By Andreas J. Grau
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Extra resources for Applications of Least-Squares Regressions to Pricing and Hedging of Financial Derivatives
For an American put option with strike K V (StT , tT ) = max(K − StT , 0) . Numerical schemes for this kind of PDE solution have been presented by several authors. An efficient solution has been presented by Forsyth and Vetzal , which will also form the foundation for the PDE reference methods used in this thesis. The next section proceeds with an overview of the numerical methods currently used in the field of derivatives pricing. 1 Overview The challenge that remains after the introduction of the Black-Scholes framework is the efficient valuation of arbitrary derivatives.
20)) as V (St , t) 1 ∂ Vt ∂Vt ∂Vt + σ 2 St2 − rVt + rSt ∂t 2 ∂St2 ∂St ≥ P (St , t), 2 ≤ 0, where at least one of the inequalities holds with equality on the complete solution. 38)): Ai i V˜j+1 − V˜ji ∆τ = σ2 i 2 (S ) 2 i+1 i−1 i i V˜j+1 − V˜j+1 V˜j+1 − V˜j+1 + S i+1 − S i S i − S i−1 + rSi i+1 i−1 V˜j+1 − V˜j+1 i − rV˜j+1 Ai , 2 32 Mathematical Foundations i except that we simply set the node values Vj+1 to the exercise value P (S i , t) if they are lower than the exercise value (see Wilmott [117, p.
5684 in this particular case. 005712) . e. 005712 · s2 ) · 0 (log(s/St )+(r−(1/2)σ 0 1 − 2σ 2 tT √ e σs 2πtT 2 )t T) 2 ds. The last integral can now be evaluated with numerical methods. 1. 84. But, as we will see in the next section, in realistic settings with many paths and more regression basis functions the Feature Extraction method converges faster to the true solution and delivers more accurate estimates than the traditional Monte Carlo method. 2 Specifications of an Asian option.