The double barrier option is characterized by pay-off with strike K, maturity T, upper Su and lower Sd barrier levels and the corresponding rebates φu and φd which can be time dependent. We divide last four quantities by strike K and introduce new variables x = ln(S/K), xu = ln(Su/K), xd = ln(Sd/K). The value of European double barrier call option U(t, x) satisfies the extended Black-Scholes eq...

This article examines the valuation problem for the European option under a general stochastic volatility in a certain approximate sense by adopting the small disturbance asymptotic theory developed by Kunitomo and Takahashi [25, 261. The option value can be decomposed into the Black and Scholes value under deterministic volatility and adjustment terms driven by the randomness of the volatility...

A master equation approach to the numerical solution of option pricing models is developed. The basic idea of the approach is to consider the Black–Scholes equation as the macroscopic equation of an underlying mesoscopic stochastic option price variable. The dynamics of the latter is constructed and formulated in terms of a master equation. The numerical efficiency of the approach is demonstrat...

We discuss two numerical methods, based on a path integral approach described in a previous paper (I), for solving the stochastic equations underlying the financial markets: the Monte Carlo approach, and the Green function deterministic numerical method. Then, we apply the latter to some specific financial problems. In particular, we consider the pricing of a European option, a zero-coupon bond...

In this paper, we modeled an artificial European option market under unknown volatility with liquidity costs using an agent-based modeling and simulation approach. The option price in the presence of liquidity costs is given by solving a partial differential equation. We proved that both unknown volatility and the unknown drift have significant effects in the pricing bias. Moreover, pricing bia...

To provide one motivation for the development of ARCH models (next handout), we briefly discuss here some difficulties associated with the Black Scholes formula, which is widely used to calculate the price of an option. For example, consider a European call option for a stock. This is the right to buy a specific number of shares of a specific stock on a specific date in the future, at a specifi...

An installment option is a European option in which the premium, instead of being paid up-front, is paid in a series of installments. If all installments are paid the holder receives the exercise value, but the holder has the right to terminate payments on any payment date, in which case the option lapses with no further payments on either side. We discuss pricing and risk management for these ...

In this paper we consider the valuation of an option with time to expiration T and pay-oo function g which is a convex function (as is a European call option), and constant interest rate r, in the case where the underlying model for stock prices (S t) is a purely discontinuous process (hence typically the model is incomplete). The main result is that, for \most" such models, the range of the va...

In this note we value and hedge European options on assets paying discrete stochastic dividends. Our hedging strategy is based only on the underlying asset, risk-free bonds and dividend strips. Our simple strategy is easily seen to be compatible with early results based, among other things, on the existence of a dividend forward contract. Such contracts, however, are not traded in the market pl...

An Asian option is a path-depending exotic option, which means that either the settlement price or the strike of the option is formed by some aggregation of underlying asset prices during the option lifetime. This thesis will focus on European style Arithmetic Asian options where the settlement price at maturity is formed by the arithmetic average price of the last seven days of the underlying ...