volatility smile Concepts

The Volatility Smile

by Emanuel Derman,Michael B.Miller · 6 Sep 2016

Asia, Wiley is globally committed to developing and marketing print and electronic products and services for our customers’ professional and personal knowledge and understanding. The Volatility Smile EMANUEL DERMAN MICHAEL B. MILLER with contributions by David Park Cover image: Under the Wave off Kanagawa by Hokusai © Fine Art Premium / Corbis Images

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products, visit www.wiley.com. Library of Congress Cataloging-in-Publication Data: Names: Derman, Emanuel, author. | Miller, Michael B. (Michael Bernard), 1973- author. Title: The volatility smile / Emanuel Derman, Michael B. Miller. Description: Hoboken, New Jersey : Wiley, 2016. | Series: The Wiley finance series | Includes index. Identifiers: LCCN 2016012191 (print) | LCCN 2016019398 (

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of Toronto. xv CHAPTER 1 Overview Financial models in light of the great financial crisis. The difficulties of option valuation. An introduction to the volatility smile. Financial science and financial engineering. The purpose and use of models. INTRODUCTION Our primary aim in this book is to provide the reader with an

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accessible, not-too-sophisticated introduction to models of the volatility smile. Prior to the 1987 global stock market crash, the Black-Scholes-Merton (BSM) option valuation model seemed to describe option markets reasonably well. After

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the crash, and ever since, equity index option markets have displayed a volatility smile, an anomaly in blatant disagreement with the BSM model. Since then, quants around the world have labored to extend the model to accommodate this anomaly

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normally distributed. More recently, JPMorgan called the events of the “London Whale” an eight-standard-deviation event (JPMorgan Chase & Co. 2013). Stock 8 THE VOLATILITY SMILE evolution, to take just one of many examples, isn’t Brownian.1 So, while financial engineers are rich in mathematical techniques, we don’t have

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simple, intuitive mental concept (e.g., volatility) to the mathematics that describes it (geometric Brownian motion and the BSM model), to a richer concept (the volatility smile), to experience-based intuition (the variation in the shape of the smile), and, finally, to a model (a stochastic volatility model, for example) that

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of relative valuation. Relative valuation is less ambitious, and that’s good. Relative valuation is especially well suited to valuing derivative securities. 12 THE VOLATILITY SMILE Why do practitioners concentrate on relative valuation for derivatives valuation? Because derivatives are a lot like molecules made out of simpler atoms, and so we

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UNDERLIERS As described earlier, replication begins with the science, the descriptive model of underlier behavior. Modern portfolio theory rests on the efficient 18 THE VOLATILITY SMILE market hypothesis (EMH), a framework that has come under renewed and very severe attack since the onset of the great financial crisis of 2007–2008

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an investor in holding an uncorrelated security with a negative expected excess return, when the investor could be holding riskless bonds instead. 32 THE VOLATILITY SMILE in the portfolio because most stocks are highly correlated with each other. Between July 2013 and July 2014, the mean volatility of stocks in the

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readily available. Even if they are less complex, avoid using securities that require you to make theoretical assumptions about their future behavior. 42 THE VOLATILITY SMILE SAMPLE PROBLEM Question: The payoff of a structured product is a piecewise-linear function of an underlying stock, S. The payoff has the following break

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to the past or realized values of the parameters. Implied variables represent the present and the imagined future. Realized variables represent the past. 52 THE VOLATILITY SMILE Throughout this book, we will use capital letters to represent marketderived prices. The price of a stock, bond, call, and put will typically be

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also quoted in terms of volatility. The contract accentuates volatility because traders and clients are more comfortable thinking in terms of volatility than 62 THE VOLATILITY SMILE Variance Swap on S&P 500 Instrument: Variance Swap Variance Buyer: EFG Fund Variance Seller: ABC Bank Trade Date: January 29, 2016 Start Date:

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and you lose $23.20: ( ) ( ) Profit2,norebal = 100 $7.8114 − $10.2033 − 54 $100 − $104 = −$239.20 + $216.00 = −$23.20 (continued) 120 THE VOLATILITY SMILE (continued) Over both periods, the rebalance strategy makes $16.58, while the norebalance strategy loses $15.42. In this case, the benefit of rebalancing was

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implied volatility for a long option position if transaction costs were 1 basis point (1 bp or 0.01%) and the traders (continued) 128 THE VOLATILITY SMILE (continued) rebalanced their hedges weekly? Daily? Assume 256 business days per year. Answer: Using Equation 7.19, for weekly rebalancing, we have: √ 2 𝜎̃ ≈ 𝜎 − k

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to use extensions of the BSM model to back out the corresponding implied volatilities, but the logic is the same. We introduced the volatility smile in Chapter 1. A volatility smile for a given underlier and expiration date is a function or graph that maps the strikes of options to their BSM implied volatilities

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market where these features first became apparent. While Figure 8.1 clearly resembles an actual smile, we saw in Chapter 1 that volatility smiles need not look anything like this. Volatility smiles can be flatter or more curved; they often look like smirks, rarely like frowns. No matter what their shape, it is

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25% 20% 15% 10% −10% Put −25% Put −40% Put ATM 40% Call Delta FIGURE 8.6 S&P 500 6-Month Volatility Smile 25% Call 10% Call 138 THE VOLATILITY SMILE that what matters for an option’s price is how likely it is to move into the money from wherever it is

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is current at-the-money implied volatility. The VIX, which is widely quoted, is, as we saw in Chapter 4, closely related. 148 THE VOLATILITY SMILE 5. There are other persistent patterns of equity index implied volatility that we briefly summarize. The volatility of implied volatility is greatest for short expirations

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in r. A normal or arithmetic Brownian motion for some variable therefore corresponds to a negatively skewed geometric Brownian motion, and hence a negatively sloped volatility smile. Expectations of changes in asset volatility as the market approaches certain significant levels can also give rise to skew structure. For example, investors’ perceptions

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are involved. The pseudo-probabilities of events are determined from market prices. The actual probabilities of human events are never truly known. 178 THE VOLATILITY SMILE SAMPLE PROBLEM Question: A dealer in state-contingent securities offers to let you buy or sell three securities, each of which pays £1 in one

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of the method of images commonly used in electrostatics, a technique that is roughly equivalent to the reflection principle in probability theory. 208 THE VOLATILITY SMILE To illustrate it, we initially make one more temporary simplification and consider a stock that undergoes arithmetic Brownian motion. The Method of Images for Arithmetic

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starting at S′ : ⎛ ⎜ ln ′ NDO (S𝜏 ) = N′ ⎜ ⎜ ⎜ ⎝ ( ) ⎞ ⎛ 1 + 𝜎2𝜏 ⎟ ⎜ ln 2 ⎟ − 𝛼N′ ⎜ √ ⎟ ⎜ 𝜎 𝜏 ⎟ ⎜ ⎠ ⎝ S𝜏 S ( ) 1 2 ⎞ 𝜎 𝜏⎟ 2 ⎟ √ ⎟ 𝜎 𝜏 ⎟ ⎠ S𝜏 S B2 + (12.3) 210 THE VOLATILITY SMILE where N′ (x) = 1 − 1 x2 √ e 2 2𝜋 is the standard normal probability density func- tion, and 𝛼 is a ratio to be determined. We

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⎟=0 N′ ⎜ √ √ ⎟ − 𝛼N ⎜⎜ ⎟ 𝜎 𝜏 𝜎 𝜏 ⎜ ⎟ ⎟ ⎜ ⎝ ⎠ ⎠ ⎝ (12.8) We can solve this equation for 𝛼 to obtain 𝛼= ( ) 2𝜇2 ( ) 2r2 −1 B 𝜎 B 𝜎 = S S (12.9) 212 THE VOLATILITY SMILE Notice again that 𝛼 is independent of the remaining time to expiration ′ vanishes on the boundary for all times, for a fixed 𝜏, so that the density

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Cd = Cerdt (13.46) Equations 13.45 and 13.46 are the appropriate binomial equations for a stock with dividends. Time-Dependent Deterministic Volatility: A Volatility Smile with Term Structure but No Skew In the last few sections we have been progressively increasing the complexity of our binomial model of stock evolution

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recall that the Breeden-Litzenberger formula is model-independent. It does not require Black-Scholes-Merton (BSM) or any other pricing model. 265 266 THE VOLATILITY SMILE In a similar fashion, the Dupire equation, which we will soon derive, describes the relationship between the local volatility of the previous chapter and the

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, there is another stochastic variable, the volatility itself, so 𝛴(S, K) = f(K/S) only if the volatility hasn’t changed stochastically. 316 THE VOLATILITY SMILE Stickiness in the Real World We can combine the linear approximations for sticky strike, sticky moneyness, and sticky local volatility into the following more general

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several subsequent chapters, we will explore specific versions of stochastic volatility, and see how different assumptions affect the shape and evolution of the volatility smile. 319 320 THE VOLATILITY SMILE Approaches to Stochastic Volatility Modeling The most obvious approach to stochastic volatility modeling is to make the stock’s volatility depend on a stochastic

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models, 𝜎 is a deterministic function of S, with ±100% correlation between S and 𝜎. With stochastic volatility, S and 𝜎 can be more flexibly 332 THE VOLATILITY SMILE correlated. We can introduce this correlation through the Brownian motion terms, expressing the correlation 𝜌 between dZ and dW through dZdW = 𝜌dt (19.28) where 𝜌 is

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Z are standard arithmetic Brownian motions, with correlation 𝜌. The volatility of the log returns of S, 𝛼S(𝛽 −1) , is determined by the 337 338 THE VOLATILITY SMILE stock price S, the constant 𝛽, and a stochastic variable, 𝛼, with 𝜉 representing the volatility of 𝛼. The parameter 𝛽 is a model parameter that lies between 0

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local volatility value 𝛴LV (S, t, K, T, 𝛼, ̄ 𝛽) plus a small correction due to 𝛼 implied volatility 𝛴SLV as 𝛴LV (S, t, K, T, 𝛼, 342 THE VOLATILITY SMILE being stochastic, so that 𝛴SLV ≡ 𝛴LV (𝛼) ̄ + (𝛴SLV − 𝛴LV (𝛼)), ̄ where we denote ̄ 𝛽) for brevity as 𝛴LV (𝛼). ̄ Thus, 𝛴LV (S, t, K, T, 𝛼, CSLV = CBSM (𝛴LV (𝛼) ̄ +

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in actual markets. Figure 22.2 shows the results of a Monte Carlo simulation for option prices and the corresponding Black-Scholes-Merton (BSM) implied volatility smiles. For the Monte Carlo simulation, we have assumed that volatility evolves according to Equation 22.1, with zero correlation between the stock price and

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: Notice that the equation for the variance of the path volatility, var[𝜎], ̄ behaves as we would expect if instantaneous volatility was mean (continued) 374 THE VOLATILITY SMILE (continued) reverting. When 𝜏 is small, it is proportional to 𝜏. When 𝜏 is large, it is inversely proportional to 𝜏. For 0.1-, 0.25-, and 1

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two different models of the smile: local volatility and stochastic volatility. If we were to calibrate both models, for example, to an observed index volatility smile with negative skew, each would produce a different evolution of volatility and a different forward skew. In addition, though both models produce the same initial

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. Chapter 2 of Fouque, Jean-Pierre, George Papanicolaou, and Ronnie Sircar. Derivatives in Financial Markets with Stochastic Volatility. Cambridge: Cambridge University Press, 2000. 382 THE VOLATILITY SMILE Lewis, Alan. Option Valuation under Stochastic Volatility. Newport Beach, CA: Finance Press, 2000. Hull, John, and Alan White. “The Pricing of Options on Assets

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that neither of the jump-diffusion volatilities was very different from the diffusion-only volatility. Later we’ll see that jumps can meaningfully impact the volatility smile, especially at short expirations, but that may require jump sizes and probabilities that are significantly greater in magnitude than what we estimate from the

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these laws; only intuition, resting on sympathetic understanding of experience, can reach them. —Albert Einstein n 1994, when researchers began attempting to explain the volatility smile, many of us hoped that there would be one better model that could replace Black-Scholes-Merton. Instead, we have ended up with a plethora

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Free Lunch Would You Like Today Sir?: Delta Hedging, Volatility, Arbitrage and Optimal Portfolios.” Wilmott (November). Andersen, Leif, and Jesper Andreasen. 2000. “Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing.” Review of Derivatives Research 4 (3): 231–262. Birru, Justin, and Stephen Figlewski. 2012. “Anatomy of a Meltdown

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options, 94 hedge ratios under, 379 implied volatility in, 80 impracticality of, 204 501 502 Black-Scholes-Merton (BSM) model (Continued) as inconsistent with volatility smiles, 163 local volatility as extension of, 303–304 stochastic volatility models in, 321–325 transaction costs in, 117, 125, 127 Black-Scholes-Merton (BSM) partial

