Ronnie Sircar

Multiscale stochastic volatility for equity, interest rate, and credit derivatives

Introduction 1. The Black-Scholes theory of derivative pricing 2. Introduction to stochastic volatility models 3. Volatility time scales 4. First order perturbation theory 5. Implied volatility formulas and calibration 6. Application to exotic derivatives 7. Application to American derivatives 8. Hedging strategies 9. Extensions 10. Around the Heston model 11. Other applications 12. Interest rate models 13. Credit risk I: structural models with stochastic volatility 14. Credit risk II: multiscale intensity-based models 15. Epilogue Bibliography Index.

Singular Perturbations in Option Pricing

After the celebrated Black--Scholes formula for pricing call options under constant volatility, the need for more general nonconstant volatility models in financial mathematics motivated numerous works during the 1980s and 1990s. In particular, a lot of attention has been paid to stochastic volatility models in which the volatility is randomly fluctuating driven by an additional Brownian motion. We have shown in [Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000; Internat. J. Theoret. Appl. Finance, 13 (2000), pp. 101--142] that, in the presence of a separation of time scales between the main observed process and the volatility driving process, asymptotic methods are very efficient in capturing the effects of random volatility in simple robust corrections to constant volatility formulas. From the point of view of PDEs, this method corresponds to a singular perturbation analysis. The aim of this paper is to deal with the nonsmoothness of the payoff...

Multiscale Stochastic Volatility Asymptotics

In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical Black--Scholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. The underlying might be the price of a stock or an index, say, and a constant volatility corresponds to a fixed standard deviation for the random fluctuations in the returns of the underlying. Modern market phenomena make it important to analyze the situation when this volatility is not fixed but rather is heterogeneous and varies with time. In previous work (see, for instance, [J. P. Fouque, G. Papanicolaou, and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000]), we considered the situation when the volatility is fast mean reverting. Using a...

Bounds and Asymptotic Approximations for Utility Prices when Volatility is Random

This paper is a contribution to the valuation of derivative securities in a stochastic volatility framework, which is a central problem in financial mathematics. The derivatives to be priced are of European type with the payoff depending on both the stock and the volatility. The valuation approach uses utility-based criteria under the assumption of exponential risk preferences. This methodology yields the indifference prices as solutions to second order quasilinear PDEs. Two sets of price bounds are derived that highlight the important ingredients of the utility approach, namely, nonlinear pricing rules with dynamic certainty equivalent characteristics, and pricing measures depending on correlation and the Sharpe ratio of the traded asset. The problem is further analyzed by asymptotic methods in the limit of the volatility being a fast mean-reverting process. The analysis relates the traditional market-selected volatility risk premium approach and the preference-based valuation techniques.

Stochastic Volatility Effects on Defaultable Bonds

This paper studies the effect of introducing stochastic volatility in the first‐passage structural approach to default risk. The impact of volatility time scales on the yield spread curve is analyzed. In particular it is shown that the presence of a short time scale in the volatility raises the yield spreads at short maturities. It is argued that combining first passage default modelling with multiscale stochastic volatility produces more realistic yield spreads. Moreover, this framework enables the use of perturbation techniques to derive explicit approximations which facilitate the complicated issue of calibration of parameters.

Optimal investment with derivative securities

Abstract.We consider an investor who maximizes expected exponential utility of terminal wealth, combining a static position in derivative securities with a traditional dynamic trading strategy in stocks. Our main result, obtained by studying the strict concavity of the utility-indifference price as a function of the static positions, is that, in a quite general incomplete arbitrage-free market, there exists a unique optimal strategy for the investor.

A Limit Theorem for Financial Markets with Inert Investors

We study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time. It turns out that semi-Markov processes are tailor made for modelling inert investors. With a suitable scaling, we show that when the price is driven by the market imbalance, the log price process is approximated by a process with long-range dependence and non-Gaussian returns distributions, driven by a fractional Brownian motion. Consequently, investor inertia may lead to arbitrage opportunities for sophisticated market participants. The mathematical contributions are a functional central limit theorem for stationary semi-Markov processes and approximation results for stochastic integrals of continuous semimartingales with respect to fractional Brownian motion.

Maturity cycles in implied volatility

Abstract.The skew effect in market implied volatility can be reproduced by option pricing theory based on stochastic volatility models for the price of the underlying asset. Here we study the performance of the calibration of the S&P 500 implied volatility surface using the asymptotic pricing theory under fast mean-reverting stochastic volatility described in [8]. The time-variation of the fitted skew-slope parameter shows a periodic behaviour that depends on the option maturity dates in the future, which are known in advance. By extending the mathematical analysis to incorporate model parameters which are time-varying, we show this behaviour can be explained in a manner consistent with a large model class for the underlying price dynamics with time-periodic volatility coefficients.

Optimal investment problems and volatility homogenization approximations

We describe some stochastic control problems in financial engineering arising from the need to find investment strategies to optimize some goal. Typically, these problems are characterized by nonlinear Hamilton-Jacobi-Bellman partial differential equations, and often they can be reduced to linear PDEs with the Legendre transform of convex duality. One situation where this cannot be achieved is in a market with stochastic volatility. In this case, we discuss an approximation using asymptotic analysis in the limit of fast mean-reversion of the process driving volatility. Simulations illustrate that marginal improvement can be achieved with this approach even when volatility is not fluctuating that rapidly.

Option Pricing Under Stochastic Volatility: The Exponential Ornstein-Uhlenbeck Model

We study the pricing problem for a European call option when the volatility of the underlying asset is random and follows the exponential Ornstein-Uhlenbeck model. The random diffusion model proposed is a two-dimensional market process that takes a log-Brownian motion to describe price dynamics and an Ornstein-Uhlenbeck subordinated process describing the randomness of the log-volatility. We derive an approximate option price that is valid when (i) the fluctuations of the volatility are larger than its normal level, (ii) the volatility presents a slow driving force toward its normal level and, finally, (iii) the market price of risk is a linear function of the log-volatility. We study the resulting European call price and its implied volatility for a range of parameters consistent with daily Dow Jones Index data.