In this thesis we study static hedging with options, an approach to market risk management in which the value of a given option is replicated by a portfolio of shorter-maturity options without requiring any portfolio rebalancing during the hedging period. We begin by introducing European options, their valuation, and the basics of option hedging. We then present the theoretical background: the Black-Scholes model for option pricing, the Markov property of asset prices, and the Greek parameters delta and gamma.
The central result is a theorem showing that the value of a European call option with maturity $T$ can be expressed as an integral of a weighted combination of European call options with shorter maturity $u < T$, where the weights are determined by the gamma of the target option evaluated at time $u$. Since the weights are independent of time and the current asset price, no portfolio rebalancing is required, which distinguishes static hedging from classical dynamic delta hedging. We also show that the same relation holds for European put options.
Since a portfolio consisting of infinitely many options is not practically feasible, the integral is approximated using the Gauss-Hermite quadrature formula, which automatically determines optimal strike prices and weights for a finite number of options. In a numerical example with four options, we achieve a relative approximation error of less than 0.06 %, confirming the efficiency of the method.
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