatility skewness, and their applications. Exercises 1. What is the difference between an American call and a European call? 2. What is the unit of

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(n) 22. Write a Python program for the graphical representation of a three-step binomial tree. [ 274 ] Python Loops and Implied Volatility In this chapter, we will study two topics: loops and implied volatility based on the European options (Black-Scholes-Merton option model) and American options. For the first topic, we have the

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for loop and while loop, the two most used loops. After presenting the definition of the implied volatility and explaining the logic behind it, we discuss three ways for its estimation: based on a for loop, on a while loop, and on a

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of an option price is an increasing function of the volatility. In particular, we will cover the following topics: • What is an implied volatility? • Logic behind the estimation of an implied volatility • Understanding the for loop, while loop, and their applications • Nested (multiple) loops • The estimation of multiple IRRs • The mechanism of a binary

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search • The estimation of an implied volatility based on an American call • The enumerate() function • Retrieving option data from Yahoo! Finance and from Chicago Board Options Exchange (CBOE) • A graphical presentation of

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put-call ratios Python Loops and Implied Volatility Definition of an implied volatility From the previous chapter, we know that for a set of input variables—S (the present stock price), X (the exercise price), T (the

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.25) 3.3040017284767735 On the other hand, if we know S, X, T, r, and c, how can we estimate sigma? Here, sigma is our implied volatility. In other words, if we are given a set values such as S=40, X=40, T=0.5, r=0.05, and c=3

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value of sigma, and it should be equal to 0.25. In this chapter, we will learn how to estimate the implied volatility. Actually, the underlying logic to figure out the implied volatility is very simple: trial and error. Let's use the previous example as an illustration. We have five values—S=40

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.30. The basic design is that after inputting 100 different sigmas, plus the first four input values shown earlier, we have 100 call prices. The implied volatility is the sigma that achieves the smallest absolute difference between the estimated call price and 3.30. Of course, we could increase the number of

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incremental value other than 1, we have to specify it as follows: >>>for i in xrange(1,10,3): print i [ 277 ] Python Loops and Implied Volatility The output values will be 1, 4, and 7. Along the same lines, if we want to print 5 to 1, that is, in descending

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order, the incremental value should be -1: >>>for j in xrange(5,1,-1): print j Estimating the implied volatility by using a for loop First, we should generate a Python program to estimate the call price based on the Black-Scholes-Merton option model

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.01: print(i,sigma, diff) In the preceding program, we used the same set of input values as the example shown earlier. Thus, our expected implied volatility is 0.25. The logic of this program is that we use the trial-and-error method to feed our Black-Scholes-Merton option model

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the difference between our estimated call price and the given call price is less than 0.01, we stop. That sigma (volatility) will be our implied volatility. The output from the earlier program is shown as follows: (49, 0.25, -0.0040060797372882817) >>> The first number, 49, is the value of the variable

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: def implied_vol_call(S,X,T,r,c): from scipy import log,exp,sqrt,stats for i in range(200): [ 279 ] Python Loops and Implied Volatility sigma=0.005*(i+1) d1=(log(S/X)+(r+sigma*sigma/2.)*T)/(sigma*sqrt(T)) d2 = d1-sigma*sqrt(T) diff=c-(S

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the previous function, the conversion criterion is when the absolute difference is less than 0.01. In a sense, the current program will guarantee an implied volatility while the previous program does not guarantee an output: def implied_vol_put_min(S,X,T,r,p): from scipy import log,exp,sqrt

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'k, implied_vol, put, abs_diff' return k,implied_vol, put_out,min_value Let's use a set of input values to estimate the implied volatility. After that, we will explain the logic behind the previous program. Assume, S=40, X=40, T=12 months, r=0.1, and the put

…

: >>>implied_vol_put_min(40,40,1.,0.1,1.501) k, implied_vol, put, abs_diff (1999, 0.2, 1.5013673553027349, 0.00036735530273501737) >>> The implied volatility is 20 percent. The logic is that we assign a big value, such as 100, to a variable called min_value. For the first sigma

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the value 1.50. We continue this way until we go though the loop. For the recorded minimum value, its corresponding sigma will be our implied volatility. We could optimize the previous program by defining some intermediate values. For example, in the previous program, we estimate ln(S/X) 10,000 times

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: def npv_f(rate, cashflows): total = 0.0 for i, cashflow in enumerate(cashflows): total += cashflow / (1 + rate)**i return total [ 281 ] Python Loops and Implied Volatility The enumerate() function would generate a pair of indices, starting from 0, and its corresponding value. With a set of input values for the discount

