stochastic volatility Concepts

Analysis of Financial Time Series

by Ruey S. Tsay · 14 Oct 2001

GARCH Model, 102 vii viii CONTENTS 3.8 The CHARMA Model, 107 3.9 Random Coefficient Autoregressive Models, 109 3.10 The Stochastic Volatility Model, 110 3.11 The Long-Memory Stochastic Volatility Model, 110 3.12 An Alternative Approach, 112 3.13 Application, 114 3.14 Kurtosis of GARCH Models, 118 Appendix A

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Markov Chain Simulation, 396 Gibbs Sampling, 397 Bayesian Inference, 399 Alternative Algorithms, 403 Linear Regression with Time-Series Errors, 406 Missing Values and Outliers, 410 Stochastic Volatility Models, 418 Markov Switching Models, 429 Forecasting, 438 Other Applications, 441 445 Preface This book grew out of an MBA course in analysis of financial

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developed Monte Carlo Markov Chain (MCMC) methods in the statistical literature and apply the methods to various financial research problems, such as the estimation of stochastic volatility and Markov switching models. The book places great emphasis on application and empirical data analysis. Every chapter contains real examples, and, in many occasions, empirical

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referred to as a simple autoregressive (AR) model of order 1 or simply an AR(1) model. This simple model is also widely used in stochastic volatility modeling when rt is replaced by its log volatility; see Chapters 3 and 10. The AR(1) model in Eq. (2.6) has several

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Nelson (1991), the conditional heteroscedastic autoregressive movingaverage (CHARMA) model of Tsay (1987), the random coefficient autoregressive (RCA) model of Nicholls and Quinn (1982), and the stochastic volatility (SV) models of Melino and Turnbull (1990), Harvey, Ruiz, and Shephard (1994), and Jacquier, Polson, and Rossi (1994). We also discuss advantages and weaknesses of

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of σt2 , whereas those in the second category use a stochastic equation to describe σt2 . The GARCH model belongs to the first category, and the stochastic volatility model is in the second category. For simplicity in introducing volatility models, we assume that the model for the conditional mean is given. However, we

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observed lagged values rt−i . Yet the volatility is a quadratic function of the lagged innovations at−i in a CHARMA model. 3.10 THE STOCHASTIC VOLATILITY MODEL An alternative approach to describe the volatility evolution of a financial time series is to introduce an innovation to the conditional variance equation of

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at ; see Melino and Turnbull (1990), Harvey, Ruiz, and Shephard (1994) and Jacquier, Polson, and Rossi (1994). The resulting model is referred to as a stochastic volatility (SV) model. Similar to EGARCH models, to ensure positiveness of the conditional variance, SV models use ln(σt2 ) instead of σt2 . A SV model is

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at the model uses two innovations t and vt . We discuss a MCMC method to estimate SV models in Chapter 10. For more discussions on stochastic volatility models, see Taylor (1994). The appendixes of Jacquier, Polson, and Rossi (1994) provide some properties of the SV model when m = 1. For instance,

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SV models often provided improvements in model fitting, but their contributions to out-of-sample volatility forecasts received mixed results. 3.11 THE LONG-MEMORY STOCHASTIC VOLATILITY MODEL More recently, the SV model is further extended to allow for long memory in volatility, using the idea of fractional difference. As stated in

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) series is a Gaussian long-memory signal plus a non-Gaussian white noise; see Breidt, Crato, and de Lima (1998). Estimation of the long-memory stochastic volatility model is complicated, but the fractional difference parameter d can be estimated by using either a quasi-maximum likelihood method or a regression method. Using

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-frequency data,” Working paper, Graduate School of Business, University of Chicago. Bai, X., Russell, J. R., and Tiao, G. C. (2001), “Kurtosis of GARCH and stochastic volatility models,” Working paper, Graduate School of Business, University of Chicago. Bollerslev, T. (1986), “Generalized autoregressive conditional heteroskedasticity.” Journal of Econometrics, 31, 307–327. Bollerslev, T

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Business & Economic Statistics, 17, 9–21. Breidt, F. J., Crato, N., and de Lima, P. (1998), “On the detection and estimation of long memory in stochastic volatility,” Journal of Econometrics, 83, 325–348. Cao, C., and Tsay, R. S. (1992), “Nonlinear time series analysis of stock volatilities,” Journal of Applied Econometrics, 7

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, N. (1994), “Multivariate stochastic variance models,” Review of Economic Studies, 61, 247–264. Jacquier, E., Polson, N. G., and Rossi, P. (1994), “Bayesian analysis of stochastic volatility models” (with discussion), Journal of Business & Economic Statistics, 12, 371–417. McLeod, A. I., and Li, W. K. (1983), “Diagnostic checking ARMA time series models

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squared-residual autocorrelations,” Journal of Time Series Analysis, 4, 269–273. REFERENCES 125 Melino, A., and Turnbull, S. M. (1990), “Pricing foreign currency options with stochastic volatility,” Journal of Econometrics, 45, 239–265. Nelson, D. B. (1990), “Stationarity and persistence in the GARCH(1, 1) model,” Econometric Theory, 6, 318–334. Nelson

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., and Tsay, R. S. (2000), “Long-range dependence in daily stock volatilities,” Journal of Business & Economic Statistics, 18, 254–262. Taylor, S. J. (1994), “Modeling stochastic volatility,” Mathematical Finance, 4, 183–204. Tong, H. (1978), “On a threshold model,” in Pattern Recognition and Signal Processing, ed. C.H. Chen, Sijhoff & Noordhoff: Amsterdam

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equity options. The inadequacy of the standard stochastic diffusion model has led to the developments of alternative continuous-time models. For example, jump diffusion and stochastic volatility models have been proposed in the literature to overcome the inadequacy; see Merton (1976) and Duffie (1995). Jumps in stock prices are often assumed to

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multivariate generalized ARCH approach,” Review of Economics and Statistics, 72, 498–505. Chib, S., Nardari, F., and Shephard, N. (1999), “Analysis of high dimensional multivariate stochastic volatility models,” Working paper, Washington University, St Louis. Harvey, A., Ruiz, E., and Shephard, N. (1995), “Multivariate stochastic variance models,” in ARCH Selected Readings, ed. R

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can focus on the probability distribution functions f (R | H, β) and f (H | ω) and the prior distribution p(β, ω). We assume 419 STOCHASTIC VOLATILITY MODELS that the prior distribution can be partitioned as p(β, ω) = p(β) p(ω), that is, prior distributions for the mean and volatility

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equations are independent. A Gibbs sampling approach to estimating the stochastic volatility in Eqs. (10.20) and (10.21) then involves drawing random samples from the following conditional posterior distributions: f (β | R, X, H, ω),

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greater than 2.52. The Ljung–Box statistics of the standardized residuals and their squared series fail to indicate any model inadequacy. Next, consider the stochastic volatility model r t = µ + at , at = h t t ln h t = α0 + α1 ln h t−1 + vt , (10.27) where vt s are

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alpha1 1.5 2.0 -10 -5 0 mu 5 10 Figure 10.4. Density functions of prior and posterior distributions of parameters in a stochastic volatility model for the monthly log returns of S&P 500 index. The dashed line denotes prior density and solid line denotes the posterior density, which

