Option Price Analysis Using Monte Carlo Simulation
Hello everyone. In today’s world of dynamic financial markets, accurately valuing options is paramount for informed decision-making. Today, I would like to use a slightly different machine-learning approach to find option prices. Through Monte Carlo Simulation, we’ll delve into sophisticated analyses to enhance our understanding of option pricing dynamics.What exactly is an option?
Detailed explanations are provided in an easy-to-understand manner on the websites of securities companies and exchanges, so I will only give an overview here. First of all, options are one type of financial derivative products called derivatives. Since it is called a derivative, it is tied to the future value (price) of some other product. The underlying asset of the derivative is called the underlying asset. For example, underlying assets include interest rates, foreign exchange rates, stock prices, economic indices, commodities, and real estate prices. In addition, candidates for the underlying asset are not limited to financial assets, but any variable that is uncertain in the future can be a candidate, so there are also things called weather derivatives that depend on the outcome of the weather.
In addition, marketability differs depending on the method of dependence on the underlying asset. Other typical derivatives include futures/forwards and swap transactions. In other words, derivatives can be identified by the underlying asset x commodity nature.
An option is a right to buy (or sell) an underlying asset at a preset strike price at a specific point in the future. First, the point is that it is a right, not an obligation. In other words, if exercising the right would result in a loss, it will be waived. In addition, the term “expiry” refers to the point at which the option is exercised in the future, and the “strike” refers to the exercise price.
Let’s explain the situation using a simple example.
Let’s assume that Mr. K was paid in company A’s stock as compensation. Suppose there is a sales restriction for one year. It’s fine if the stock price continues to rise, but if it falls, it’s a loss. Also, even if the stock price is ultimately higher than it is now, it is not a good idea to get excited and depressed about the rise and fall of the stock price in the meantime. Therefore, by purchasing options, you can enjoy the benefits of rising stock prices and limit losses due to stock declines, even if you pay a certain amount of compensation. Here, we will purchase a put option from a financial institution that allows us to sell the stock for 100 a year from now at 100 ₹per share. By the way, the right to sell is called a “put” and the right to buy is called a “call.”
For example, if the price goes up to 80 ₹ per share after one year, if there were no option, the loss would be 100–80 = 20 ₹, but by exercising the option right, you can sell the A share for 100 ₹. This means that you made a profit of 20 ₹ from the option.
On the other hand, consider the case where the price becomes 120 ₹ after one year. If you exercise the option, you would sell something worth 120 ₹ for 100 ₹. In such cases, there is no need to exercise the option right. In other words, although the profit from the option is zero, the A shares that you own can be sold at 120 ₹ per share, so you can enjoy the profit from the stock price appreciation. In other words, there was no loss in either the stock price rise/fall scenario. The act of reducing asset price fluctuations, or market risk, in this way is called hedging.
So how should we estimate the value of this put option? At least it’s clear that it’s not free. No matter how the stock price moves, you will not incur a loss from options. In other words, if the option were free, it would be impossible because the initial capital would be zero and the probability of loss would be zero and there would be a guaranteed profit (this situation is called arbitrage). However, it doesn’t necessarily mean that it is too expensive. Ultimately, it seems reasonable to think that it is determined by how the underlying asset price will fluctuate in the future, that is, by the probability distribution of future stock prices.
Monte Carlo Method
Monte Carlo simulation has many applications in financial practice related to the analysis of complex random processes, including financial planning, valuation of risks (VaR), estimating the value of complex options and other financial assets.
Having understood the basics of probability distributions , we are now ready to learn about a computer method in which probability distributions play an important role. This method is called Monte Carlo simulation , statistical modeling or simulation modeling (English: ‘Monte Carlo simulation’ ).
Monte Carlo simulation in finance involves using a computer to simulate the functioning of a complex financial system.
A characteristic feature of Monte Carlo simulation is the generation of a large number of random samples from a given probability distribution or distributions characterizing the risk in the system under consideration.
