White Papers
COMPUTING GREEKS: MALLIAVIN CALCULUS VS THE FINITE DIFFERENCE SCHEME
A derivative is a financial contract whose value is derived from an underlying security; a stock or bond, for example. In this paper we discuss two fundamental mathematical problems with derivatives: pricing and hedging.
While the price of publicly traded derivatives is set by the laws of supply and demand, many derivatives are private contracts in which both parties would like to be assured that a 'fair' price has been reached. The determination of this fair price is known as option pricing.
Hedging is a strategy that financial institutions put in place, to optimize risk return ratios. The risk is assumed by the financial institution, which would now like to take a position in the underlying security (and perhaps in other instruments too), to minimize its own exposure.
Another issue of hedging relates to portfolio management. As a protection against price fluctuations in the portfolio's components, a number of options must be incorporated. The question is how many options should be held? In an arbitrage-free, complete market model, definitive solutions exist to address both hedging and pricing issues. Arbitrage-free pricing theory is related to risk-neutral valuation, which itself involves expectations of the options' discounted payoff. Since it is not always possible to use analytical solutions, numerical methods are required. In this case, one of the following approaches is employed: binomial trees; partial differential equation resolution; or Monte Carlo methods.
With the first two approaches, the computation increases exponentially in line with the scale of the problem, whereas the Monte Carlo approach involves a linear increase. This is significant when considering multi-asset options such as basket options.
This paper considers a basket of assets, with the Monte Carlo method applied to demonstrate its slow convergence (the standard error of a Monte Carlo analysis decreases with the square root of the sample size, i.e. the number of simulations); and to show the problems of tracking errors in the estimates. To determine the hedging parameters, the 'finite difference scheme' is used; but this can pose problems for discontinuous payoffs. The Monte Carlo method is compared with Malliavin2 calculus; the latter enables explicit probabilistic formulae to be written for Greeks. These provide weighted expectations of the original discounted payoff.
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Computing Greeks: Malliavin Calculus vs The Finite Difference Scheme
