Monte Carlo In Derivative Investment

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Introduction to Monte Carlo method to option evaluation.

Aiming to answer question “c”, this paper proposes the simulation model for pricing the European Call option. Of course, this is the type of security, for which Black-Scholes formula provides an exact answer, so there is no real need to use the simulation to price it. On the other hand, as long as we know the exact solution, it becomes possible to check the accuracy of our simulation results.

In the Black-Scholes world-view, a fair value for an option is the present value of an option payoff at expiration under risk-neutral random walk for underlying asset prices. Therefore the general approach to using Monte Carlo simulation to find the price of the option is straightforward:

1. Using the risk free neasure, simulate sample paths of the underlying asset prices over the relevant time horizon
2. Evaluate the discounted cash flows of a security on each sample path.
3. Average the dicounted cash flows over sample paths.(Charnes, 2000)

The European call option solved by Monte Carlo simulation relies on the following process followed by the underlying asset S:

S(t +t) = S(t) Exp [( -^2/2) t+t]

where S(t) denotes the value of S at time t,  is a random sample from normal distribution with mean zero and standard deviation 1.0 (Hull, 2000)

The following code in VBA demostrates the practical application of the Monte-Carlo algrithm (Caflisch, 2001) for the European option evaluation constructing a complete path for S using N random samples:

Function MCCall(r As Double, sigma As Double, k As Double, s0 As Double, t As Double, N As Long) As Double

Dim s As Double
Dim discountFactor As Double
Dim growthFactor As Double
Dim variance As Double
Dim v As Double
Dim vi As Double
Dim dN As Double
Dim i As Long

discountFactor = Exp(-r * t)
gr...