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posterior means of the parameters change slightly, but the series of posterior means of h t are stable. 10.7.2 Multivariate Stochastic Volatility Models In this subsection, we study multivariate stochastic volatility models using the Cholesky decomposition of Chapter 9. We focus on the bivariate case, but the methods discussed also apply to

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, where b2t and q21,t can be interpreted as the residual and least squares estimate of the linear regression a2t = q21,t a1t + b2t . 425 STOCHASTIC VOLATILITY MODELS The conditional covariance matrix of at is parameterized by {g11,t , g22,t } and {q21,t } as σ11,t σ12,t 0 1

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γ = (γ , γ , σ 2 ). The augmented parameωi = (αi0 , αi1 , σiv 0 1 u ters are Q, G1 , and G2 . To estimate such a bivariate stochastic volatility model via Gibbs sampling, we use results of the univariate model in the previous subsection and two additional conditional posterior distributions. Specifically, we can draw

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. Estimation of Bivariate Volatility Models for Monthly Log Returns of IBM Stock and the S&P 500 Index from January 1962 to December 1999. The Stochastic Volatility Models Are Based on the Last 1000 Iterations of a Gibbs Sampling with 1300 Total Iterations. (a) Bivariate GARCH(1, 1) model with time-

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1.5si2 ], where si2 is the sample variance of the log return rit . Posterior means and standard errors of the “traditional” parameters of the bivariate stochastic volatility model are given in Table 10.2(b). To check for convergence of the Gibbs sampling, we ran the procedure several times with different starting

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sampling results is clearly seen. It is informative to compare the GARCH model with time-varying correlations in Eqs. (10.33)–(10.36) with the stochastic volatility model. First, as expected, the mean equations of the two models are essentially identical. Second, Figure 10.8 shows the time plots of fitted volatilities

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for IBM stock return. The upper panel is for the GARCH model, and the lower panel shows the posterior mean of the stochastic volatility model. The two models show similar volatility characteristics; they exhibit volatility clusterings and indicate an increasing trend in volatility. However, the GARCH model produces

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bivariate GARCH model produces a spurious volatility peak. This spurious peak is induced by its dependence on IBM returns and does not appear in the stochastic volatility model. Indeed, the fitted volatilities of S&P 500 index return by 429 40 60 0 • 0.0 0.2 0.4 0.6

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0.4 0.6 0.8 cor Figure 10.7. Scatterplots of posterior means of various statistics of two different Gibbs samples for the bivariate stochastic volatility model for monthly log returns of IBM stock and the S&P 500 index. The x-axis denotes results based on 500 + 3000 iterations

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and the y-axis denotes results based on 300 + 1000 iterations. The notation is defined in the text. the bivariate stochastic volatility model are similar to that of the univariate analysis. Fourth, Figure 10.10 shows the time plots of fitted conditional correlations. Here the two models

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the GARCH model are relatively smooth and positive with a mean value 0.55 and standard deviation 0.11. However, the correlations produced by the stochastic volatility model vary markedly from one month to another with a mean value 0.57 and standard deviation 0.17. Furthermore, there are isolated occasions in

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60 80 120 160 (a) GARCH with time-varying correlation 1970 1980 year 1990 2000 1990 2000 vol 20 40 60 80 120 160 (b) Stochastic volatility 1970 1980 year Figure 10.8. Time plots of fitted volatilities for monthly log returns of IBM stock from 1962 to 1999: (a) a GARCH

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model with time-varying correlations, and (b) a bivariate stochastic volatility model estimated by Gibbs sampling with 300 + 1000 iterations. Tsay (1994) discuss a Gibbs sampling procedure to estimate such a model when the volatility in

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40 60 80 100 (a) GARCH with time-varying correlation 1970 1980 year 1990 2000 1990 2000 0 vol 20 40 60 80 100 (b) Stochastic volatility model 1970 1980 year Figure 10.9. Time plots of fitted volatilities for monthly log returns of the S&P 500 index from 1962 to

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1999: (a) a GARCH model with time-varying correlations, and (b) a bivariate stochastic volatility model estimated by Gibbs sampling with 300 + 1000 iterations. another is governed by P(st = 2 | st−1 = 1) = e1 , P(st = 1 | st−

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0.8 (a) GARCH with time-varying correlation 1970 1980 year 1990 2000 1990 2000 -0.4 0.0 cor 0.4 0.8 (b) Stochastic volatility model 1970 1980 year Figure 10.10. Time plots of fitted correlation coefficients between monthly log returns of IBM stock and the S&P 500

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undex from 1962 to 1999: (a) a GARCH model with time-varying correlations, and (b) a bivariate stochastic volatility model estimated by Gibbs sampling with 300 + 1000 iterations. S = (s1 , s2 , . . . , sn ) contains the augmented parameters. The volatility vector H = (h 2 , . . . , h

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forecasting period. In a sense, forecasting here is done by using the fitted model to simulate realizations for the forecasting period. We use the univariate stochastic volatility model to illustrate the procedure; forecasts of other models can be obtained by the same method. 439 500 1000 1500 2000 (a) squared log

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returns, (b) the GARCH-M model in Eq. (10.42), and (c) the twostate Markov switching GARCH-M model in Eq. (10.40). Consider the stochastic volatility model in Eqs. (10.20) and (10.21). Suppose that there are n returns available and we are interested in predicting the return rn+i

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forecasts of the return and its volatility for five forecast horizons starting with December 1999. Both the GARCH model in Eq. (10.26) and the stochastic volatility model in Eq. (10.27) are used in the forecasting. The volatility forecasts of the GARCH(1, 1) model increase gradually with the forecast

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of S&P 500 Index. The Data Span Is From January 1962 to December 1999 and the Forecast Origin Is December 1999. Forecasts of the Stochastic Volatility Model Are Obtained by a Gibbs Sampling with 2000 + 2000 Iterations. (a) Horizon GARCH SVM Log return 1 0.66 0.53 2 0.

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compute the Value at Risk of a long position worth $1 million with probability 0.01 for the next trading day. 6. Build a bivariate stochastic volatility model for the monthly log returns of General Motors stock and the S&P 500 index for the sample period from January 1950 to December

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sampling methods using Markov chains and their applications,” Biometrika, 57, 97–109. Jacquier, E., Polson, N. G., and Rossi, P. E. (1994), “Bayesian analysis of stochastic volatility models” (with discussion), Journal of Business & Economic Statistics, 12, 371–417. Jones, R. H.(1980), “Maximum likelihood fitting of ARMA models to time series with

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Linear time series, 27 Liquidity, 179 Ljung–Box statistic, 25, 87 multivariate, 308 Local linear regression, 143 Log return, 4 Logit model, 209 Long-memory stochastic volatility, 111 time series, 72 Long position, 5 Marginal distribution, 7 Markov process, 395 Markov property, 29 Markov switching model, 135, 429 Martingale difference, 93 Maximum

