erf = x => {
const sign = Math.sign(x);
const ax = Math.abs(x);
const t = 1 / (1 + 0.3275911 * ax);
const y = 1 - (((((1.061405429*t - 1.453152027)*t) + 1.421413741)*t - 0.284496736)*t + 0.254829592)*t*Math.exp(-ax*ax);
return sign * y;
}
// Standard normal CDF
N = z => 0.5 * (1 + erf(z / Math.SQRT2));
// Black-Scholes pricing function
blackScholes = (S, K, T, r, sigma, q) => {
const d1 = (Math.log(S / K) + (r - q + 0.5 * sigma * sigma) * T) / (sigma * