Black-Scholes model and Monte Carlo simulation
Options are financial derivatives based on the value of underlying securities. They give the buyer the right to buy (call options) or sell (put options) the underlying asset at a pre-determined price within a specific timeframe. There are also two basic styles of options: American and European. American options can be exercised any time before the expiration date of the option, whereas European options can only be exercised on the expiration date. This blog digged into an option-pricing model to understand the evaluation of European options.
Black-Scholes model
The Black-Scholes model or Black-Scholes-Merton model is a mathematical model for pricing an options contract. The model is based on geometric Brownian motion (GBM) with constant drift and volatility. The Black–Scholes equation is a second order partial differential equation, which gives a theoretical estimate of the price of European options over time.
where S(t) is the spot price of the underlying asset at time t; V(S, t) is the price of the option as a function of the underlying asset S, at time t; r is the risk-free interest rate; 𝜎 is the volatility of returns of the underlying asset; t is current time in years. By solving the equation for the corresponding terminal and boundary conditions, we have the solution as the price of a European call option C(S, t) and the price of a European put option P(S, t) are:
where
where K is the strike price or exercise price of the option; N(･) denotes the cumulative distribution function (CDF) of the standard normal distribution; T is expiry or maturity and T-t is time to maturity in years.
The underlying stock S follows a GBM, which satisfies the stochastic differential equation (SDE):
where 𝜇 represents the drift rate of growth of the underlying stock, and W denotes a Weiner process (one-dimensional Brownian motion). By using Ito’s lemma, the analytic solution of the SDE can be solved as:
where Z is a standard normal random variable.
The model assumes underlying asset prices follow a lognormal distribution. Asset prices are often observed to have significant right skewness and some degree of kurtosis. This implies the market is pricing in a greater likelihood of a high volatility move to the downside.
Monte Carlo simulation
Monte Carlo methods can be used in finance to analyse the value of an option with multiple sources of uncertainty. First, the price of the underlying asset is simulated by random number generation for a number of paths or trajectories. After repeatedly simulating the trajectories and computing appropriate averages, the estimated price of options can be obtained, which is consistent with the analytical results from Black-Scholes model.
For European option, the buyer elects to exercise the option on the maturity date T, the seller is obligated to sell/buy the inderlying stock to/from the buyer at the strike price K. Thus, the call/input option price at time t = 0 is
According to the sample code above, the simulated paths over time can be plotted to understand the behavior of this stochastic process.
