Explicit pathwise expansion for multivariate diffusions with applications

In this paper, we derive a generic decomposition of the option pricing formula for models with finite activity jumps in the underlying asset price process (SVJ models). This is an extension of the well-known result by Alòs [(2012) A decomposition formula for option prices in the Heston model and applications to option pricing approximation, Finance and Stochastics 16 (3), 403–422, doi: https://doi.org/10.1007/s00780-012-0177-0 ] for Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ] SV model. Moreover, explicit approximation formulas for option prices are introduced for a popular class of SVJ models — models utilizing a variance process postulated by Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ]. In particular, we inspect in detail the approximation formula for the Bates [(1996), Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options, The Review of Financial Studies 9 (1), 69–107, doi: https://doi.org/10.1093/rfs/9.1.69 ] model with log-normal jump sizes and we provide a numerical comparison with the industry standard — Fourier transform pricing methodology. For this model, we also reformulate the approximation formula in terms of implied volatilities. The main advantages of the introduced pricing approximations are twofold. Firstly, we are able to significantly improve computation efficiency (while preserving reasonable approximation errors) and secondly, the formula can provide an intuition on the volatility smile behavior under a specific SVJ model.

A quantitative analysis on the pricing of forward starting options under stochastic volatility and stochastic interest rates is performed. The main finding is that forward starting options not only depend on future smiles, but also directly on the evolution of the interest rates as well as the dependency structures among the underlying asset, the interest rates, and the stochastic volatility: compared to vanilla options, dynamic structures such as forward starting options are much more sensitive to model specifications such as volatility, interest rate, and correlation movements. We conclude that it is of crucial importance to take all these factors explicitly into account for a proper valuation and risk management of these securities. The performed analysis is facilitated by deriving closed-form formulas for the valuation of forward starting options, hereby taking the stochastic volatility, stochastic interest rates as well the dependency structure between all these processes explicitly into account. The valuation framework is derived using a probabilistic approach, enabling a fast and efficient evaluation of the option price by Fourier inverting the forward starting characteristic functions. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:103–125, 2011

A quantitative analysis on the pricing of forward starting options under stochastic volatility and stochastic interest rates is performed. The main finding is that forward starting options not only depend on future smiles, but also directly on the evolution of the interest rates as well as the dependency structures between the underlying asset, the interest rates and the stochastic volatility: compared to vanilla options, dynamic structures such as forward starting options are much more sensitive to model specifications such as volatility, interest rate and correlation movements. We conclude that it is of crucial importance to take all these factors explicitly into account for a proper valuation and risk management of these securities. The performed analysis is facilitated by deriving closed-form formulas for the valuation of a forward starting options, hereby taking the stochastic volatility, stochastic interest rates as well the dependency structure between all these processes explicitly into account. The valuation framework is derived using a probabilistic approach, enabling a fast and efficient evaluation of the option price by Fourier inverting the forward starting characteristic functions.

This article addresses the pricing problem of forward start options within the framework of a jump diffusion model, incorporating Vasicek stochastic interest rate, Heston stochastic volatility, and arbitrary jump size distributions. The study begins by applying a measure transformation to reformulate the expectation of the discounted payoff into the difference between the probabilities of two random events under new probability measures. Subsequently, the characteristic function is derived using the properties of Ito’s integral, compound Poisson processes, and the Feynman-Kac theorem. By leveraging the relationship between the characteristic function and the distribution function, the pricing formula for forward start options is established. The rationality and practicality of the proposed model are thoroughly demonstrated. Numerical experiments are conducted to analyze the accuracy and computational efficiency of the pricing formula. Additionally, sensitivity analyses are performed on key parameters to evaluate their influence on the option pricing outcomes.

We consider the problem of pricing European forward starting options in the presence of stochastic volatility. By performing a change of measure using the asset price at the time of strike determination as a numeraire, we derive a closed-form solution within Heston’s stochastic volatility framework applying distribution properties of the volatility process. In this paper we develop a new and more suitable formula for pricing forward starting options. This formula allows to cover the smile effects observed in a Black-Scholes environment, in which the extreme exposure of forward starting options to volatility changes is ignored.

