<Text-field layout="Heading 1" style="Heading 1">Introduction</Text-field>

<Text-field layout="Heading 1" style="Heading 1">Solution 2: Monte Carlo simulation </Text-field>

Monte Carlo is one of the most popular methods for pricing securities whose prices are given in terms of expected values. The approach consists of the following steps:

simulate sample paths of the underlying assets

evaluate the discounted cash flows of the derivative security on each sample path

average the discounted cash flows over the sample paths.

To generate sample paths, we will use Marsaglia's random variate generator, implemented as an external C++ routine:

void MCInitialize() { Ziggurat::Reset(); }

Maple's interface to the above C++ routine is defined in the Initialization section:

The following C++ function generates a sample stock path.

void MCSamplePath(double *S, int N, double T, double S0, double r, double sigma) { double h = 1./(N-1); double C = exp(r*h-sigma*sigma*h/2); S[0] = S0; for (int i = 0; i < N-1; i++) { S[i+1] = S[i]*C*exp(sigma*sqrt(h)*Ziggurat::Next()); } }

We have defined a corresponding Maple routine called MCSamplePath that invokes this C++ routine:

T := 1.: sigma := 0.2: r := 0.15: S0 := 100.:

Generate a sample path

MCSamplePath(S, N, T, S0, r, sigma);

Plot the sample path.

plots[pointplot]([seq([(i-1)*T/N, S[i]], i = 1..N)], connect = true);

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K := 100.;

NiM+SSJLRzYiJCIkKyIiIiE=

The estimated price of the option using 10 sample paths

ArithmeticAsian(T, S0, r, sigma, K, 10^4, 10, MCSamplePath);

NiMkIit0TUYnbyghIio=

And using 10,000 sample paths. We note that the estimated price is closer but not identical to the results obtained by the PDE approach.

ArithmeticAsian(T, S0, r, sigma, K, 10^4, 10^4, MCSamplePath);

NiMkIispXHp6RSkhIio=

One of the main weaknesses of the MC approach is that it tends to waste points due to clustering. This compromises performance when the sample size is small (or the dimension is large). We can often improve the performance by holding more regularly distributed (usually deterministic) points using a technique called Quasi Monte Carlo simulation, which we do using the C++ routine QMCSamplePath.

P := Array(1..4): for j from 1 to 4 do QMCSamplePath(S2, N2, T, S0, r, sigma); P[j] := plots[pointplot]([seq([(i-1)*T/N2, S[i]], i = 1..N2)], connect = true); end do: plots[display](convert(P, list));

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Using the Quasi Monte Carlo simulation method, we arrive at virtually the same price as with the PDEs approach.

ArithmeticAsian(T, S0, r, sigma, K, N, 10^5, QMCSamplePath);

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