Option Pricing Using Monte Carlo Simulations

I, Gaurav Malik, along with my colleague Gaurav Bansal studied variation between traded price of a Call Option on Nifty Index vis-a-vis price from Black Scholes Model and from Monte Carlo Simulations which were run using a Python code available on Github.

The purpose of this study is to analyze how the number of simulations affect the accuracy of the option price and compare the simulated prices with actual traded price. In theory, as we increase the number of simulations, price accuracy should increase, best being at infinite number of runs. But in reality, time and computational power is a big factor and the number of runs is limited to the point beyond which increasing the simulations will result in minimal gains in accuracy.

The theory of option pricing continues to evolve since 1900s when Louis Bachelier published his famous work on option pricing. The price of the options trading on the stock exchange(s) is determined by market forces.

The Black-Scholes Model is one of the most popular method for option pricing. The Black-Scholes Model depends on fixed inputs and has three major assumptions, namely; i) underlying prices are lognormal, ii) the volatility of the underlying is constant and iii) the risk-free rate is constant.

The call option price using Black-Scholes Model is given by:

where

In a risk-neutral world the stock price follows a Generalised Wiener Process i.e., it has a constant drift (r) and a variability in the path followed by stock (S), i.e., prices have a random walk, dz along this drift.

where dz over a small time period is equivalent to

With the use of Ito’s Lemma, the stock price at time T can be written as:

The expected payoff from the option is computed as the risk-neutral mean of the maxima of the underlying asset’s values at the expiration of the option minus the exercise price ‘K’. The risk-neutral mean is then discounted at the risk-free rate to get the estimated value of the call option.

The parameters of the call option used for comparing the traded price with the price given by Black-Scholes Model and Monte Carlo Simulations are as under:

Strike Price - 12000

Risk-free rate - 10%

Maturity Date - 27 June 2019

Valuation Date - 7 June 2019

Implied Volatility (as per NSE) - 10.33%

Traded Price - ? 85.70

The price of the call option calculated using Black Scholes Model (BSM) is ? 85.03 which is 0.79% less than the actual traded price.

The volatility in the Monte Carlo Simulation is considered constant and equal to the implied volatility provided by NSE. Since, it is a European Call option it has a single maturity date and hence, the number of time steps considered is 1.

The process is run for multiple numbers of simulations. For each simulation 5 loops are run for 10 times. Thus, the result is 50 option prices with same parameters and the mean of these 50 prices is considered as the option price at that particular simulation number.

Standard Error is calculated as 

Rate of Decrease in standard error is calculated as 

The results from the Monte Carlo Simulation are:

From the results of Monte Carlo Simulations, following observations can be made: -

• Increase in the number of simulations, results in decrease in the variation between the Monte Carlo Simulated price and the actual traded price.
• The rate of decrease in variation decreases with increase in number of simulations. This indicates that an optimal number of simulations can be run based on the time taken to run a simulation.
• The Monte Carlo Simulated Price is very close to the traded price when number of simulations is high (>60000).
• While the Black-Scholes Model provides only one option price with a variation ~0.8%, the Monte Carlo simulations reduces the variation to 0.02% in 100,000 runs. This variation can further decrease using better computational software and models.
• The increase in number of simulations increases the time taken to run the code.

 Thus, there is a trade-off between the time taken to run the simulation and an accurate option price. Optimal number of simulations must be run in order to balance this trade off.

References

• The Complete Guide to Option Pricing Formulas by Espen Gaarder Haug, Second Ed.
• Options, Futures and Other Derivatives by John C. Hull, 9th Ed.
• National Stock Exchange of India, https://www.nseindia.com/