139 Computation of the Derivative Price For many popular European derivatives

139 computation of the derivative price for many

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13.9 Computation of the Derivative Price For many popular European derivatives such as puts, call and binaries, as well as for some exotics like barriers, there is a formula for their price obtained from direct calculation of the expectation in the risk-neutral world. When there is no simple formula, there are three common methods to approximate the price using a computer: Binomial tree approximation Monte Carlo simulation (ORF 409) Numerical solution of a PDE. In the first we go back to the tree models with a “large” number of periods N and solve the problem (by running back through the tree) as an approximation to the continuous-time model. Here the continuous-time, continuous-state GBM model is approximated by a discrete time, discrete state model. However this can only be used when dynamic programming applies, for example with barrier options, but not with Asian options because the latter is strongly path-dependent. In the second, we simulate paths of the GBM process in the risk-neutral world and estimate the expected payoff by the average at the end of those paths. The approximation is discrete time, continuous state (up to machine precision of the computer). The stock price paths are generated at the discrete times t n by the iteration S t n +1 = S t n + rS t n Δ t + σS t n Δ t ε n +1 , 60
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where { ε n } N - 1 n =1 is a sequence of independent N (0 , 1) distributed random generated numbers. Then given we run M paths where M = 10 , 000 or 50 , 000 for example, let H ( k ) denote the payoff of the option the holder would receive at time T if the stock followed the k th simulation path. (For example, if the option is European, so H = h ( S T ), then H ( k ) = h ( S ( k ) T )). We approximate IE Q { h ( S T ) } ≈ 1 M M summationdisplay k =1 H ( k ) . The third method involves discretizing the Black-Scholes PDE with finite-difference ap- proximations to the partial derivatives. We won’t describe this further here. 13.10 More Complicated Financial Market Models The Fundamental Theorem of Asset Pricing, namely that absence of arbitrage is equivalent to the existence of a risk-neutral world in which the discounted price processes of traded securities are martingales, holds for quite general stochastic stock price models under quite mild assumptions. For example, the stock price may have jumps, or the volatility may be stochastic, or it may be driven by a different type of building block than Brownian motion in which returns are not normal. However, there is no guarantee that there is a unique risk-neutral world. In the case of the Black-Scholes model, there is a unique Q . In more realistic and complex models, there is typically an infinite family of possible risk-neutral worlds. In the unique case, the market is said to be complete and all derivatives can be hedged by trading the underlying stock. Here enforcing no arbitrage leads to a unique derivative price. In the other case, we have an incomplete market : just enforcing absence of arbitrage does not uniquely nail down derivatives prices, there may be many (usually a whole interval) that are consistent with no arbitrage. This situation usually occurs when the model has unhedgeable
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  • Fall '11
  • Dividend, Mathematical finance, Black–Scholes

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