Visualizing Black-Scholes Option Pricing
Equity options are financial derivatives allowing investors to obtain risk-return profiles not possible through the simple purchase of equity.
Although options have existed for decades, the amount of options bought and sold has increased dramatically in 2020. In fact, the world’s largest equity derivatives clearing organization reported that 545,272,318 contracts were cleared in May 2020; this is a 17.3 percent increase from May 2019.
How can an investor determine the ‘fair price’ of an option contract? One way to estimate the market value of an option contract is to use the Black-Scholes formula.
Although many methods can be used to value options, none are more popular than one notorious model published by Fischer Black and Myron Scholes in 1973. The model, now known as Black-Scholes, yields a formula used to calculate the theoretical value of a European call or put option. According to Black-Scholes, the value of a call or put is a function of a few variables:
- Strike price
- Underlying asset price
- Time until expiration
- Asset price volatility
- Risk free rate
To better understand the relationships that determine the price of a call option, I created visualizations using Python.
First, I calculated a theoretical call option price for an asset with varying price and strike. I used a constant risk free rate of 0.01, an implied volatility of 0.50, and a maturity of 1 year until expiry.
From the plot above, it is clear that a higher strike price leads to lower call option price. Oppositely, a higher asset price leads to higher call option price. It is important to note, however, that neither of these relationships are perfectly linear.
Rotating the plot slightly gives more insight:
Next, I held the strike price constant and varied implied volatility. For this trial, I held the strike price at a constant value of 1.
We can tell that implied volatility is positively correlated to call options price.
The relationship between implied volatility and option price can be used to calculate vega — a metric commonly used to evaluate option profitability. Specifically, vega quantifies an option’s price sensitivity to changes in the implied volatility of the underlying asset.
Next, I held the asset price constant and varied the risk free rate of return. The risk-free rate of return is the theoretical return of an investment with minimal risk. The risk-free rate is often calculated as the difference between the rate of inflation and the yield of a low-risk bond. For this trial, I held the asset price at a constant value of 1.
Although risk-free rate and option price are positively correlated, the gradient is so small that it is barely noticeable. In other words, small changes in risk-free rate have minimal effects on option pricing. This is mostly because the range of values that could feasibly be used as a risk free rate is slim. Although the risk free rate constantly varies, it rarely is outside of the range plotted above. As a result, it is often okay to assume a constant risk-free rate when modeling option prices on short timeframes.
