Computing Option Price and Greeks using Monte Carlo Simulation
I successfully completed a comprehensive project focused on pricing a plain Vanilla European option using Monte Carlo simulation in Python.
The objective was to demonstrate the impact of increasing simulations on option pricing accuracy while studying derivatives, Option Types (Asian, American, European), Payoffs, Black Scholes model, Binomial Tree method (for American options), Option Greeks, Variance Reduction Methods, Put-Call Parity, Interest Rate term structure, Yield Curve, and Bootstrapping using Newton Raphson method.
What is Monte Carlo Simulation?
Monte Carlo simulation is a computational technique used to estimate outcomes or analyze the behavior of complex systems or processes. It involves running numerous iterations or simulations based on random sampling to model the uncertainty and variability present in the system.
In the context of finance, Monte Carlo simulation is commonly used to assess the risk and value of financial instruments, such as options or portfolios. By generating a large number of random scenarios for different input variables, such as asset prices or interest rates, Monte Carlo simulation can provide a range of possible outcomes and their associated probabilities.
This method is mostly used when there is no analytical solution for the given problem.
About the project : -
To begin, I carefully selected a risk-free rate, exploring the use of the entire term structure and understanding forward rate computation.
The data was fetched using the Yahoo Finance API. The fetched data included Risk free rate, Implied Volatility, Strike Price and Stock Price.
These values were processed using Monte Carlo Simulation in order to obtain simulations, using which means were calculated and which were taken as payoffs for the call and put option. These payoffs were then discounted using the risk free rate to get the output for the subsequent number of simulations.
Comparing the Monte Carlo simulation results with Black Scholes formula, I determined convergence points and plotted graphs to visualize convergence.
Confidence intervals were computed to validate the Black Scholes model within a certain range, considering standard errors. For reference, Black Scholes Model is an analytical solution to the option pricing problem.
What are Option Greeks?
Option Greeks are a set of parameters or metrics used to measure the sensitivity of options to changes in various factors. They help traders and investors understand the risk and potential profitability of options positions.
I empirically calculated Option Greeks and compared them with theoretical values, referring to the below giving table.
What is Put-Call Parity?
Put-Call Parity is a fundamental concept in options pricing theory that establishes a relationship between the prices of put options and call options with the same underlying asset, strike price, and expiration date. Put-Call Parity states that under certain assumptions, the price of a European call option plus the present value of the strike price equals the price of a European put option plus the current price of the underlying asset.
Mathematically, the Put-Call Parity relationship can be expressed as:
C + PV(X) = P + S
Where: C = Price of the call option, PV(X) = Present value of the strike price (X), P = Price of the put option and S = Current price of the underlying asset
I validated put-call parity, ensuring consistency in pricing calculations. Detailed documentation captured all steps, equations, assumptions, and data sources used.
Moving forward, I plan to extend the project by pricing Asian options using Monte Carlo simulation with arithmetic and geometric averaging. Validating the results against closed-form solutions will further enhance accuracy. Exploring variance reduction techniques, such as control variates or antithetic variates, will optimize computational efficiency.
Throughout this project, I received invaluable guidance and mentorship, contributing to my growth in quantitative finance. The experience enhanced my analytical skills, Python proficiency, and practical understanding of option pricing complexities. I look forward to applying this knowledge to future endeavors in the field.