Monte Carlo simulation has several very different uses. One use case is in financial planning.
Stanford University researcher Sam Savage provided the following fairly accurate analogy for the role of this method:
“What do you make sure you do before you climb up the ladder? You shake it to test its stability — that’s a Monte Carlo simulation.”
Just as shaking a ladder helps us assess the risks of falling down a ladder, Monte Carlo simulation allows us to experiment with a proposed financial strategy or policy before it is actually implemented. For example, investment performance can be assessed against a benchmark or commitment.
The key to this simulation and the Monte Carlo method is the probability distribution for the various sources of risk (including interest rates and equity market rates of return , in this case).
The consequences of pension fund investment policy decisions can be assessed using modeling for a certain period. The experiment can be repeated for a different set of assumptions.
How Does Monte Carlo Simulation Work?
Understanding the mechanics of Monte Carlo simulation is crucial before delving into its implementation. In essence, Monte Carlo simulation generates a set of random variables that mimic the characteristics of the risk factors being simulated.
The simulation generates a large number of potential outcomes along with their respective probabilities, allowing for the modeling of realistic scenarios such as stock prices, option prices, and probabilities.
It’s important to note that Monte Carlo simulations can be computationally demanding and slow, particularly when a large number of scenarios are generated.
Now, let’s explore how we can utilize Monte Carlo simulation to price a European call option and implement its algorithm in Python.
Pricing a European Call Option Using Monte Carlo Simulation
Let’s begin by examining the renowned Black-Scholes-Merton formula (1973):
S(t) = Stock price at time t
r = Risk free rate
σ = Volatility
Z(t) = Brownian motion
Our objective is to solve the equation above to obtain an explicit formula for S(t).
We employed the Euler Discretization Scheme to solve the stochastic equation above. The solution is given by the expression:
Let’s apply the logarithm function to equation 3–2 above, which will enable a faster implementation in Python. The vectorization process using the numpy package in Python would easily handle the log version of the solution above.
Monte Carlo Implementation in Python
This dashboard allows users to price stock options using either the Black-Scholes model or Monte Carlo simulations.
Users can input data such as the option type, strike price, underlying stock ticker symbol, risk-free rate, volatility, expiry date, and number of simulations.
The dashboard also displays historical stock prices for the underlying stock.
The current output from the simulation shows that the call option price is 0.0 and the put option price is 164.2849.
Conclusion
This blog post explores option price analysis using Monte Carlo simulation as an alternative to traditional methods. It covers the basics of options, introduces Monte Carlo simulation’s application in finance, and explains its mechanics. The post outlines the pricing of a European call option using the Black-Scholes-Merton formula and presents a Python implementation for simulating option prices. It concludes with a dashboard allowing users to input option details and view historical stock prices, providing practical insights into financial modeling and analysis.
References
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[3] Y. Lai and J. Zhang, “Pricing Lookback Options under Normal Inverse Gaussian Model by Variance Reduction and Randomized Quasi-Monte Carlo Methods,” 2014 Seventh International Joint Conference on Computational Sciences and Optimization, Beijing, China, 2014, pp. 435–439, doi: 10.1109/CSO.2014.89.
[4] Y. Lai and J. Zhang, “Pricing Lookback Options under Normal Inverse Gaussian Model by Variance Reduction and Randomized Quasi-Monte Carlo Methods,” 2014 Seventh International Joint Conference on Computational Sciences and Optimization, Beijing, China, 2014, pp. 435–439, doi: 10.1109/CSO.2014.89.
[5] Z. Hui, C. Kai and Z. Ziting, “Comparative analysis of Asian and European options based on Monte Carlo simulation,” 2021 International Conference on Computer, Blockchain and Financial Development (CBFD), Nanjing, China, 2021, pp. 356–361, doi: 10.1109/CBFD52659.2021.00079.