Exact explicit solution of the log‐normal stochastic volatility (SV) option model has remained an open problem for two decades. In this paper, I consider the case where the risk‐neutral measure induces a martingale volatility process, and derive an exact explicit solution to this unsolved problem which is also free from any inverse transforms. A representation of the asset price shows that its distribution depends on that of two random variables, the terminal SV as well as the time average of future stochastic variances. Probabilistic methods, using the author's previous results on stochastic time changes, and a Laplace–Girsanov Transform technique are applied to produce exact explicit probability distributions and option price formula. The formulae reveal interesting interplay of forces between the two random variables through the correlation coefficient. When the correlation is set to zero, the first random variable is eliminated and the option formula gives the exact formula for the limit of the Taylor series in Hull and White's (1987) approximation. The SV futures option model, comparative statics, price comparisons, the Greeks and practical and empirical implementation and evaluation results are also presented. A PC application was developed to fit the SV models to current market prices, and calculate other option prices, and their Greeks and implied volatilities (IVs) based on the results of this paper. This paper also provides a solution to the option implied volatility problem, as the empirical studies show that, the SV model can reproduce market prices, better than Black–Scholes and Black‐76 by up to 2918%, and its IV curve can reproduce that of market prices very closely, by up to within its 0.37%.

In this paper, we propose two new representation formulas for the conditional marginal probability density of the multi-factor Heston model. The two formulas express the marginal density as a convolution with suitable Gaussian kernels whose variances are related to the conditional moments of price returns. Via asymptotic expansion of the non-Gaussian function in the convolutions, we derive explicit formulas for European-style option prices and implied volatility. The European option prices can be expressed as Black–Scholes style terms plus corrections at higher orders in the volatilities of volatilities, given by the Black–Scholes Greeks. The explicit formula for the implied volatility clearly identifies the effect of the higher moments of the price on the implied volatility surface. Further, we derive the relationship between the VIX index and the variances of the two Gaussian kernels. As a byproduct, we provide an explanation of the bias between the VIX and the volatility of total returns, which offers support to recently proposed methods to compute the variance risk premium. Via a series of numerical exercises, we analyse the accuracy of our pricing formula under different parameter settings for the one- and two-factor models applied to index options on the S&P500 and show that our approximation substantially reduces the computational time compared to that of alternative closed-form solution methods. In addition, we propose a simple approach to calibrate the parameters of the multi-factor Heston model based on the VIX index, and we apply the approach to the double and triple Heston models.

We propose a new class of models for pricing forward starting options. We assume that the asset price is a nonlinear function of a CIR process, time changed by a composition of a Levy subordinator and an absolutely ´ continuous process. The new models introduce the nonlinearity in both drift and diffusion components of the underlying process and can capture jumps and stochastic volatility in a flexible way. By employing the spectral expansion technique, we are able to derive the analytical formulas for the forward starting option prices. We also implement a specific model numerically and test its sensitivity to some of the key parameters of the model.

Vanilla options become effective immediately after they are entered, while some exotic options will only come to effective some time after they are bought or sold. Forward starting options are one kind of such exotic options started actually at some pre-specified future date. This paper presents an extension of regime-switching jump diffusion model, in which the parameters are driven by a continuous time and stationary Markov chain on a finite state space, by introducing the Wishart process into the instantaneous variance-covariance matrix of the risky asset price and stochastic interest rate. We derive the discounted conditional joint characteristic function and the forward characteristic function of the log-asset price and its the instantaneous variance-covariance process, and thereby the price of forward starting options are well evaluated by the probabilistic approach combined with the Fourier-cosine (COS) method. We also provide efficient Monte Carlo simulation of this proposed model, and simulated solutions to forward starting options pricing within a two-state regime switching framework. Numerical results show that the COS method is accurate and efficient for pricing forward starting options. Finally, we analyse impacts of some main parameters (especially, parameters in the Wishart stochastic volatility) in this proposed model on option prices and Δ values. Also, we consider the forward implied volatility. Furthermore, the forward starting options under the regime-switching jump diffusion model with Wishart stochastic volatility and stochastic interest rate which we derived are more generalized than those recently appeared in the derivatives pricing literature, and thus have wider application.