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Value) become zero. The Python program to estimate multiple IRRs is shown as follows: def IRRs_f(cash_flows): n=1000 [ 283 ] Python Loops and Implied Volatility r=range(1,n) epsilon=abs(mean(cash_flows)*0.01) irr=[-99.00] j=1 npv=[] for i in r: npv.append(0) lag

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to an infinitive loop. For such cases, we could use Ctrl + C or Ctrl + Enter to stop such an infinitive loop: [ 285 ] Python Loops and Implied Volatility i=1 While i<5: Print i >>> If these commands do not work, then use Ctrl + Alt + Del to launch the Task Manager, choose Python

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.5,0.05,0.2) >>>round(put,2) 1.77 >>> The following program uses both a while loop and the put option to estimate the implied volatility. Here, we assume that the previous European put option function is included in the p4y.py master program (module): import p4f S=40;K=40

…

,T,r,sigma) i+=1 print('i, implied-vol, diff') print(i,sigma, diff) [ 286 ] Chapter 10 From the following output, we know that the implied volatility is 0.2, the same as we estimated using the Black-Scholes-Merton call option model. Again, we could verify this using 0.2 as

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zero. A sample implementation of the sign() function is given as follows: >>>sign(-2) -1 >>>sign(2) 1 >>>sign(0) 0 [ 287 ] Python Loops and Implied Volatility Nested (multiple) for loops For a two-dimensional matrix, you need two loops with variables i and j shown as follows: n1=2 n2=3

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return implied_vol To test the program, we could estimate an American call by inputting a set of values, including sigma, and then estimate the implied volatility as follows: >>>binomialCallAmerican(150,150,2./12.,0.003,0.2) 4.908836114170818 >>>implied_vol_American_call(150,150,2./12.,0.003,4.91

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, in seconds, is required to finish a program. The function used is time.clock(): import time start = time.clock() n=10000000 [ 289 ] Python Loops and Implied Volatility for i in range(1,n): k=i+i+2 diff= (time.clock() - start) print round(diff,2) The total time we need to finish

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the previous meaningless loop is about 1.59 seconds. The mechanism of a binary search To estimate the implied volatility, the logic underlying the earlier methods is to run the Black-Scholes-Merton option model a hundred times and choose the sigma value that achieves

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] if midval.values < target: my_min = mid + 1 elif midval.values > target: my_max = mid - 1 else: return mid raise ValueError [ 291 ] Python Loops and Implied Volatility In the previous program, x.iloc[mid] gives us the value since x is in a Data.Frame format: >>>x.iloc[600] 0 600 Baasha

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related codes as follows: >>>stocks[0:10] array(['000001.SS', 'A', 'AA', 'AAPL', 'BC', 'BCF', 'C', 'CNC', 'COH', 'CPI'], dtype=object) >>> [ 293 ] Python Loops and Implied Volatility Assignment through a for loop The following program assigns values to a variable: >>>x=[0.0 for i in xrange(5)] >>>x [0.0, 0

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46.75 50 0 0 13 December 125.00 (IBM1313L125-4) 0 0 46.45 50.45 0 0 [ 295 ] Vol 4184836 Python Loops and Implied Volatility 13 Dec 125.00 (IBM1313L125-8) 0 0 46.2 50.3 0 0 13 Dec 125.00 (IBM1313L125-A) 0 0 46.5 50

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data from Yahoo! Finance: >>>from pandas.io.data import Options >>>ticker='IBM' >>>x = Options(ticker) >>>calls, puts = x.get_options_data() [ 297 ] Python Loops and Implied Volatility We can use the head() and tail() functions to view the first and last several lines of the retrieved data: >>>calls.head() Strike Symbol Last

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0.00 16.15 16.70 23 467 4 175 IBM140222C00175000 12.20 0.04 11.95 12.50 5 767 >>> [ 299 ] Python Loops and Implied Volatility Retrieving the current price from Yahoo! Finance Using the following Python program, we can retrieve the current stock prices for a given set of ticker

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(0,1.5) plot(x, y, 'b-') plot(x, y2,'r') show() The corresponding graph is shown in the following figure: [ 301 ] Python Loops and Implied Volatility The put-call ratio for a short period with a trend Based on the preceding program, we could choose a shorter period with a trend

