Question

In Financial Derivatives:

a). Explain what we mean by a “geometric Brownian motion” (GBM) and if it is a reasonable representation of the “real world” behaviour of stock prices. Explain what information is required to simulate daily “real world” stock prices (over several periods) using a GBM.

b) Based on part (a), explain the steps required to price a one year, plain vanilla put option (on a stock) using Monte Carlo Simulation (MCS) and state any assumptions used. Explain the sources of the error in the MCS estimate of the put premium and explain how any errors might be reduced.

Answer 1

A.) A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift.

Geometric Brownian motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock price behavior. ... A GBM process only assumes positive values, just like real stock prices. A GBM process shows the same kind of 'roughness' in its paths as we see in real stock prices.

Following information is required for Simulate stock prices:

S= underlying price (per share)

X = strike price (per share)

σ = volatility (% p.a.)

r = continuously compounded risk-free interest rate (% p.a.)

q = continuously compounded dividend yield (% p.a.)

t = time to expiration (% of year)

b). Monte Carlo simulations are used to model the probability of different outcomes in a process that cannot easily be predicted due to the intervention of random variables. It is a technique used to understand the impact of risk and uncertainty in prediction and forecasting models

Three steps are required in the simulation process:

1 – sampling on random input variables X,

2 – evaluating model output Y, and

3 – statistical analysis on model output. We will focus our discussions on independent random variables. However, Monte Carlo simulation is applicable for dependent variables.