A closed-form pricing formula for forward start options under a regime-switching stochastic volatility model

Barrier option pricing under the 2-hypergeometric stochastic volatility model

This paper uses an extension of the equilibrium model of Lucas (1978) to study the valuation of options on the market portfolio with return predictability, endogenous stochastic volatility and interest rates. Equilibrium conditions imply that the mean-reverting of the rate of dividend growth induces the predictable feature of the market portfolio. Although the actual drift of the price for the market portfolio does not explicitly enter into the option price formula when the equivalent martingale pricing principle is used, parameters underlying the predictable feature affect option prices through their influence on endogenized volatility and interest rates. Equilibrium conditions also review that there is strong interdependence between the equilibrium price process for the market portfolio and its volatility process, both of which are induced by the process for aggregate dividend. Closed-form pricing formulas for options on the market portfolio incorporate both stochastic volatility and stochastic interest rates. With realistic parameter values, numerical examples show that stochastic volatility and stochastic interest rates are both necessary for correcting the pricing biases generated by the Black-Scholes model. In addition, Closed-form solutions for European bond option prices are obtained, which encompass the Vasicek (1977) model and the Cox-Ingersoll-Ross (1985) model. In this sense, the current model provides a consistent way to price options written on the market portfolio and the bonds.

Approximations to transition densities are needed for estimation of most multivariate diffusion models used in finance. I test a variety of proxies within Markov chain Monte Carlo estimation of a multivariate term structure model with nonlinear dynamics. Realistic parameters for the simulation experiments are obtained from estimation on monthly US interest rate data. The resulting processes are close to being explosive under the pricing probability measure. A simple Gauss proxy performs surprisingly well. Closed-form likelihood expansions exhibit the best results with respect to parameter estimates and computational speed. A full Bayesian approach together with importance sampling promises to be very well suited for larger-dimensional problems.

This paper develops a complex Fourier series (CFS) expansion pricing formula for evaluating European-type (call/put) options -- asset-or-nothing options, cash-or-nothing options, forward-start options, ratchet options and vanilla options -- when these risky assets are driven by Levy processes, time-changed Levy processes or stochastic volatility with or without jumps. This novel pricing formula is an extension of the original CFS expansion method proposed in Chan (2016). The major contribution of this paper is to derive a CFS pricing formula for both forward-start and ratchet options. In this paper, we also conduct an error-bound analysis of the CFS expansion pricing formula. In most cases, we can obtain an exponential convergence rate when we have an appropriate definite integration range for pricing European-type options. Moreover, based on the pricing formula, we can also derive different option Greeks. Finally, in follow-up papers, we will present the application of this pricing formula to options with early-exercise features (such as American options and barrier options) and path-dependent features (such as Asian options).

Recent studies have shown that stochastic volatility in a continuous-time framework provides an excellent fit for financial asset returns when combined with finite-activity Merton's type compound Poisson Jump-diffusion models. However, we demonstrate that stochastic volatility does not play a central role when incorporated with infinite-activity Levy type pure jump models such as variance-gamma and normal inverse Gaussian processes to model high and low frequency historical time-series SP500 index returns. In addition, whether sources of stochastic volatility are diffusions or jumps are not relevant to improve the overall empirical fits of returns. Nevertheless, stochastic diffusion volatility with infinite-activity Levy jumps processes considerably reduces SP500 index call option in-sample and out-of-sample pricing errors of long-term ATM and OTM options, which contributed to a substantial improvement of pricing performances of SVJ and EVGSV models, compared to constant volatility Levy-type pure jumps models and/or stochastic volatility model without jumps. Interestingly, unlike asset returns, whether pure Levy jumps specifications are finite or infinite activity is not an important factor to enhance option pricing model performances once stochastic volatility is incorporated. Option prices are computed via improved Fast Fourier Transform algorithm using characteristic functions to match arbitrary log-strike grids with equal intervals with each moneyness and maturity of actual market option prices considered in this paper.