John G. O'Hara

FFT based option pricing under a mean reverting process with stochastic volatility and jumps

Numerous studies present strong empirical evidence that certain financial assets may exhibit mean reversion, stochastic volatility or jumps. This paper explores the valuation of European options when the underlying asset follows a mean reverting log-normal process with stochastic volatility and jumps. A closed form representation of the characteristic function of the process is derived for the computation of European option prices via the fast Fourier transform.

SOLVING THE ASIAN OPTION PDE USING LIE SYMMETRY METHODS

Asian options incorporate the average stock price in the terminal payoff. Examination of the Asian option partial differential equation (PDE) has resulted in many equations of reduced order that in general can be mapped into each other, although this is not always shown. In the literature these reductions and mappings are typically acquired via inspection or ad hoc methods. In this paper, we evaluate the classical Lie point symmetries of the Asian option PDE. We subsequently use these symmetries with Lies systematic and algorithmic methods to show that one can obtain the same aforementioned results. In fact we find a familiar analytical solution in terms of a Laplace transform. Thus, when coupled with their methodic virtues, the Lie techniques reduce the amount of intuition usually required when working with differential equations in finance.

Symmetry analysis of a model of stochastic volatility with time-dependent parameters

We provide the solutions for the Heston model of stochastic volatility when the parameters of the model are constant and when they are functions of time. In the former case, the solution follows immediately from the determination of the Lie point symmetries of the governing 1+1 evolution partial differential equation. This is not the situation in the latter case, but we are able to infer the essential structure of the required nonlocal symmetry from that of the autonomous problem and hence can present the solution to the nonautonomous problem. As in the case of the standard Black-Scholes problem the presence of time-dependent parameters is not a hindrance to the demonstration of a solution.

Pricing Extendible Options Using the Fast Fourier Transform

This paper applies the fast Fourier transform (FFT) approach, within the Black-Scholes framework, to the valuation of options whose time to maturity can be extended to a future date (extendible options). We determine the valuation of the extendible options as sums of expectations of indicator functions, leading to a semianalytic expression for the value of the options over a range of strikes. Compared to Monte Carlo simulation, numerical examples demonstrate that the FFT is both computationally more efficient and higher in accuracy.

Efficient pricing of discrete arithmetic Asian options under mean reversion and jumps based on Fourier-cosine expansions

We propose an efficient pricing method for arithmetic Asian options based on Fourier-cosine expansions. In particular, we allow for mean reversion and jumps in the underlying price dynamics. There is an extensive body of empirical evidence in the current literature that points to the existence and prominence of such anomalies in the prices of certain asset classes, such as commodities. Our efficient pricing method is derived for the discretely monitored versions of the European-style arithmetic Asian options. The analytical solutions obtained from our Fourier-cosine expansions are compared to the benchmark fast Fourier transform based pricing for the examination of its accuracy and computational efficiency.

Power option pricing via Fast Fourier Transform

The basis of the option universe has been the European option, and much literature has been devoted to the extension of this option to create many new exotic options, including some with nonlinear payoffs. In this work, we study a European-style power option pricing, under a constant volatility dynamics, using the risk-neutral valuation within the Black-Scholes framework. Apart from applying the closed-form solution, we price the power option using the Fast Fourier Transform (FFT) technique which requires an analytical characteristic function of the power option. The resulting approximations are then compared with other numerical methods such as the Monte Carlo simulations, which show promising results and demonstrate the efficiency of the FFT technique as it can compute option prices for a whole range of strike prices. Besides, we show that there exists a relationship between the power call option and the power put option that is similar to the put-call parity relationship of vanilla options. We also find a transformation between the underlying asset and the power contract which enables us to obtain the pricing formulas of the power options from the vanilla options, as well as simplify the Greeks for power options. In addition to the Greeks derived from the closed-form solution, we present the Greeks using the pricing formula obtained from a characteristic function.

An Analytic Formula for the Price of American Exchange Options

In this article we demononstrate how to find an analytic solution to the pricing of American exchange options using homotogy methods. As an aside we derive formula for American call and put options when the underlying pays a continuous dividend. These solutions are given in terms of power series.

Iterative Approaches to Convex Minimization Problems

Abstract The aim of this paper is to generalize the results of Yamada et al. [Yamada, I., Ogura, N., Yamashita, Y., Sakaniwa, K. (1998). Quadratic approximation of fixed points of nonexpansive mappings in Hilbert spaces. Numer. Funct. Anal. Optimiz. 19(1):165–190], and to provide complementary results to those of Deutsch and Yamada [Deutsch, F., Yamada, I. (1998). Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19(1&2):33–56] in which they consider the minimization of some function θ over a closed convex set F, the nonempty intersection of N fixed point sets. We start by considering a quadratic function θ and providing a relaxation of conditions of Theorem 1 of Yamada et al. (1998) to obtain a sequence of fixed points of certain contraction maps, converging to the unique minimizer of θover F. We then extend Theorem 2 and obtain a complementary result to Theorem 3 of Yamada et al. (1998) by replacing the condition on the parameters by the more general condition lim n→∞λ n /λ n+N = 1. We next look at minimizing a more general function θ than a quadratic function which was proposed by Deutsch and Yamada (1998) and show that the sequence of fixed points of certain maps converge to the unique minimizer of θ over F. Finally, we prove a complementary result to that of Deutsch and Yamada (1998) by using the alternate condition on the parameters.

Highly Efficient Option Valuation Under the Double Jump Framework with Stochastic Volatility and Jump Intensity Based on Shannon Wavelet Inverse Fourier Technique

In this paper, we explore the highly efficient valuation of financial options under a double exponential jump framework, with stochastic volatility and jump intensity. In particular, we investigate both the accuracy and efficiency of pricing options using the novel Shannon wavelet inverse Fourier technique (SWIFT). Resulting prices are compared to the benchmark Fast Fourier Transform (FFT) and, its more recent alternative, the Fourier Cosine (COS) expansion prices. We demonstrate that not only is the SWIFT method more efficient, it is also accurate with exponential error convergence for both call and put valuations. Finally, further evidence of model robustness and stability is presented through a price sensitivity analysis, where we investigate the significant impact of changing model parameters to the resulting option values.

A path-independent approach to integrated variance under the CEV model

In this paper, a closed form path-independent approximation of the fair variance strike for a variance swap under the constant elasticity of variance (CEV) model is obtained by applying the small disturbance asymptotic expansion. The realized variance is sampled continuously in a risk-neutral market environment. With the application of a Brownian bridge, we derive a theorem for the conditionally expected product of a Brownian motion at two different times for arbitrary powers. This theorem enables us to provide a conditional Monte-Carlo scheme for simulating the fair variance strike. Compared with results in the recent literature, the method outlined in our paper leads to a simplified approach for pricing variance swaps. The method may also be applied to other more sophisticated volatility derivatives. An empirical comparison of this model with the Heston model and a conditional Monte Carlo scheme is also presented using option data on the S&P 500.