Monte Carlo Based Numerical Pricing of Multiple Strike-Reset by Stavros Christodoulou

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1) θ=θ0 A sufficient condition for this interchange according to [11] is given by the following theorem. Theorem 1. Let h(s) be a function of s where s ∈ S ⊆ R, Θ a neighbourhood of θ0 , with (pθ (s))θ∈Θ a family of densities such that ❼ pθ (s) > 0 for s ∈ S and ∀ θ ∈ Θ Pθ -almost surely. ❼ pθ (s) is differentiable with respect to θ for s ∈ S and ∀θ ∈ Θ Pθ -almost surely. ❼ ∃ν, µ such that 1 µ + 1 ν = 1, and a function N (s) such that ∀θ ∈ Θ we have 1. Eθ [|h(s)|µ ] < ∞ 2. Eθ [|N (s)|ν ] < ∞ θ (s) pθ (s) 3.

The first inequality arises since the value of the Strike Reset Put option should be less than the value of the corresponding look-back option. The third inequality is Jensen’s inequality. The final inequality arises since St2 is log normally distributed and the maximum of a log normal distribution has finite first moments. As a result if we choose µ = 2 we obtain that E |h (S1 ) |2 ≤ E MT2 < ∞. This finishes the proof as all the conditions of Theorem 1 are met. For the Gamma of the Strike Reset option we do not provide a proof for this interchange of integration and differentiation but similar arguments can be used to obtain the required result.

C; end end %Calculates the option Price. ˆ2 −1)/(sigmaˆ2*S0ˆ2*dt) −... Z 1/(sigma*S0ˆ2*sqrt(dt))) 44 CHAPTER 5. APPENDIX B 45 Function ForwardValuation is the implementation of the Strike Reset Put option Forward valuation Algorithm. Simillarly with the previous functions it takes as inputs the initial underlying asset price S0 , the number of sampled trajectories N , the number of possible exercise dates M , the number of strike reset rights L, the volatility of the underlying asset sigma, the risk free rate r, the dividends rate q, the time to maturity T and finally the initial strike value k0.