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, 138 Square root of time rule, 260 Standard Brownian motion, 61 State-space model nonlinear, 145 Stationarity, 23 weak, 300 Stochastic diffusion equation, 226 INDEX Stochastic volatility model, 110, 418 multivariate, 424 Structural form, 310 Student-t distribution standardized, 88 Survival function, 286 Tail index, 271 Threshold, 131 Threshold autoregressive model multivariate

The Volatility Smile

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

on Local Volatility Models 303 CHAPTER 18 Patterns of Volatility Change 309 CHAPTER 19 Introducing Stochastic Volatility Models 319 CHAPTER 20 Approximate Solutions to Some Stochastic Volatility Models 337 CHAPTER 21 Stochastic Volatility Models: The Smile for Zero Correlation 353 CHAPTER 22 Stochastic Volatility Models: The Smile with Mean Reversion and Correlation 369 CHAPTER 23 Jump-Diffusion Models

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, in Chapters 14 through 24, we explore more advanced option models consistent with the smile. These models can be grouped into three families: local volatility, stochastic volatility, and jump-diffusion. While these newer models address many of the shortcomings of the Black-Scholes-Merton model, they are themselves imperfect. As markets evolve

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dividends and a riskless rate of 0%. CHAPTER 10 A Survey of Smile Models An overview of models consistent with the smile. Local volatility models, stochastic volatility models, jump-diffusion models. In the presence of a smile, the BSM model produces incorrect hedge ratios and exotic option values. AN OVERVIEW OF SMILE

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BSM assumptions that is certainly violated by actual underliers is the assumption that the volatility of the underlier is constant over time. Volatility fluctuates. In stochastic volatility models, there are two random processes, one for the stock itself, and another for the volatility or variance of the stock. These two random processes

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derive an 168 THE VOLATILITY SMILE arbitrage-free formula for option values. We will do exactly this in a later chapter. The main problem with stochastic volatility models is that we don’t really know the appropriate stochastic differential equation for volatility. An additional objection is that, while volatility is stochastic, its

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same stochastic process. But volatility can also change for other reasons, independent of changes in the underlier. In the next chapter, we will formally introduce stochastic volatility models, which allow volatility to change independently. END-OF-CHAPTER PROBLEMS 18-1. Assume that the sticky strike rule is true, and that implied volatility

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approaches we will consider, and we will use both of them, as well as more heuristic approaches, to understand the effects of stochastic volatility on the volatility smile. By approaching stochastic volatility from different starting points, we can learn much.1 1 There are (at least) two other approaches, which we now briefly mention

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BSM volatility stochastic.2 This treatment is not theoretically rigorous but is nevertheless very useful for gaining intuition about the effects of stochastic volatility models. A HEURISTIC APPROACH FOR INTRODUCING STOCHASTIC VOLATILITY INTO THE BLACK-SCHOLES-MERTON MODEL In this section we use the BSM formula to understand the qualitative behavior of the smile

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a negative correlation between the index and its volatility in order to reflect the observed skew. With this intuition established, we now proceed to examine stochastic volatility more rigorously. The Extended Black-Scholes-Merton Model: A Stochastic Differential Equation for Volatility We begin this section by exploring how we can model the

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increases, the range gets smaller. Figure 19.4 shows the region encompassing ±1 standard deviation for an Ornstein-Uhlenbeck process and Brownian motion. 329 Introducing Stochastic Volatility Models ± FIGURE 19.4 ± 2 Schematic Illustration of the Standard Deviation of Yt SAMPLE PROBLEM Question: Assume that volatility can be described by the following

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it is in the second part. Symmetric shocks and no shocks are not necessarily equivalent in a mean reversion model. Introducing Stochastic Volatility Models 331 A Survey of Some Stochastic Volatility Models Most stochastic volatility models assume traditional geometric Brownian motion for the stock price: dS = 𝜇dt + 𝜎dZ S (19.23) If the volatility term 𝜎 is

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model, 𝜎 was independent of and uncorrelated with S. In local volatility 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

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can often be written as averages over a distribution of BSM prices with a range of volatilities, which makes analysis easier and more intuitive. Introducing Stochastic Volatility Models 335 END-OF-CHAPTER PROBLEMS 19-1. Assume that volatility can be described by the following meanreverting discrete time series model: ) ( d𝜎t = 𝜎t

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to the BSM price based on the current SPX level and volatility. Assume zero dividends and riskless rate. CHAPTER 20 Approximate Solutions to Some Stochastic Volatility Models Adding stochastic volatility to the local volatility model. A negative local volatility skew picks up convexity. The partial differential equation for options with stochastic stock volatility. The

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a wide range of strikes, varying only from 18.76% to 19.14%. Approximate Solutions to Some Stochastic Volatility Models 341 Now let’s switch on the stochastic volatility by letting 𝜉 be small but nonzero. The stochastic volatility term 𝛼 in Equation 20.4 will now fluctuate over time and make the skew stochastic. When the value

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CBSM || ⎥ ⎢( 𝛴 )2 | 2 1 | LV 𝜕𝜎 ≈ ⎢ (𝛼𝜉) ̄ 2𝜏 ⎥ | ⎥ 2 ⎢ 𝛼̄ 𝜕CBSM || ⎥ ⎢ | 𝜕𝜎 |𝜎=𝛴LV ⎦ ⎣ 2 | 𝜕 CBSM | | 2 1 2 | 𝜉2𝜏 ≈ 𝛴LV 𝜕𝜎 | 2 𝜕CBSM || 𝜕𝜎 ||𝜎=𝛴LV (20.11) Approximate Solutions to Some Stochastic Volatility Models 343 Using our formulas for vega and volga from Chapter 19, 𝜕 2 CBSM 𝜕𝜎 2 = 1 𝜎 𝜕CBSM 𝜕𝜎 [ 1 𝜎2𝜏 ] ( ( ))2 S 𝜎2𝜏 ln − K 4

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examine these models in greater mathematical detail. We begin by deriving a partial differential equation for the value of an option in the presence of stochastic volatility by extending the BSM riskless-hedging argument. This section follows closely a derivation from Wilmott (1998). Assume the following general stochastic evolution process for a

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different, too. We didn’t change the option payoffs, but we did change the world they inhabit. Approximate Solutions to Some Stochastic Volatility Models 351 THE CHARACTERISTIC SOLUTION TO THE STOCHASTIC VOLATILITY MODEL Just as the solution to the BSM equation is the risk-neutral discounted expected value of the option’s payoffs, so

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over the mixture transforms into an integral. Denoting the probability density function of the path volatilities as 𝜙 (𝜎), ̄ we have ∞ CSV = ∫0 CBSM (𝜎)𝜙( ̄ 𝜎) ̄ d𝜎̄ (21.12) Stochastic Volatility Models: The Smile for Zero Correlation 357 Let’s assume that the volatility of volatility is small, and then perform a ̄ second-order Taylor expansion

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𝜏 1 (22.2) We can use this equation, together with some intuition about the way mean-reverting volatility evolves, to understand how the smiles in stochastic volatility models behave for very short and very long expirations. Volatility versus Path Volatility In standard Brownian motion, the diffusion process causes the variance or range

