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Wilmott mag article on Crank Nicholson

Wilmott mag article on Crank Nicholson - Efficient...

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2 Wilmott magazine Christian Wallner Ostpreußenstraße 6A 21391 Reppenstedt GERMANY [email protected] Uwe Wystup HfB—Business School of Finance and Management Sonnemannstrasse 9–11 60314 Frankfurt am Main GERMANY http://www.mathfinance.de Efficient Computation of Option Price Sensitivities for Options of American Style an example. For contract parameters maturity in years T , strike K and put/call indicator φ , which is + 1 for a call and 1 for a put, the payoff of the option is [ φ( S T K ) ] + = max[0 , φ( S T K ) ] . (2) We denote by V ( t , x ) the value of an American style put or call at time t if the spot S t takes the value x . It is well known (see e.g. Karatzas and Shreve 1998) that in this model the value at time zero is given by V ( 0 , S 0 ) = sup τ T IE [ e r d τ [ φ( S τ K ) ] + ] , (3) 1. Introduction We examine which is a suitable method to compute Greeks for American style call and put options in the Black-Scholes model. We choose an exchange rate for the underlying following a geometric Brownian motion, dS t = S t [ ( r d r f ) dt + σ dW t ] , (1) under the risk-neutral measure. As usual r d denotes the domestic interest rate, r f the foreign interest rate, σ the volatility. The analysis we do is also applicable to equity options, but we take the foreign exchange market as Abstract: No front-office software can survive without providing derivatives of option prices with respect to underlying market or model parameters, the so called Greeks. If a closed form solution for an option exists, Greeks can be computed analytically and they are numerically stable. However, for American style options, there is no closed-form solution. The price is computed by binomial trees, finite difference methods or an analytic approximation. Taking derivatives of these prices leads to instable numerics or mis- leading results, specially for Greeks of higher order. We compare the computation of the Greeks in various pricing methods and conclude with the recommendation to use Leisen-Reimer trees . Keywords: American Options, Greeks, Leisen-Reimer trees. JEL classification: C63, F31
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Wilmott magazine 3 where T is the set of all stopping times taking values in [0 , T ] . A closed- form solution for this optimization problem has not yet been found. 1.1 Option Price Sensitivities Option price sensitivities, the so-called Greeks of option values are deriva- tives with respect to market variables or model parameters. The most commonly used Greeks are listed in Table 1. Numerous relationships and properties of the Greeks for European style options are presented in Reiss and Wystup (2001). Other relevant publications include the work by Carr (2001), Broadie and Glasserman (1996) in the case of Monte Carlo simula- tions, Pelsser and Vorst (1994) in the case of binomial trees, the work by Eric Benhamou (2003) and (2004), who uses Malliavin calculus, and the contribution by Rogers and Stapleton (1998) using binomial trees with a random number of steps. Joubert and Rogers (1997) use a lookup table for a fast, accurate and inelegant valuation of American options. Formulae
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