Sequential Monte Carlo Methods for Option Pricing

In the following paper we provide a review and development of sequential Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte Carlo-based algorithms, that are designed to approximate expectations w.r.t a sequence of related probability measures. These approaches have been used, successfully, for a wide class of applications in engineering, statistics, physics and operations research. SMC methods are highly suited to many option pricing problems and sensitivity/Greek calculations due to the nature of the sequential simulation. However, it is seldom the case that such ideas are explicitly used in the option pricing literature. This article provides an up-to date review of SMC methods, which are appropriate for option pricing. In addition, it is illustrated how a number of existing approaches for option pricing can be enhanced via SMC. Specifically, when pricing the arithmetic Asian option w.r.t a complex stochastic volatility model, it is shown that SMC methods provide additional strategies to improve estimation.Comment: 37 Pages, 2 Figure

Approximate option pricing

As increasingly large volumes of sophisticated options (called derivative securities) are traded in world financial markets, determining a fair price for these options has become an important and difficult computational problem. Many valuation codes use the binomial pricing model, in which the stock price is driven by a random walk. In this model, the value of an n-period option on a stock is the expected time-discounted value of the future cash flow on an n-period stock price path. Path-dependent options are particularly difficult to value since the future cash flow depends on the entire stock price path rather than on just the final stock price. Currently such options are approximately priced by Monte carlo methods with error bounds that hold only with high probability and which are reduced by increasing the number of simulation runs. In this paper the authors show that pricing an arbitrary path-dependent option is {number_sign}-P hard. They show that certain types f path-dependent options can be valued exactly in polynomial time. Asian options are path-dependent options that are particularly hard to price, and for these they design deterministic polynomial-time approximate algorithms. They show that the value of a perpetual American put option (which can be computed in constant time) is in many cases a good approximation to the value of an otherwise identical n-period American put option. In contrast to Monte Carlo methods, the algorithms have guaranteed error bounds that are polynormally small (and in some cases exponentially small) in the maturity n. For the error analysis they derive large-deviation results for random walks that may be of independent interest

A Non-Gaussian Option Pricing Model with Skew

Closed form option pricing formulae explaining skew and smile are obtained within a parsimonious non-Gaussian framework. We extend the non-Gaussian option pricing model of L. Borland (Quantitative Finance, {\bf 2}, 415-431, 2002) to include volatility-stock correlations consistent with the leverage effect. A generalized Black-Scholes partial differential equation for this model is obtained, together with closed-form approximate solutions for the fair price of a European call option. In certain limits, the standard Black-Scholes model is recovered, as is the Constant Elasticity of Variance (CEV) model of Cox and Ross. Alternative methods of solution to that model are thereby also discussed. The model parameters are partially fit from empirical observations of the distribution of the underlying. The option pricing model then predicts European call prices which fit well to empirical market data over several maturities.Comment: 37 pages, 11 ps figures, minor changes, typos corrected, to appear in Quantitative Financ

Pricing Currency Options with a Market Model of Interest Rates under Jump-Diffusion Stochastic Volatility Processes of Spot Exchange Rates

This paper proposes a pricing method of currency options with a market model of interest rates. Using a simple approximation and a Fourier transform method, we derive a formula of the option pricing under jump-diffusion stochastic volatility processes of spot exchange rates. As an application, we apply the formula to the calibration of volatility smiles in the JPY/USD currency option market. Moreover, using the approximate prices as a control variate, we achieve substantial variance reduction in Monte Carlo simulation.

"Pricing Currency Options with a Market Model of Interest Rates under Jump-Diffusion Stochastic Volatility Processes of Spot Exchange Rates"

This paper proposes a pricing method of currency options with a market model of interest rates. Using a simple approximation and a Fourier transform method, we derive a formula of the option pricing under jump-diffusion stochastic volatility processes of spot exchange rates. As an application, we apply the formula to the calibration of volatility smiles in the JPY/USD currency option market. Moreover, using the approximate prices as a control variate, we achieve substantial variance reduction in Monte Carlo simulation.