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a coefficient proportional to 1/𝜏, so that we can write var[𝜎] ̄ = const/𝜏. As 𝜏 → ∞, we then have const ̄ 𝜎 lim 𝛴 ≈ 𝜎̄ − 8 𝜏→∞ (22.6) At long expirations, the stochastic volatility model with mean reversion and zero correlation converges to an implied volatility function that is independent of moneyness. Asymptotically, there is no smile. Why is

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the correction term in Equation 22.6 negative? Why does stochastic volatility lower the implied volatility from the nonstochastic case? The reason, as before, is that the option price CBSM (𝜎) is a concave function of 𝜎 as 𝜏 → ∞, and

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negative deviations in volatility less likely, consistent with a flatter smile. Σ Short expiration Long expiration Strike FIGURE 22.1 𝜌=0 The Smile for Stochastic Volatility Model with 373 Stochastic Volatility Models: The Smile with Mean Reversion and Correlation 21.4% Implied Volatility 21.2% 21.0% 20.8% 20.6% 20.4% 20

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expiration increases, but changes in the level of the index have no impact on the 90–100 strike skew. NONZERO CORRELATION IN STOCHASTIC VOLATILITY MODELS We’ve shown that stochastic volatility models lead to a symmetric smile when there is no correlation between the stock price and its volatility. That’s not a bad

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as the market rises. As a result, the correct hedge ratio is smaller than the BSM delta. Stochastic volatility: In an extended BSM stochastic volatility model, implied volatility is a function of K/S and the instantaneous stochastic volatility itself. When the skew is negative and S is held constant, implied volatility increases as K decreases

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–300. Gatheral, Jim. The Volatility Surface: A Practitioner’s Guide. Hoboken, NJ: John Wiley & Sons, 2006. Heston, Steven. “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” Review of Financial Studies 6, no. 2 (1993): 327–343. Wilmott is perhaps the easiest place to start

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-DIFFUSION SMILE Jump-diffusion models can produce very steep short-term smiles, similar to those observed in equity index option markets. Recall that extended BSM stochastic volatility models, by contrast, have difficulty producing a very steep short-term smile unless the volatility of volatility is extremely large. The Full Jump-Diffusion Model

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impact of individual jumps on the terminal stock distribution is overwhelmed by the diffusion process whose variance grows linearly with time. Recall that mean-reverting stochastic volatility models also produce flat long-term smiles. As illustrated in Figure 24.3, a Poisson distribution of jumps superimposed on a diffusion process produces a

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Fear or Worldwide Concern?” Journal of Derivatives 13 (2): 8–21. Fouque, Jean-Pierre, George Papanicolaou, and Ronnie Sircar. 2000. Derivatives in Financial Markets with Stochastic Volatility. Cambridge, UK: Cambridge University Press. Gabaix, Xavier, Parameswaran Gopikrishnan, Vasiliki Plerou, and H. Eugene Stanley. 2003. “Understanding the Cubit and Half-Cubic Laws of Financial

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Interest Rates: A Discrete Time Approximation.” Journal of Financial and Quantitative Analysis 25:419–440. Heston, Steven. 1993. “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” Review of Financial Studies 6 (2): 327–343. Hodges, Hardy M. 1996. “Arbitrage Bounds on the Implied Volatility

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: John Wiley & Sons. Keynes, John Maynard. 1936. The General Theory of Employment, Interest and Money. London: Macmillan. Lee, Roger. 2001. “Implied and Local Volatilities under Stochastic Volatility.” International Journal of Theoretical and Applied Finance 4 (1): 45–89. Leland, Hayne E. 1985. “Option Pricing and Replication with Transaction Costs.” Journal of Finance

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40:1283–1301. Lewis, Alan. 2000. Option Valuation under Stochastic Volatility. Newport Beach, CA: Finance Press. Malz, Allan M. 1997. “Option-Implied Probability Distributions and Currency Excess Returns.” Federal Reserve Bank of New York, Staff Reports

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Page numbers followed by f refer to figures Absolute valuation, 11–12 Analytical approximation: of jump-diffusion, 410–416 of smile for geometric Brownian motion stochastic volatility with zero correlation, 363–368 of transaction costs, 123–124 Anderson, Leif, 409 Andreasen, Jesper, 409 Apple Inc., 20 Approximate static hedge, 42–44 Arbitrage

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model (CAPM), 32 INDEX CBOE (Chicago Board Options Exchange), 82 CEV (constant elasticity of variance) model, 166–167 Chain rule: hedge ratio from, 169 in stochastic volatility models, 342 Chicago Board Options Exchange (CBOE), 82 Collars, 38–40 Compensated process, 400–401 Compensation, for jumps, 387–391 Constant elasticity of variance (CEV

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–42 Geometric Brownian motion (GBM): assumed, in Black-Scholes-Merton model, 133 for interest rates, 151 method of images for, 209–211 in stochastic volatility models, 331, 332, 342 stochastic volatility with zero correlation, 362–368 stock prices not following, 3 in valuation of variance, 76 valuing down-and-out barrier option under, with

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–166 Limitations: of diversification, 32 of replication, 16–17 Linear average approximation, 261 Lo, Andrew, 13 Local variance, 279–280 Local volatility: extension of, with stochastic volatility models, 320 implied vs., 257–262, 278–286 Local volatility function, 164–165 506 Local volatility models, 164–167, 249–308 advantages of, 303–304

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model for variance of, 306–308 with positive convexity, 48f selection of proper hedge ratio for, 170 selection of volatility for hedging, 203–204 in stochastic volatility models, 321–322 in stock-only hedge portfolios, 381 when hedging with implied volatility, 101–103 when hedging with realized volatility, 94–100 Pseudo-probability

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options with, 169–171 jump-diffusion models, 168. See also Jump-diffusion models local volatility models, 164–167. See also Local volatility models stochastic volatility models, 167–168. See also Stochastic volatility models valuing exotic options with, 171–173 S&P 500, and implied distribution, 183, 184f Spread: bid-ask, 117 butterfly, see Butterfly

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, 321–325 characteristic solution to, 351–352 extending Black-Scholes-Merton model to, 344–350 extending local volatility models to, 337–344 geometric Brownian motion stochastic volatility with zero correlation, 362–368 hedge ratios in, 379 mean-reverting volatility with zero correlation, 369–375 nonzero correlation in, 375–376, 377f–378f and

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risk-neutral valuation, 332–334 survey of, 331–332 two-state stochastic path volatility, 360–362 and volatility change patterns, 317 510 Stochastic volatility models (Continued) zero correlation smile and moneyness, 353–356 zero correlation smile as symmetric, 356–360 Stock(s): with continuous known dividend yield, 240–242

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volatility sensitivity of options, 57–60 when volatility is stochastic, 74–75 Vasiçek interest rate model, 334 Vega: of European option, 58, 59f in stochastic volatility models, 343 of vanilla European options, 113 Velocity, 286 VIX volatility index, 82 VOD (Vodafone), 148, 149f Vodafone (VOD), 148, 149f Volatility: instantaneous, 364 and

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, 309–310 stickiness in the real world, 316–317 sticky delta rule, 311–314 512 Volatility change 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