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corresponding graph is shown in the following figure: Summary In this chapter, we introduced different types of loops. Then, we demonstrated how to estimate the implied volatility based on a European option (Black-Scholes-Merton option model) and on an American option. We discussed the for loop and the while loop, and

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. The mean of those discounted terminal values using the risk-free rate as our discount rate would be our option price. [ 303 ] Python Loops and Implied Volatility Exercises 1. How many types of loops are present in Python? What are the differences between them? 2. What are the advantages of using a

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for loop versus a while loop? What are the disadvantages? 3. Based on a for loop, write a Python program to estimate the implied volatility. For a given set of values S=35, X=36, rf=0.024, T=1, sigma=0.13, and c=2.24, what is the

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different volatilities based on the Black-Scholes-Merton option model's call and put? 6. For a stock with multiple calls, we could estimate its implied volatility based on its call or put. Based on the Black-Scholes-Merton option model, could we get different values? 7. When estimating a huge number

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,3.3) Find reasons and modify this program accordingly. 11. From this chapter, we learn that we could use the following program to estimate an implied volatility based on the Black-Scholes-Merton option model: def implied_vol_put_min(S,X,T,r,p): from scipy import log,exp,sqrt,stats

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_pickle('c:/temp/putsFeb2014.pickle') Volatility smile and skewness Obviously, each stock should possess just one volatility. However, when estimating implied volatility, different strike prices might offer us different implied volatilities. More specifically, the implied volatility based on out-of-the-money options, atthe-money options, and in-the-money options might be quite different. Volatility

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(y,m,d) # get exact expiring date T=(exp_date-today).days/252.0 # T in years # Step 4: run a loop to estimate the implied volatility n=len(calls.Strike) # number of strike strike=[] # initialization implied_vol=[] # initialization call2=[] # initialization x_old=0 # used when we choose the first strike for

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Symbols _ expression 36 52-week high and low trading strategy 196 %d 40 % operator 29 A ActivatePython installation URL 124 American call used, for estimating implied volatility 288, 289 American option about 242 versus European option 242 Amihud's model for illiquidity (2002) 198 Anaconda Python, launching from 96, 97 URL 96

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lookback options, pricing with 342, 343 floor function using 28 for loop about 277 assigning through 294 IRR, estimating via 282, 283 used, for estimating implied volatility 278, 279 from math import * 82 F-test 194 functions activating, import function used 48 default input values, using for 45 defining, from Python editor

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, 362 volatility smile 360, 362 W web page data, retrieving from 180, 181 web page examples URL 163 while loop about 284 used, for estimating implied volatility 286, 287 X xlim() function 130 x.sum() dot function 107 [ 385 ] Y Yahoo! Finance current price, retrieving from 300 different expiring dates 299 historical

Hedge Fund Market Wizards

by Jack D. Schwager  · 24 Apr 2012  · 272pp  · 19,172 words

an above-average probability of increased future volatility. Do you favor long-dated options? Often, the longer the duration of the option, the lower the implied volatility, which makes absolutely no sense. We recently bought far out-of-themoney 10-year call options on the Dow as an inflation hedge

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. Implied volatility on the index is very low. The Dow companies would be in the best position to pass along higher prices. There is also an interest

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the forward rate of over 12 percent. In other words, the option prices implicitly assumed the 6-month forward rate as the expected level. The implied volatility at the time was around 100 basis points normalized, which meant the market was assigning the odds of nothing happening for the next six months

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dollar would be down sharply and vice versa. At the time, we happened to be looking for an efficient way of getting short the euro. Implied volatility on the euro puts was expensive. We cheapened the premium substantially by taking our exposure through a worst of option, which is an exotic option

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volatility, a nonsensically improbable event had just occurred. That’s past tense. How does that relate to a trade you did? If the three-month implied volatility says that the price move that has just occurred was a three and a half standard deviation event, we are going to like the odds

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in this chapter before our conversation related to Cornwall’s short trade in collaterized debt obligations (CDOs). 5Mai explained that the typical quoting convention for implied volatility in interest rate markets, known as “normalized volatility,” is the number of absolute basis points reflecting a one-standard-deviation event, as opposed to the

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standard convention of quoting implied volatility in other asset classes in terms of percentage changes in the underlying security. Normalized volatility of 100 basis points equals a much smaller volatility, as

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. The future volatility estimate implied by market prices (i.e., option premiums), which may be higher or lower than the historical volatility, is called the implied volatility. *This appendix was originally published in Market Wizards (1989). About the Author Mr. Schwager is a recognized industry expert in futures and hedge funds and

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