Monte Carlo Documentation

Overview

The Monte Carlo Simulation is a numerical method used for estimating the fair value of options by simulating the price paths of the underlying asset over time. Unlike the Black-Scholes model, Monte Carlo does not rely on a closed-form solution. Instead, it generates random price paths based on the asset's volatility and risk-free rate, computes the payoff for each simulated path, and then takes the average discounted payoff.

Key Concepts in Monte Carlo Simulation

• Random Walk: The Monte Carlo simulation uses a random walk to generate different price paths for the asset. This is done by sampling from a normal distribution to account for the randomness of stock price movements.


• Risk-Neutral Valuation: The simulation assumes that the asset price follows a risk-neutral process, meaning that investors are indifferent to risk. The simulation discounts future payoffs back to the present value using the risk-free interest rate.


• Multiple Simulations: Monte Carlo simulation runs multiple trials (simulations) to estimate the option's price. Each trial represents a possible future scenario for the asset's price, and the more trials you run, the more accurate the estimate.


• Time Steps: The total time to expiration is divided into discrete time steps, and at each step, the asset's price is updated based on the volatility and risk-free rate. The granularity of time steps can affect the precision of the result.

Function Parameters

• S: Current price of the underlying asset.
• K: Strike price of the option.
• T: Time to expiration in years.
• r: Risk-free interest rate.
• σ: Volatility of the underlying asset (annualized standard deviation of returns).
• Option type: The type of option, either 'call' or 'put'.
• num_simulations: Number of simulation paths. More simulations increase accuracy but require more computational power.
• num_steps: Number of time steps per simulation. More steps provide a finer resolution of the price path but may increase computational time.
• epsilon: A small value used for numerical calculation of the option's Greeks.

Explanation of Key Calculations

1. Price Paths: At each time step, the asset's price is updated according to the following formula:
   • \( S_{t+1} = S_t \times \exp\left((r - 0.5\sigma^2)\Delta t + \sigma \sqrt{\Delta t} \cdot z\right) \)
   where \(z\) is a random sample from a standard normal distribution, and \(\Delta t\) is the time step.
2. Payoff Calculation: For each price path, the payoff is calculated at expiration based on the option type:
   • Call Option: \( \text{payoff} = \max(S_T - K, 0) \)
   • Put Option: \( \text{payoff} = \max(K - S_T, 0) \)
3. Discounting the Payoffs: After computing the payoff for each simulation, the values are discounted to the present using the risk-free rate:
   • Discounted Payoff: \( e^{-rT} \cdot \text{payoff} \)
4. Estimating Option Price: The option price is the average of the discounted payoffs across all simulations:
   • Option Price: \( \frac{1}{n} \sum_{i=1}^n \left(e^{-rT} \cdot \text{payoff}_i\right) \)
5. Greek Calculation: The Greeks (Delta, Gamma, Vega, Theta, Rho) can be computed using finite difference methods by shocking the input parameters (e.g., increasing \(S\) or \(\sigma\)) and rerunning the simulation to observe the effect on the option price.