Mathematical Finance: Core Theory, Problems and Statistical Algorithms

by Nikolai Dokuchaev · 24 Apr 2007

. (2004). An Introduction to Financial Option Valuation. Cambridge: Cambridge University Press. [8] Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities. Journal of Finance 42, 281–300. [9] Karatzas, I. and Shreve, S.E. (1998). Methods of Mathematical Finance. New York: SpringerVerlag. [10] Korn, R. (2001

Market Risk Analysis, Quantitative Methods in Finance

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

yield using a numerical method. When we make realistic assumptions about the evolution of the underlying price, such as that the price process has a stochastic volatility, then the only way that we can find a theoretical price of an American option is using a numerical method such as finite differences or

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algorithm, to find the maximum or minimum value of the function in the feasible domain. Other financial applications of constrained optimization include the calibration of stochastic volatility option pricing models using a least squares algorithm and the estimation of the parameters of a statistical distribution. In the majority of cases no analytic

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is generated. For pricing and hedging options it is common to simulate price paths of assets following alternative asset price diffusions, perhaps with mean reversion, stochastic volatility or jumps. Simulations are particularly useful when volatility is assumed to be stochastic.34 Simulation is a crude but sure method to obtain option prices

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Siddique, S. (2000) Conditional skewness in asset pricing tests. Journal of Finance 54, 1263–1296. Heston, S. (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6(2), 327–343. Hodges, S.D. (1997) A generalisation of the Sharpe ratio

Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues

by Alain Ruttiens · 24 Apr 2013 · 447pp · 104,258 words

further, it makes thus sense to now model t, to (try to) reduce this forecast error. There are two ways: either, by a stochastic equation (stochastic volatility model, cf. Chapter 12, Section 12.2), or, similarly as for the conditional mean, by a linear (auto)regression, that is, by an ARCH model

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more than two assets, via correlation matrixes. Currently, the main trails followed by researchers consist of looking for multivariate GARCH models11 or for a multivariate stochastic volatility model, generalizing the Heston model (cf. Section 12.2) in a matrix process of n Wiener processes, leading to a (complex) stochastic correlation model that

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. KUMAR, A.S. LESNIEWSKI, D.E. WOODWARD, “Managing smile risk”, Wilmott Magazine, July 2002, pp. 84–108. 8 See A. LEWIS, The mixing approach to stochastic volatility and jump models, Wilmott.com, March 2002. Let us also mention the dynamic model developed by A. SEPP, which involves the VIX spot, the underlying

Derivatives Markets

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

for why Volatility is not Constant, the Leverage Effect 16.8.3 Modeling Changing Volatility, the Deterministic Volatility Model 16.8.4 Modeling Changing Volatility, Stochastic Volatility Models 16.9 Why Black–Scholes is Still Important CHAPTER 17 RISK-NEUTRAL VALUATION, EMMS, THE BOPM, AND BLACK–SCHOLES 17.1 Introduction 17.1

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lemma; 33. Black–Scholes derived from Bachelier, illustrating the important connection between these two models; 34. modeling non-constant volatility: the deterministic volatility model and stochastic volatility models; 35. why Black–Scholes is still important; 36. and a final synthesis chapter that includes a discussion of the different senses of risk-neutral

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for why Volatility is not Constant, the Leverage Effect 16.8.3 Modeling Changing Volatility, the Deterministic Volatility Model 16.8.4 Modeling Changing Volatility, Stochastic Volatility Models 16.9 Why Black–Scholes is Still Important In this chapter we are going to give an introduction to continuous-time finance. This can

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in modeling varying volatility is that it generates option pricing models that are complete in the sense we described. 16.8.4 Modeling Changing Volatility, Stochastic Volatility Models Stochastic volatility (SVOL) models are beyond the scope of this text. However, a few comments may indicate the flavor of this approach. In the deterministic volatility

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Constant. 19. Economic Reasons for why Volatility is not Constant, the Leverage Effect. 20. Modeling Changing Volatility, the Deterministic Volatility Model. 21. Modeling Changing Volatility, Stochastic Volatility Models. 22. Why Black–Scholes is Still Important. ■ END OF CHAPTER EXERCISES FOR CHAPTER 16 1. In this exercise, you will calculate the historical sigma

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modeling 586–7, 587–8; deterministic volatility model 586–7; economic reasons for inconsistency of volatility 586; empirical features of volatility 585; leverage effect 586; stochastic volatility (SVOL) models 586–7 non-dealer intermediated plain vanilla swaps 281–4 non-hedgeable risks 599–601 non-replicability: contingent claims, extra risks and 600

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modeling 586–7, 587–8; deterministic volatility model 586–7; economic reasons for inconsistency of volatility 586; empirical features of volatility 585; leverage effect 586; stochastic volatility (SVOL) models 586–7; numeraire 554; definition and pricing a standard 551–3; pricing European options under shifted arithmetic Brownian motion (ABM) with no drift

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contracting 22; Heston volatility model 587–8; historical volatility estimator method 583–4; implied volatility estimator method 585; risk in risk-neutral valuation of 600; stochastic volatility (SVOL) models 586–7; volatility estimation in Black-Scholes model 583–5; see also non-constant volatility models Wall Street Journal 6, 165, 334 wealth

The Concepts and Practice of Mathematical Finance

by Mark S. Joshi · 24 Dec 2003

raised by pricing in a model that does not allow perfect hedging. We continue our study of alternative models in Chapter 16 where we introduce stochastic volatility. We develop pricing approaches using PDE and Monte Carlo techniques for vanilla and exotic options. In Chapter 17, we introduce the Variance Gamma model

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88 Practicalities a crash, thereby driving the cost of such options up. We discuss the mathematics of such models in Chapter 15. 4.5.2 Stochastic volatility A further source of uncertainty in modelling stock prices is volatility. As we mentioned above, stock price distributions tend to have fat tails. One way

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which means that assumptions on the behaviour of future smiles strongly affect the price of exotic options. Weak static replication is generally not applicable in stochastic volatility models. 10.9 Exercises Exercise 10.1 A range-accrual option pays £1 at expiry for each day the underlying has spent between two

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these properties are retained by models allowing the volatility to be a deterministic function of time and spot. However, the second property fails for fully stochastic volatility models, as future prices will depend on the stochasticallyevolved value of volatility. On the other hand, for jump-diffusion models, the first property fails

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the interested reader to [125] for one approach to implementing the method. This model is sometimes called the Dupire model, [51], or a restricted stochastic volatility model, as the volatility is a deterministic function of the stochastic stock price. An alternative interpretation is that it is a tree with nodes varying

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to realize that whilst making instantaneous volatility stochastic makes implied volatility stochastic, the relationship between the two is not straightforward. Indeed in the presence of stochastic volatility, it is necessary to redevelop the pricing formula to take account of the added uncertainty, and one then needs to plug the instantaneous volatility into

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volatility a. The Brownian motions WO and W(2) may be correlated or uncorrelated as we choose. Many of the issues which arise with stochastic-volatility models are similar to those involved with jump-diffusion models. Both models imply an incomplete market and hence an infinity of prices. Pricing formulas can

