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Lecture 5 - Monte-Carlo approach to the valuation of real...

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Monte-Carlo approach to the valuation of real options Stochastic Processes for Oil Prices Geometric Brownian Motion Simulation The real simulation of a GBM uses the real drift a. The price at future time t is given by: P t = P 0 exp{ (a - 0.5 s 2 ) D t + s N(0, 1) } s is the volatility of P With real drift use a risk-adjusted (to P) discount rate The risk-neutral simulation of a GBM uses the risk-neutral drift a’ = r - d . The price at t is: P t = P 0 exp{ (r - d - 0.5 s 2 ) D t + s N(0, 1) } d is the convenience yield of P With risk-neutral drift, the correct discount rate is the risk-free interest rate. Mean Reversion Process Consider the arithmetic mean reversion process The solution is given by the equation with stochastic integral: Where h is the reversion speed. The variable x(t) has normal distribution with mean and variance given by: We want a mean reversion process for the oil prices P with lognormal distribution with mean E[P(T)] = exp{E[x(T)]} Risk-Neutral Mean Reversion Process for P The risk-neutral process for the variable x(t), considering the AR(1) exact discretization (valid even for large Δt) is:
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