Monte Carlo Simulation for Valuing Performance-Based Stock Options

Monte Carlo Simulation

The Monte Carlo simulation can also be used in the analysis of option pricing. Monte Carlo simulations of option pricing rely on certain assumptions. An important assumption of a Monte Carlo model is that the underlying stock’s price follows a Geometric Brownian Motion (“GBM”) stochastic process. The GBM assumes that the stock price of a company follows a random walk. As a simple explanation, the GBM forecasts a constant drift in the stock price determined by volatility and a “shock” determined by randomness. A key difference between Monte Carlo simulation and other methods is the addition of a probability distribution as an additional input. Probability distributions include normal, lognormal, uniform, triangular, etc. A lognormal distribution is recommended for option pricing analysis. Since the lognormal distribution is positively skewed, it can be used to represent values that don’t go below zero and have unlimited upside potential. As share prices of companies have a floor value of zero, a lognormal distribution is preferred.

Performance conditions

There are certain ESOPs which are issued with performance vesting conditions For example, an ESOP may vest when the revenue of the company crosses a certain threshold Performance metrics like revenue, operating profits, market capitalization etc. can be best modelled using a Monte Carlo simulation.

Example of the Performance Based Stock Options

The options had the following terms:

1.????? Grant Date: April 1, 2024

2.????? Exercise price/Strike Price: Share Price as on April 1, 2024

3.????? Vesting Condition: On achievement of 2x of April 1, 2024, price per share of the company.

Valuation Methodology

We used a Monte Carlo simulation model which is implemented in a risk-neutral framework where the underlying (common stock in this case) price is assumed to follow a Geometric Brownian motion with a shift equal to the risk-free rate.?

??t = ??0 ?????? ((??f –??2/2) ?? + ??√????)

Where: ??t = Stock price at the end of the period

??0 = Stock price at the beginning of the period

??f = Risk-free rate

?? = Annualized standard deviation based on the logarithmic return of daily stock prices

?? = Time until expiration

?? = Standard normal random variable

Monte Carlo simulation is most commonly used when there’s a path dependency property required to determine the final payoff as in the case of many types of exotic options and market condition-based stock awards.

A future value is calculated for each iteration of the Monte Carlo simulation model based on a payoff. Future values are discounted to the Valuation Date with the risk-free rate. Ultimately, the estimated fair value of the instrument is the average of all iterations of the Monte Carlo simulation model.

In the case of the Options, since there are uncertainties around the payoff as well as the timing of the payoff based on certain market vesting conditions as described previously, we deemed the Monte Carlo simulation as the most appropriate approach. In each iteration, we estimated the time to vest based on the simulated stock prices and the Market Condition described previously as well as the stock price at the time of vesting (if it vests) to calculate the payoff which was then discounted back to the present using the risk-free rate. The final value was calculated by averaging the present value of the payoffs across 1,00,000 iterations.

Monte Carlo Simulation, like any financial model, presents a mix of advantages and challenges. On the one hand, it is a powerful tool for assessing risk and uncertainty, allowing companies to simulate a wide range of potential outcomes by running numerous scenarios. This flexibility makes it particularly valuable in complex financial environments where traditional models might fall short.