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that is, the market does not change its choice of measure. We explore all these issues in this chapter. 16.2 Risk-neutral pricing with stochastic-volatility models Given that the spot and the instantaneous variance evolve according to (16.2), what are the risk-neutral measures? Invoking the multi-dimensional version

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reverts to the level V, at rate ? . A meanreverting variance process is appealing because volatility tends to have a natural 16.3 Monte Carlo and stochastic volatility 391 level which is occasionally perturbed. In particular, a turbulent period will eventually subside, and the volatility will fall back to the background level. However

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allowed us to evolve the spot to the expiry time of a vanilla option in a single jump whilst ignoring the values in between. For stochastic volatility models, the stochastic differential equation is generally not solvable and Monte Carlo simulation is therefore much more cumbersome. Any Ito process can be simulated

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a question of computing the moments of the integral of a log-normal process. This is possible analytically. 16.4 Hedging issues One reason why stochastic-volatility models tend to be more popular than jumpdiffusion models is that they allow the illusion of hedgeability. There are two sources of uncertainty, the

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market's choice of risk-neutral measure not changing - the market must not be fickle. Thus whilst only one option is required for hedging in stochastic-volatility models, hedgeability really depends upon the assumption that the market does not change its risk-preferences, as we saw for jump-diffusion models. 16.

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correct boundary value by construction and since multiplication by f (1;) commutes with differentiation in the other variables, (16.26) is satisfied by Of. 398 Stochastic volatility We can now price any option for which we know the fundamental transform. We simply numerically invert the Fourier transform at the appropriate value of

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implied volatilities. On the other hand, increasing the stock price leads to lower volatility, and hence lower prices and implied volatilities. The marked difference between stochastic-volatility smiles and jump-diffusion smiles is in their time decay. The amount of stochasticity in the volatility increases over time and this leads to long

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to price path-dependent exotics, we may need to add in an extra 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

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with appropriate parameters. On the other hand, it is difficult to price exotic options, and the hedging is really too good to be true. Stochastic-volatility models introduces smiles by letting volatility be a stochastic quantity. Real-world volatility is mean-reverting. Any drift can be chosen for the volatility in

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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.

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uncorrelated then the spot can be long-stepped and the price of a vanilla option can be written as an integral over Black-Scholes prices. Stochastic-volatility smiles tend to be shallow relative to jump-diffusion smiles for short maturities and relatively steep for long maturities. 16.9 Further reading The

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are studied and solved. If you want to implement transform-based solutions to stochastic volatility models this is the book to buy. 400 Stochastic volatility The transform approach to stochastic-volatility pricing was started by Heston, [72]. A quite general jump-diffusion and stochastic-volatility model, which probably pushes the transform technique as far as it will go

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been developed by Duffle, Pan & Singleton, [50]. In [101], the transform technique is extended to cover a large class of models. An alternate approach to stochastic volatility models using ideas from ergodic theory has been developed by Fouque, Papanicolaou & Sircar, [55]. Their model relies on the volatility having a very fast mean

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reversion which means that the only effective state variable is spot. The book is interesting, readable and accessible. One of the first papers on stochastic volatility was by Hull & White, [75], where they developed a price in the uncorrelated case by moment-matching the density of the total variance along

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make the 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 SABR model [65]; this model involves a log-normal volatility with no drift and a correlated CEV

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of the information arriving in a given period should only depend on the length of that period. Note that this is quite different from stochastic-volatility models where an increase in volatility persists and keeps the stock more volatile until the volatility returns randomly or mean-reverts to its previous level

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in favour of the model. However, as with jump-diffusion models there is a tendency for the smile to flatten too quickly. This contrasts with stochastic volatility models, where the total amount of variation of volatility increases over time and for certain parameter sets, smiles flatten quite slowly. If we take

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18.3 Dynamics implied by models We have studied a number of alternative models in varying detail. We recall them here (i) jump-diffusion, (ii) stochastic-volatility, (iii) Variance Gamma, (iv) displaced diffusion, that is a Black-Scholes type model in which the underlying plus a constant is log-normal instead of

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the shallowest is five years. Spot is 100 and jumps are asymmetric with mean ratio equal to 0.8. 18.3.2 Stochastic-volatility smiles If we use a stochastic-volatility model with constant parameters the model is of log-type and again everything is defined relative to the current value of spot and

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is uncorrelated with spot. The reversion speed is 1 and the volatility of variance is 1. Initial volatility is 10%. The principal difference between stochastic-volatility and jump-diffusion smiles is that there is an implicit assumption that the volatility has not changed in (18.4). A big difference between jump

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smile is symmetric, unlike a jump-diffusion smile. However, skewness can be introduced using the 9 parameter. For a single fixed maturity, Variance Gamma and stochastic-volatility smiles look very similar. As with jump-diffusion models, this smile will be much sharper for small values of T - t than large ones.

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-constant. It is less clear whether it is sticky or floating. We can achieve a good match to it using an uncorrelated mean-reverting stochastic-volatility process and maintain time-homogeneity. 18.5 Hedging The pricing of exotic options is not just about finding prices that are compatible with market dynamics

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the fit should also change slightly. This is also related to uniqueness of fits. If a model has many parameters - for example a jump-diffusion stochastic-volatility model with all parameters time-dependent, then it is possible to get similar qualities of fits with vastly different parameter sets. This implies instability in

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To price this option correctly, you need a model which captures the variation in volatility well. The 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

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accurately, we will need a model that accurately reproduces random changes in skew. We could again use a model with stochastic jump-intensity or a stochastic-volatility model in which correlation between spot and volatility is stochastic. These cliquet-related products exemplify the need to capture the smile dynamics well when pricing

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not it predicts the future will be different from the present. An important criterion for selecting a model is its performance at hedging. Jump-diffusion, stochastic-volatility and Variance Gamma models predict floating smiles. Displaced-diffusion predicts a stickier smile. When pricing an exotic option we should be careful to examine what

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the performance of various option-pricing models at hedging vanilla options on the S&P is carried out. The authors find that sophisticated models, particularly stochastic-volatility models, perform better than the Black-Scholes model. Appendix A Financial and mathematical jargon Finance is full of arbitrary terms that appear to make little

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see what sort of smiles are implied and look at pricing variations for exotic options. Vanilla options Implement a pricer for vanilla options for a stochastic-volatility model with uncorrelated volatility and spot. Implement a Monte Carlo pricer also and check they give the same answers. Do the Monte Carlo pricer in

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does the shape change with maturity? Compare with jump-diffusion smiles. Exotic options Write a pricer for Asian options using a Monte Carlo implementation of stochastic volatility. With parameters as in the previous setting, spot equal to 100 and strike equal to 100, price a one-year Asian call option with

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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 14: Variance Gamma 457 Exotic options Write a pricer for Asian options using a

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D. Brigo, F. Mercurio, Interest Rate Models - Theory and Practice, Springer Verlag, 2001. [25] M. Britten-Jones, A. Neuberger, Option prices, implied price processes and stochastic volatility, Journal of Finance 55(2), April 2000, 839-66. [26] M. Broadie, P. Glasserman, Estimating security derivative prices by simulation, Management Science 42, 1996, 269

