Simulation
Monte Carlo simulation (MCS) is among the most important numerical techniques in
finance, if not the most important and widely used. This mainly stems from the fact that it
is the most flexible numerical method when it comes to the evaluation of mathematical
expressions (e.g., integrals), and specifically the valuation of financial derivatives. The
flexibility comes at the cost of a relatively high computational burden, though, since often
hundreds of thousands or even millions of complex computations have to be carried out to
come up with a single value estimate.
Random Variables
Consider, for example, the Black-Scholes-Merton setup for option pricing (cf. also
Chapter 3). In their setup, the level of a stock index ST at a future date T given a level S 0 as
of today is given according to Equation 10-1.
Equation 10-1. Simulating future index level in Black-Scholes-Merton setup
The variables and parameters have the following meaning:
ST
Index level at date T
r
Constant riskless short rate
σ
Constant volatility (= standard deviation of returns) of S
z
Standard normally distributed random variable
This simple financial model is easily parameterized and simulated as follows:
In [ 10 ]: S0 = 100 # initial value
r = 0.05 # constant short rate
sigma = 0.25 # constant volatility
T = 2.0 # in years
I = 10000 # number of random draws
ST1 = S0 * np.exp((r - 0.5 * sigma ** 2 ) * T
+ sigma * np.sqrt(T) * npr.standard_normal(I))
The output of this simulation code is shown in Figure 10-3:
In [ 11 ]: plt.hist(ST1, bins= 50 )
plt.xlabel(‘index level’)
plt.ylabel(‘frequency’)
plt.grid(True)
