- American Options - Pricing Methods and Spreadsheets
- Pricing American Call Options by the Black-Scholes
Under this scenario, we will use the Geometric Brownian Motion of the stock price which implies that the stock follows a random walk. Random walk Random Walk Theory The Random Walk Theory or the Random Walk Hypothesis is a mathematical model of the stock market. Proponents of the theory believe that the prices of means that the future stock prices cannot be predicted by the historical trends because the price changes are independent of each other.
American Options - Pricing Methods and Spreadsheets
Before we start discussing different option pricing models, we should understand the concept of risk-neutral probabilities, which are widely used in option pricing and may be encountered in different option pricing models.
Pricing American Call Options by the Black-Scholes
Nevertheless, these assumptions can be relaxed and adjusted for special circumstances if necessary. In addition, we could easily use this model to price options on assets other than stocks (currencies, futures ).
Unlike the simulation in a binomial model, in continuous time simulation, we do not need to simulate the stock price in each period, but we need to determine the stock price at the maturity, S(T) , using the following formula:
The free boundary problem for pricing an American call option consists in finding a function V (t, S) and the early exercise boundary function Sf such that V solves the Black-Scholes PDE on a time depending domain:
Under the binomial model, we consider that the price of the underlying asset will either go up or down in the period. Given the possible prices of the underlying asset and the strike price of an option, we can calculate the payoff of the option under these scenarios, then discount these payoffs and find the value of that option as of today.
In this article, we will discuss two scenarios: simulation in the binomial model with many periods and simulation in continuous time.
The overall value of an option is actually determined by six factors: strike price, the current market price of an underlying stock, dividend yield, prime interest rate, proximity to the expiration date, and the volatility of the stock prices over the course of the option.
Under the binomial model, we consider the variants when the asset (stock) price either goes up or down. In the simulation, our first step is determining the growth shocks of the stock price. This can be done through the following formulas:
After finding future asset prices for all required periods, we will find the payoff of the option and discount this payoff to the present value. We need to repeat the previous steps several times to get more precise results and then average all present values found to find the fair value of the option.