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fine structure of asset returns: an empirical investigation, Journal of Business 75(2), 2002, 305-32. [34] P. Can, H. Geman, D. Madan, M. Yor, Stochastic volatility for Levy processes, Mathematical Finance 13, 2003, 345-82. [35] T. Chan, Pricing contingent claims on stocks driven by Levy processes, Annals of Applied Probability

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of interest rates: a new methodology for contingent claims valuation, Econometrica 60, 1992, 77-105. [72] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies 6(2), 1993, 327-43. [73] S. Hodges, A. Neuberger, Rational bounds for exotic

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time-varying skewness in foreign exchange returns, Journal of Business and Economic Statistics 20(3), 2002, 390-411. [84] C.S. Jones, The dynamics of stochastic volatility, Journal of Econometrics 116(1), 2003,181-224. [85] M. Joshi, Pricing path-dependent exotic options using replication methods, QUARC Royal Bank of Scotland working

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Finance, forthcoming. References 530 [91] M. Joshi, The convergence of binomial trees for pricing the American put, preprint, 2007. [92] M. Joshi, R. Rebonato, A stochastic-volatility displaced-diffusion extension of the LIBOR market model, QUARC Royal Bank of Scotland working paper, 2001. [93] M. Joshi, J. Theis, Bounding Bermudan swaptions in

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[99] H.E. Leland, Option pricing and replication with transaction costs, Journal of Finance 40, 1985, 1283-301. [100] A.L. Lewis, Option Valuation under Stochastic Volatility, Finance Press, 2000. [101] A.L. Lewis, A simple option formula for general jump-diffusion and other exponential Levy processes, preprint www.optioncity.net, 2001

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foreign exchange, 413 FX, 424 interest-rate, 355-357, 424 jump-diffusion, 378, 415 sticky, 88, 413-414 sticky-delta, 413 538 smile (cont.) stochastic volatility, 398, 416 time dependence, 414-415 Variance Gamma, 406,417 smile dynamics Deiman-Kani, 420 displaced-diffusion, 420 Dupire model, 420 equity, 421 FX, 424

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interest-rate, 424 jump-diffusion, 415 market, 413-415 model, 415-421 stochastic volatility, 416 Variance Gamma, 417 smoothing operator, 120 spectral theory, 228 speculator, 12, 18 split share, see share split spot price, 31 square root of a

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, 389 and risk-neutral pricing, 390-393 implied, 400 pricing by Monte Carlo, 391-394 pricing by PDE and transform methods, 395-398 stochastic volatility smiles, see smile, stochastic volatility stock, 6-7, 433 stop loss hedging strategy, 18 stopping time, 143, 286, 346 straddle, 182, 257 Index lower bound via local optimization,

Mathematical Finance: Theory, Modeling, Implementation

by Christian Fries · 9 Sep 2007

prices. However, this insight is almost useless, since: • Most models are not able to reproduce arbitrarily given prices exactly. – – Extended models (e.g. models with stochastic volatility), which allow for a calibration to more than one option price per maturity, do this in an approximative way, i.e. the residual error of

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the pricing of complex derivatives (which justifies the effort), but not primarily as interpolation method to price European options. Examples of such model extensions are stochastic volatility or jump-diffusion extensions of the LIBOR Market Model, [21]. • Some models even require a continuum of European option prices K 7→ V(T, K

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]- 13th December 2006 http://www.christian-fries.de/finmath/ 17.1. LIBOR MARKET MODEL Further generalization of the model consider non deterministic σi , i.e. stochastic volatility models. In this case the terminal LIBOR distribution no longer correspond to the ones of the Black model, which is, of course, intended. Equation (17

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+1 ) , PMarket (T i+1 )(T i+1 − T i ) The parameters σi may well be stochastic processes. In this cases σi is called a stochastic volatility model. 265 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd/2.5/deed.en Comments welcome. ©2004

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design in Figure 17.4 would make sense if one would like to explore many combinations of different volatility and correlation models. C| 13 A stochastic volatility model would result in a stochastic covariance model. 277 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd

Frequently Asked Questions in Quantitative Finance

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

Paul Wilmott On Quantitative Finance by Paul Wilmott Advanced Modelling in Finance Using Excel and VBA by Mary Jackson and Mike Staunton Option Valuation under Stochastic Volatility by Alan Lewis The Concepts and Practice of Mathematical Finance by Mark Joshi C++ Design Patterns and Derivatives Pricing by Mark Joshi Heard on the

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rates via mean-reverting random walks. Higher-dimensional versions of BM can be used to represent multi-factor random walks, such as stock prices under stochastic volatility. One of the unfortunate features of BM is that it gives returns distributions with tails that are unrealistically shallow. In practice, asset returns have tails

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you go beyond basic Black-Scholes it becomes more useful. For example, suppose you want to derive the valuation partial differential equations for options under stochastic volatility. The stock price follows the real-world processes, dS = µS dt + σ S dX1 anddσ = a(S, σ , t)dt + b(S, σ , t)dWX2

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model and its ilk. References and Further Reading Joshi, M 2003 The Concepts and Practice of Mathematical Finance. CUP Lewis, A 2000 Option Valuation under Stochastic Volatility. Finance Press Neftci, S 1996 An Introduction to the Mathematics of Financial Derivatives. Academic Press What are the Greeks? Short Answer The ‘greeks’ are the

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way. Or what about a closed form involving a subtle integration in the complex plane that must ultimately be done numerically? That is the Heston stochastic volatility model. If closed form is so appreciated, is it worth spending much time seeking them out? Probably not. There are always new products being invented

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and 2. Wilmott magazine, May and July Haug, EG 2006 The complete Guide to Option Pricing Formulas. McGraw-Hill Lewis, A 2000 Option Valuation under Stochastic Volatility. Finance Press What are the Forward and Backward Equations? Short Answer Forward and backward equations usually refer to the differential equations governing the transition probability

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-diffusion models can do a good job of capturing steepness in volatility skews and smiles for short-dated option, something that other models, such as stochastic volatility, have difficulties in doing. References and Further Reading Cox, J & Ross, S 1976 Valuation of Options for Alternative Stochastic Processes. Journal of Financial Econometrics 3

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, for example, and is also often seen in fixed-income derivatives pricing. Let’s work with the stochastic volatility model to get inspiration. Suppose we have a lognormal random walk with stochastic volatility. This means we have two sources of randomness (stock and volatility) but only one quantity with which to hedge (stock). That

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-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 most popular model of

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often have several parameters which can either be chosen to fit historical data or, more commonly, chosen so that theoretical prices calibrate to the market. Stochastic volatility models are better at capturing the dynamics of traded option prices better than deterministic models. However, different markets behave differently. Part of this is because

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(February) Dupire, B 1994 Pricing with a smile. Risk magazine 7 (1) 18-20 (January) Heston, S 1993 A closed-form solution for options with stochastic volatility with application to bond and currency options. Review of Financial Studies 6 327-343 Javaheri, A 2005 Inside Volatility Arbitrage. John Wiley & Sons Lewis, A

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2000 Option valuation under Stochastic Volatility. Finance Press Lyons, TJ 1995 Uncertain Volatility and the risk-free synthesis of derivatives. Applied Mathematical Finance 2 117-133 Rubinstein, M 1994 Implied binomial