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replication valuation, 15 and theory, 5–8 504 Financial theory, 5–8 Fisk-Stratonovich integral, 424, 427 Foreign exchange (FX) options: jumps in, 383 volatility smile in, 149–150 Formal proof, of Dupire’s equation, 275–277 Forward approach, to stochastic integration, 425–426 Forward integrals, 427–429 Forward Itô integrals

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158–160 Instantaneous variance, 364, 380 Instantaneous volatility, 364 Integration by parts, 427–429 Interest rates: modeling of, 164 Vasiçek interest rate model, 334 volatility smile of, 151 Intuition, and financial models, 11 Irrational exuberance, 311 Itô integrals, backward, 92–93, 421–429 Itô’s lemma: changes in option values

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Out-of-the-money options: deltas of, 141, 142 local and implied volatilities in, 262 payoffs of, 148 Parameter(s): implied volatility as, 50 of volatility smile, 131–136 Partial differential equation (PDE), 395–398 Partial differential equation (PDE) model, 125–128 Path-dependent options: dynamic hedging for, 204 lookback options

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, 144–145, 158 sticky strike rule, 310–311 Strike price: and implied volatility, 132 relative, 136 Strong replication, 204 Strong static replication, 204 Swaption volatility smile, 151 Taylor series expansion: of the call price, 45–46 INDEX in Jarrow-Rudd convention, 232 in jump-diffusion models, 401, 412 in jump modeling

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implied volatility, 134 and moneyness, 137 and replicating portfolios, 81 short, and implied volatility behavior, 282–286 Time to maturity, 138–139 Trading consequences, of volatility smile, 151–152 Trading desks, relative valuation used by, 12 Transaction costs, 117–129 analytical approximation of, 123–124 effects of, 117–120 partial differential equation

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patterns (Continued) sticky strike rule, 310–311 and stochastic volatility models, 317 Volatility paths, 355 Volatility points, 135 Volatility sensitivity, of options, 57–60 Volatility smile, 131–152 and delta, 140–143 in equity indexes, 144–148 in foreign exchange options, 149–150 graphing of, 136–139 in individual equities, 148

Stigum's Money Market, 4E

by Marcia Stigum and Anthony Crescenzi · 9 Feb 2007 · 1,202pp · 424,886 words

-the-money and out-of-the-money options are typically higher than the implied volatilities of at-the-money options. This is known as the volatility smile. Typically, the implied volatility also depends on other characteristics of the option such as its maturity. SHORTCOMINGS OF BLACK-SCHOLES One of the failures of

A Primer for the Mathematics of Financial Engineering

by Dan Stefanica · 4 Apr 2008

of the money or deep in the money options is higher than the implied volatility of at the money options. This phenomenon is called the volatility smile. Another possible pattern for implied volatility is the volatility skew, when, e.g., the implied volatility of deep in the money options is smaller than

The Misbehavior of Markets: A Fractal View of Financial Turbulence

by Benoit Mandelbrot and Richard L. Hudson · 7 Mar 2006 · 364pp · 101,286 words

is one of the liveliest subdisciplines in mathematical finance. The most common approach is to try merely fixing the old formula. Software to correct the “volatility smile,” the U-shaped pattern that Black-Scholes volatility errors often trace on graph paper, is now standard. Many adopt the GARCH methods mentioned earlier; while

Corporate Finance: Theory and Practice

by Pierre Vernimmen, Pascal Quiry, Maurizio Dallocchio, Yann le Fur and Antonio Salvi · 16 Oct 2017 · 1,544pp · 391,691 words

or far in-the-money is higher than the implied volatility recalculated on the basis of at-the-money options. This phenomenon is called the volatility smile (because when we draw volatility on a chart as a function of strike price, it looks like a smile). We will see in the following

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company Vishny, R. visibility volatility behavioural finance capital employed debt securities investment risk options and price speculative valuation underlying asset value of option volatility risk volatility smile volume growth volumes/volume effects financial market restrictions gross margin liquidity measured in terms of voluntary offers voting caps, shareholders voting rights bondholders capital structure

The Myth of the Rational Market: A History of Risk, Reward, and Delusion on Wall Street

by Justin Fox · 29 May 2009 · 461pp · 128,421 words

nature of financial market risk. This realization came quickly to some options traders. After October 19, options prices displayed what came to be called a “volatility smile.” By turning the Black-Scholes equation around, one can calculate the implied volatility of any stock from the price of its options. Put options allow

