Monte Carlo methods, like the mathematics of Measure & Probability is based on the analogy between probability and volume, except that the intuitive notion of probability is used in reverse.[ref]Monte Carlo Methods in Financial Engineering – Paul Glasserman [/ref]
Monte Carlo uses this identity in reverse calculating the volume of a set by interpreting the volume as a probability.
In practice, once a model for an underlying asset has been established, such as geometric Brownian motion, it is relatively easy to compute the Monte Carlo valuation of any derivative. The practical steps involved may be summarised as follows
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- Decide on a pricing model for the dynamics of the underlying asset. For a stock this would typically be $dS= \mu S_t dt + \sigma S_t dW_t$
- Express the model in a discrete form with the geometric Brownian motion(s) $W_t$ as discrete standard Brownian motion $\sqrt{\Delta t} Z_t$
- Simulate values of the dynamics $dS$ by simulating a large number of Brownian motion paths randomly. Each simulation returns random values of $S$ at discreet times $t + \Delta t$ where $\Delta t$ is the time step (interval) that occurs $n$ times between now $t=0$ and the expiration date $t=1$ for a standard Brownian motion.
- In the above step, the probablility is cumulative such that at the final timestep $t=1$ of the simulation, the pay-off should hold true. Otherwise, ignore simulations that do not conform to the option pay-off. For a call, this would be simulations that result in $S(T) – K < 0$.
- Take an average of all the simulations that conform to the option pay-off.
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