A Simple Option Formula for General Jump-Diffusion and other Exponential Levy Processes

Option values are well-known to be the integral of a discounted transition density times a payoff function; this is just martingale pricing. It's usually done in 'S-space', where S is the terminal security price. But, for Levy processes the S-space transition densities are often very complicated, involving many special functions and infinite summations. Instead, we show that it's much easier to compute the option value as an integral in Fourier space - and interpret this as a Parseval identity. The formula is especially simple because (i) it's a single integration for any payoff and (ii) the integrand is typically a compact expressions with just elementary functions. Our approach clarifies and generalizes previous work using characteristic functions and Fourier inversions. For example, we show how the residue calculus leads to several variation formulas, such as a well-known, but less numerically efficient, 'Black-Scholes style' formula for call options. The result applies to any European-style, simple or exotic option (without path-dependence) under any Lévy process with a known characteristic functionoption pricing, jump-diffusion, Levy processes, Fourier, characteristic function, transforms, residue, call options, discontinuous, jump processes, analytic characteristic, Levy-Khintchine, infinitely divisible, independent increments

OPTION PRICING UNDER LÉVY PROCESSES: A UNIFYING FORMULA

A new option pricing formula is presented that unifies several results of the existing literature on pricing exotic options under Lèvy processes. To demonstrate the flexibility of the formula a few examples are given which provide new valuation formulas within the Lévy frameworkLévy processes, pseudo differential operators, option pricing

Pricing Discretely Monitored Asian Options by Maturity Randomization

We present methodologies to price discretely monitored Asian options when the underlying evolves according to a generic Levy process. For geometric Asian options we provide closed-form solutions in terms of the Fourier transform and we study in particular these formulas in the Levy-stable case. For arithmetic Asian options we solve the valuation problem by recursive integration and derive a recursive theoretical formula for the moments to check the accuracy of the results. We compare the implementation of our method to Monte Carlo simulation implemented with control variates and using different parametric Levy processes. We also discuss model-risk issues

Pricing discretely monitored Asian options under Levy processes

We present methodologies to price discretely monitored Asian options when the underlying evolves according to a generic Lévy process. For geometric Asian options we provide closed-form solutions in terms of the Fourier transform and we study in particular these formulas in the Lévy-stable case. For arithmetic Asian options we solve the valuation problem by recursive integration and derive a recursive theoretical formula for the moments to check the accuracy of the results. We compare the implementation of our method to Monte Carlo simulation implemented with control variates and using different parametric Lévy processes. We also discuss model risk issues

On Duality Principle in Exponentially Lévy Market

This paper describes the effect of duality principle in option pricing driven by exponentially Lévy market model. This model is basically incomplete - that is; perfect replications or hedging strategies do not exist for all relevant contingent claims and we use the duality principle to show the coincidence of the associated underlying asset price process with its corresponding dual process. The condition for the ‘unboundedness’ of the underlying asset price process and that of its dual is also established. The results are not only important in Financial Engineering but also from mathematical point of view

Additive logistic processes in option pricing

In option pricing, it is customary to first specify a stochastic underlying model and then extract valuation equations from it. However, it is possible to reverse this paradigm: starting from an arbitrage-free option valuation formula, one could derive a family of risk-neutral probabilities and a corresponding risk-neutral underlying asset process. In this paper, we start from two simple arbitrage-free valuation equations, inspired by the log-sum-exponential function and an ℓp vector norm. Such expressions lead respectively to logistic and Dagum (or “log-skew-logistic”) risk-neutral distributions for the underlying security price. We proceed to exhibit supporting martingale processes of additive type for underlying securities having as time marginals two such distributions. By construction, these processes produce closed-form valuation equations which are even simpler than those of the Bachelier and Samuelson–Black–Scholes models. Additive logistic processes provide parsimonious and simple option pricing models capturing various important stylised facts at the minimum price of a single market observable input