Tools for Computational Finance

by Rüdiger Seydel · 2 Jan 2002 · 313pp · 34,042 words

a ﬁgure, joining the points with straight lines. (You will notice a convex shape of the curve. This shape has lead to call this phenomenon volatility smile.) Table 1.3. Calls on the DAX on 4. Jan 1999 K V 6000 80.2 6200 47.1 6300 35.9 6350 31.3

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three main principles of numerical analysis, namely order (of convergence), stability, and eﬃciency. A Crank-Nicolson variant has been developed that is consistent with the volatility smile, which reﬂects the dependence of the volatility on the strike [AB97]. Notes and Comments 175 In view of the representation (4.12) the Crank-Nicolson

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] F. AitSahlia, P. Carr: American options: A comparison of numerical methods. In [RT97] (1997) p. 67-87. [AnA00] L. Andersen, J. Andreasen: Jump diﬀusion process: Volatility smile ﬁtting and numerical methods for option pricing. Review Derivatives Research 4 (2000) 231-262. [AB97] L.B.G. Andersen, R. Brotherton-Ratcliﬀe: The equity option

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volatility smile: an implicit ﬁnite-diﬀerence approach. J. Computational Finance 1,2 (1997/1998) 5–38. [AnéG00] T. Ané, H. Geman: Order ﬂow, transaction clock, and

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Vieta 17 Volatility 5–6, 9, 15, 17, 34, 36–40, 43, 52, 54, 58, 92, 102, 107, 116, 164, 212, 224, 226, 235, 246 Volatility smile 55, 174 Von Neumann stability 225, 227, 235–237 Wave 226, 230–231 Wave number 225–226, 231 Wavelet 206 Weak convergence 94–95, 100

Derivatives Markets

by David Goldenberg · 2 Mar 2016 · 819pp · 181,185 words

. This turns out not to be empirically true, at least since the market crash of 1987, and it generates a volatility smile, and its variations. A vast literature has developed around explaining volatility smiles and its variations. We can’t cover that here, but we can look at the economic reasons for expecting σ

Analysis of Financial Time Series

by Ruey S. Tsay · 14 Oct 2001

motion fails to explain some characteristics of asset returns and the prices of their derivatives (e.g., the “volatility smile” of implied volatilities; see Bakshi, Cao, and Chen, 1997, and the references therein). Volatility smile is referred to as the convex function between the implied volatility and strike price of an option. Both out

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-ofthe-money and in-the-money options tend to have higher implied volatilities than at-the-money options especially in the foreign exchange markets. Volatility smile is less pronounced for equity options. The inadequacy of the standard stochastic diffusion model has led to the developments of alternative continuous-time models. For

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model enjoys several nice properties. The returns implied by the model are leptokurtic and asymmetric with respect to zero. In addition, the model can reproduce volatility smile and provide analytical formulas for the prices of many options. The model consists of two parts, with the first part being continuous and following a

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from discrete observations,” Scandinavian Journal of Statistics, 24, 1–19. Kou, S. (2000), “A jump diffusion model for option pricing with three properties: Leptokurtic feature, volatility smile, and analytic tractability,” working paper, Columbia University. Lo, A. W. (1988), “Maximum likelihood estimation of generalized Ito’s processes with discretely sampled data,” Econometric Theory

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, 279 Vector AR model, 309 Vector ARMA model, 322 marginal models, 327 Vector MA model, 318 Volatility, 79 Volatility equation, 82 Volatility model, factor, 383 Volatility smile, 244 White noise, 26 Wiener process, 223 generalized, 225

How I Became a Quant: Insights From 25 of Wall Street's Elite

by Richard R. Lindsey and Barry Schachter · 30 Jun 2007

in the December 1995 issue of Risk magazine. 17. It was actually published twice. First in Goldman Sachs Quantitative Strategies Research Notes, January 1994, “The Volatility Smile and Its Implied Tree,” Derman and Kani, and then later in a paper called “Riding on a Smile,” Risk, 7, no. 2 (1994), pp. 32

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11:41 note s 18. This was also published twice. First in Goldman Sachs Quantitative Strategies Research Notes, February 1996, “Implied Trinomial Trees of the Volatility Smile,” Derman, Kani, and Chriss, and then in a paper by the same name published in the Summer 1996 in the Journal of Derivatives. 19. I