Svetlana Boyarchenko

OPTION PRICING FOR TRUNCATED LÉVY PROCESSES

A general class of truncated Levy processes is introduced, and possible ways of fitting parameters of the constructed family of truncated Levy processes to data are discussed. For a market of a riskless bond and a stock whose log-price follows a truncated Levy process, TLP-analogs of the Black–Scholes equation, the Black–Scholes formula, the Dynkin derivative and the Lelands model are obtained, a locally risk-minimizing portfolio is constructed, and an optimal exercise price for a perpetual American put is computed.

Perpetual American Options Under Lévy Processes

We consider perpetual American options, assuming that under a chosen equivalent martingale measure the stock returns follow a Levy process. For put and call options, their analogues for more general payoffs, and a wide class of Levy processes that contains Brownian motion, normal inverse Gaussian processes, hyperbolic processes, truncated Levy processes, and their mixtures, we obtain formulas for the optimal exercise price and the fair price of the option in terms of the factors in the Wiener--Hopf factorization formula, i.e., in terms of the resolvents of the supremum and infimum processes, and derive explicit formulas for these factors. For calls, puts, and some other options, the results are valid for any Levy process. We use Dynkins formula and the Wiener--Hopf factorization to find the explicit formula for the price of the option for any candidate for the exercise boundary, and by using this explicit representation, we select the optimal solution. We show that in some cases the principle of the smooth fit fails and suggest a generalization of this principle.

American Options: The EPV Pricing Model

We explicitly solve the pricing problem for perpetual American puts and calls, and provide an efficient semi-explicit pricing procedure for options with finite time horizon. Contrary to the standard approach, which uses the price process as a primitive, we model the price process as the expected present value of a stream, which is a monotone function of a Levy process. Certain processes exhibiting mean-reverting, stochastic volatility and/or switching features can be modelled in this way. This specification allows us to consider assets that pay no dividends at all when the level of the underlying stochastic factor is too low, assets that pay dividends at a fixed rate when the underlying stochastic process remains in some range, or capped dividends.

Perpetual American options under Levy processes

We consider perpetual American options assuming, that under a chosen equivalent martingale measure the shock returns follow a Levy process. For put and call options, their analogs for more general payoffs, and a wide class, of Levy processes, which contains Brownian motion, normal inverse Gaussian processes, hyperbolic processes, truncated Levy processes and their mixtures, we obtain formulas for the optimal exercise price and the fair price of the option in terms of the factors in the Wiener-Hopf factorization formula, i.e., in terms of the resolvents of the supremum and infimum processes, and derive explicit formulas for these factors. For calls, puts and some other options, the results are valid for any Levy process.

Pricing of perpetual Bermudan options

Abstract We consider perpetual Bermudan options and more general perpetual American options in discrete time. For wide classes of processes and pay‐offs, we obtain exact analytical pricing formulae in terms of the factors in the Wiener‐Hopf factorization formulae. Under additional conditions on the process, we derive simpler approximate formulae.

American Options in Regime-Switching Models

The pricing problem for American options in Markov-modulated Levy models is solved. The early exercise boundaries and prices are calculated using a generalization of Carrs randomization procedure for regime-switching models. The pricing procedure is efficient even if the number of states is large, provided the transition rates are not large w.r.t. the riskless rates. The payoffs and riskless rates may depend on a state. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modeled as finite-state Markov chains.

American options: The EPV pricing model

SummaryWe explicitly solve the pricing problem for perpetual American puts and calls, and provide an efficient semi-explicit pricing procedure for options with finite time horizon. Contrary to the standard approach, which uses the price process as a primitive, we model the price process as the expected present value of a stream, which is a monotone function of a Lévy process. Certain processes exhibiting mean-reverting, stochastic volatility and/or switching features can be modeled this way. This specification allows us to consider assets that pay no dividends at all when the level of the underlying stochastic factor is too low, assets that pay dividends at a fixed rate when the underlying stochastic process remains in some range, or capped dividends.

Preemption Games Under Levy Uncertainty

We study a stochastic version of Fudenberg–Tiroles preemption game. Two firms contemplate entering a new market with stochastic demand. Firms differ in sunk costs of entry. If the demand process has no upward jumps, the low cost firm enters first, and the high cost firm follows. If leaders optimization problem has an interior solution, the leader enters at the optimal threshold of a monopolist; otherwise, the leader enters earlier than the monopolist. If the demand admits positive jumps, then the optimal entry threshold of the leader can be lower than the monopolists threshold even if the solution is interior; simultaneous entry can happen either as an equilibrium or a coordination failure; the high cost firm can become the leader. We characterize subgame perfect equilibrium strategies in terms of stopping times and value functions. Analytical expressions for the value functions and thresholds that define stopping times are derived.

Exit Problems in Regime-Switching Models

This paper provides a general framework for pricing of perpetual American and real options in regime-switching Levy models. In each state of the Markov chain, which determines switches from one Levy process to another one, the payoff stream is a monotone function of the Levy process labelled by the state, which allows for additional switching within each state of the Markov chain (payoffs can be different in different regions of the real line). As applications, we solve exit problems for a price-taking firm.

Efficient Variations of the Fourier Transform in Applications to Option Pricing

In this paper, we clarify the relationships among popular methods for pricing European options based on the Fourier expansion of the payoff function (iFT method) and the simlified trapezoid rule. We suggest new variations that allow us to decrease the number of terms by a factor of between five and ten (when the iFT requires several dozen terms), or even by a factor of several dozen or a hundred (when the iFT may need thousands or millions of terms). We also give efficient recommendations for an (approximately) optimal choice of parameters for each numerical scheme.