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...

Standard derivative pricing theory is based on the assumption of agents acting as price takers on the market for the underlying asset. We relax this hypothesis and study if and how a large agent whose trades move prices can replicate the payoff of a derivative security. Our analysis extends prior work of Jarrow to economies with continuous security trading. We characterize the solution to the h...

We study the Black-Scholes equation in stochastic volatility models. In particular, we show that the option price is the unique classical solution to a parabolic differential equation with a certain boundary behaviour for vanishing values of the volatility. If the boundary is attainable, then this boundary behaviour serves as a boundary condition and guarantees uniqueness in appropriate functio...

‚robert merton, who’s best known for his academic work in developing option-pricing models, published his first scholarly paper on a topic far removed from finance: Gulliver’s Travels. While an undergraduate at Columbia University in New York, he wrote a piece for a literature class that analyzed the physical impossibility of a flying island as described in Jonathan Swift’s classic work. Merton...

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...

G. Linde, M. Åberg. 2004: Pricing European Call Options Using a Pseudospectral Method. Written in English. Uppsala, Sweden. The aim of the project is to price European call options with a pseudospectral method (PS). An option is a financial asset that can be resembled to a lottery coupon. In a predefined time in the future the option is either worthless or worth more than it was bought for. Bla...

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...