elle
(Elle)
#1
Valuation
One of the most important applications of Monte Carlo simulation is the valuation of
contingent claims (options, derivatives, hybrid instruments, etc.). Simply stated, in a risk-
neutral world, the value of a contingent claim is the discounted expected payoff under the
risk-neutral (martingale) measure. This is the probability measure that makes all risk
factors (stocks, indices, etc.) drift at the riskless short rate. According to the Fundamental
Theorem of Asset Pricing, the existence of such a probability measure is equivalent to the
absence of arbitrage.
A financial option embodies the right to buy (call option) or sell (put option) a specified
financial instrument at a given (maturity) date (European option), or over a specified
period of time (American option), at a given price (the so-called strike price). Let us first
consider the much simpler case of European options in terms of valuation.
European Options
The payoff of a European call option on an index at maturity is given by h(ST) ≡ max(ST –
K,0), where ST is the index level at maturity date T and K is the strike price. Given a, or in
complete markets the, risk-neutral measure for the relevant stochastic process (e.g.,
geometric Brownian motion), the price of such an option is given by the formula in
Equation 10-10.
Equation 10-10. Pricing by risk-neutral expectation
Chapter 9 briefly sketches how to numerically evaluate an integral by Monte Carlo
simulation. This approach is used in the following and applied to Equation 10-10.
Equation 10-11 provides the respective Monte Carlo estimator for the European option,
where is the ith simulated index level at maturity.
Equation 10-11. Risk-neutral Monte Carlo estimator
Consider now the following parameterization for the geometric Brownian motion and the
valuation function gbm_mcs_stat, taking as a parameter only the strike price. Here, only
the index level at maturity is simulated:
