A Fourier approach to valuating derivative assets
Sammanfattning: This paper valuates two different financial contracts, the European Call and the Spread option using the Fourier transform. In the European Call case the underlying asset is modelled by the geometric Brownian motion stochastic differential equation. All necessary conditions in order for the transform to exists are examined and it turns out that the payoff needs to be scaled by an exponential factor which includes a constant a where a < 0. Later an optimization problem is defined in order to find the a which yields the best numeric integration. At the end the Fourier method is compared against the Black Scholes formula yielding a difference with 10 −15 in magnitude. In the Spread option case the underlying assets are modelled by a two-dimensional Heston model with three volatilities, one for each asset and one for how they effect each other. Here the payoff need to be scaled by two different exponential factors each including one constant, call them a and b where a < 0 and b < 0. Again an optimization problem is defined in order to find the a,b which yields the best numeric integration. The Fourier method for this case is compared against a Monte Carlo simulation with and without a control variate.
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