In this paper we present an adaptive technique to solve the multidimensional Black-Scholes equation. The number of grid-points required for a given tolerance of the local discretization errors is reduced substantially when compared to a standard equidistant grid. Using our adaptive methods in space and time we have control of the local discretization errors and can refine the grid where needed ...

We construct a discrete time self-financing portfolio comprised of call options short and stock shares long which is riskless and grows at a fixed rate of return. It is also shown that when shorting periods tend to zero then so devised portfolio turns into the Black-Scholes bond replication. Unlike in standard approach the analysis presented here requires neither Ito Calculus nor solving the He...

We generalize the classical binomial approach of the model of Black and Scholes to a Markov binomial approach. This leads to a new formula for the cost of an option.

In the Black-Scholes model, consider the problem of selecting a change of drift which minimizes the variance of Monte Carlo estimators for prices of path-dependent options. Employing Large Deviations techniques, the asymptotically optimal change of drift is identified as the solution to a one-dimensional variational problem, which may be reduced to the associated Euler-Lagrange differential equ...

This paper deals with the efficient valuation of American options. We adopt Heston’s approach for a model of stochastic volatility and derive a generalized Black Scholes equation. This leads to a parabolic boundary value problem with a free boundary, the optimal exercise price of the option. For its efficient numerical solution, we employ, among other multiscale methods, a monotone multigrid me...

We study the applicability of meshfree approximation schemes for the solution of multi-asset American option problems. In particular, we consider a penalty method which allows us to remove the free and moving boundary by adding a small and continuous penalty term to the Black-Scholes equation. A comparison with results obtained recently by two of the authors using a linearly implicit finite dif...

In the present paperwe provide an analytical solution for pricing discrete barrier options in the Black-Scholes framework. We reduce the valuation problem to a Wiener-Hopf equation that can be solved analytically. We are able to give explicit expressions for the Greeks of the contract. The results from our formulae are compared with those from other numerical methods available in the literature...

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...