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-smile effect into an option-pricing model, and still have no arbitrage. The most popular are, in order of complexity, as follows• Deterministic volatility surface • Stochastic volatility • Jump diffusion The deterministic volatility surface is the idea that volatility is not constant, or even only a function of time, but a known function

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exactly at an instant in time, but it does a very poor job of capturing the dynamics, that is, how the data change with time. Stochastic volatility models have two sources of randomness, the stock return and the volatility. One of the parameters in these models is the correlation between the two

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the smile. As a rule one pays for convexity. We see this in the simple Black-Scholes world where we pay for gamma. In the stochastic volatility world we can look at the second derivative of option value with respect to volatility, and if it is positive we would expect to have

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using a formula that is only correct for constant volatility.) Figure 2-10: ∂2V/∂σ2 versus strike. Stochastic volatility models have greater potential for capturing dynamics, but the problem, as always, is knowing which stochastic volatility model to choose and how to find its parameters. When calibrated to market prices you will still usually

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hedge perfectly. So expect it to appear in the following situations:• When you have a stochastic model for a quantity that is not traded. Examples: stochastic volatility; interest rates (this is a subtle one, the spot rate is not traded); risk of default. • When you cannot hedge. Examples: jump models; default models

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approach to find a person’s own price for an instrument rather than the market’s. References and Further Reading Ahn, H & Wilmott, P 2003b Stochastic volatility and mean-variance analysis. Wilmott magazine November 2003 84-90 Markowitz, H 1959 Portfolio Selection: efficient diversification of investment. John Wiley & Sons Wilmott, P 2006

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quantify the relevant market price of risk. This is a way of consistently relating prices of derivatives with the same source of unhedgeable risk, a stochastic volatility for example. Both the equilibrium and no-arbitrage models suffer from problems concerning parameter stability. In the fixed-income world, examples of equilibrium models are

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more accurately. Typical approaches include the deterministic or local volatility models, in which volatility is a function of asset and time, σ(S, t), and stochastic volatility models, in which we represent volatility by another stochastic process. The latter models require a knowledge or specification of risk preferences since volatility risk cannot

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these effects also require a knowledge or specification of risk preferences. It is theoretically even harder to hedge options in these worlds than in the stochastic volatility world. To some extent the existence of other traded options with which one can statically hedge a portfolio of derivatives can reduce exposure to assumptions

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solutions can be found. Since the model requires small volatility of volatility it is best for interest rate derivatives. Fast drift and high volatility in stochastic volatility models: These are a bit more complicated, singular perturbation problems. Now the parameter is large, representing both fast reversion of volatility to its mean and

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Further Reading Hagan, P, Kumar, D, Lesniewski, A & Woodward, D 2002 Managing smile risk. Wilmott magazine, September Rasmussen, H & Wilmott, P 2002 Asymptotic analysis of stochastic volatility models. In New Directions in Mathematical Finance, Ed. Wilmott, P & Rasmussen, H, John Wiley & Sons Whalley, AE & Wilmott, P 1997 An asymptotic analysis of an

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of Payoff(S1, . . . , Sd) at time T: where Σ is the correlation matrix and there is a continuous dividend yield of Di on each asset. Stochastic volatility If the risk-neutral volatility is modelled bydσ = (p − λq) dt + q dX2, where λ is the market price of volatility risk, with the stock

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opposed to risk-neutral, data well. Asymptotic analysis If the volatility of volatility is large and the speed of mean reversion is fast in a stochastic volatility model, with a correlation ρ, then closed-form approximate solutions (asymptotic solutions) of the pricing equation can be found for simple options for arbitrary functions

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pricing and the term structure of interest rates: a new methodology. Econometrica 60 77-105 Heston, S 1993 A closed-form solution for options with stochastic volatility with application to bond and currency options. Review of Financial Studies 6 327-343 Ho, T & Lee, S 1986 Term structure movements and pricing interest

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2000 Option valuation under Stochastic Volatility. Finance Press Merton, RC 1973 Theory of rational option pricing. Bell Journal of Economics and Management Science 4 141-83 Merton, RC 1974 On the

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, RC 1976 Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3 125-44 Rasmussen, H & Wilmott, P 2002 Asymptotic analysis of 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

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in the need to analyze and develop ever more complex ‘what if’ scenarios. Option Valuation under Stochastic Volatility by Alan Lewis “This exciting book is the first one to focus on the pervasive role of stochastic volatility in option pricing. Since options exist primarily as the fundamental mechanism for trading volatility, students of the

Tools for Computational Finance

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

. The rate of return µ of S is zero; dW (1) and dW (2) may be 40 Chapter 1 Modeling Tools for Financial Options correlated. The stochastic volatility σ follows the mean volatility ζ and is simultaneously perturbed by a Wiener process. Both σ und ζ provide mutual mean reversion, and stick together

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as constant but is modeled by some SDE (such as equation (1.40)), then a system of SDEs must be integrated. An example of a stochastic volatility is provided by Example 1.15, compare Figure 3.1. In such cases the Black-Scholes equation may not help and a Monte Carlo simulation

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]. on Section 6.2: To see how the multidimensional volatilities of the model enter into a lumped volatility, consult [Shr04]. Other multidimensional PDEs arise when stochastic volatilities are modeled with SDEs, see [BaR94], [ZvFV98a], [Oo03], [HiMS04]. A list of exotic options with various payoﬀs is in Section 19.2 of [Deu01]. Also

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-228. [BaS01] I. Babuška, T. Strouboulis: The Finite Element Method and its Reliability. Oxford Science Publications, Oxford (2001). [BaR94] C.A. Ball, A. Roma: Stochastic volatility option pricing. J. Financial Quantitative Analysis 29 (1994) 589-607. [Bar97] G. Barles: Convergence of numerical schemes for degenerate parabolic equations arising in ﬁnance theory

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. Sci. Comp. 23 (2002) 20952122. [FVZ99] P.A. Forsyth, K.R. Vetzal, R. Zvan: A ﬁnite element approach to the pricing of discrete lookbacks with stochastic volatility. Applied Math. Finance 6 (1999) 87–106. [FoLLLT99] E. Fournié, J.-M. Lasry, J. Lebuchoux, P.-L. Lions, N. Touzi: An application of Malliavin calculus

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. Press, Cambridge (2004). N. Hilber, A.-M. Matache, C. Schwab: Sparse Wavelet Methods for Option Pricing under Stochastic Volatility. Report, ETH-Zürch (2004). N. Hofmann, E. Platen, M. Schweizer: Option pricing under incompleteness and stochastic volatility. Mathem. Finance 2 (1992) 153–187. P. Honoré, R. Poulsen: Option pricing with EXCEL. in [Nie02]. J

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Asian options. J. Computational Finance 1,2 (1997/98) 39–78. R. Zvan, P.A. Forsyth, K.R. Vetzal: Penalty methods for American options with stochastic volatility. J. Comp. Appl. Math. 91 (1998) 199-218. R. Zvan, P.A. Forsyth, K.R. Vetzal: Discrete Asian barrier options. J. Computational Finance 3,1