By Björn Lutz
The subject of this e-book is the advance of pricing formulae for ecu variety derivatives on resources with mean-reverting habit, in particular commodity derivatives. For this type of resources, comfort yield results result in mean-reversion below the risk-neutral degree. Mean-reversion within the log-price procedure is mixed with different stochastic components equivalent to stochastic volatility, jumps within the underlying and the fee approach and a stochastic aim point in addition to with deterministic seasonality results. one other concentration is on numerical algorithms to calculate the Fourier indispensable in addition to to combine structures of standard differential equations.
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Extra info for Pricing of Derivatives on Mean-Reverting Assets
The maximum degree for a formula with n points is 2 n 1. The quadrature rule has small approximation errors if the integrand is a continuous and differentiable function. In the case of indefinite integrals in the interval Œ0; 1/, we also have a truncation error. 72) denotes the summarized error. The weighting function is only required to be nonnegative; it can also be unity. Other weighting functions are used when the approximation works better for the product of weighting function and modified integrand than for the original integrand.
34 3 Fundamentals of Derivative Pricing num´eraire. Under this measure, all assets are quoted in terms of the underlying, and the price of the underlying is therefore 1. t; T / with time to maturity T t and a face value of 1 as num´eraire. Hence, Q2 is also called the T-forward measure. The equivalent martingale measures Q1 and Q2 are characterized via their Radon-Nikodym derivatives with respect to Q45 RT exp. XT / d Q1 D Z1 D RT dQ EQ exp. XT / d Q2 D exp. 45) 1 . 45) is just St since the underlying is expected to grow with rate r under Q and therefore, RT the Radon-Nikodym derivative is Z1 D exp.
For n D N , we have performed a full circle and obtain the same result as for n D 0. Due to this circular convolution property, we have also F n D FN n . Each element exp. 2 Ni m n / is used N times and can therefore be stored to speed up the computation. 58) to our integration problem. 57). t; T; n/ wn . 54 Note that both indices n (which determines the value of the integration parameter ) and m (which determines the strike of the call option) take N values from zero to N 1. To discretize the integral, it is truncated at some large value max , for which the function value of the integrand is satisfactorily small, say 10 6 .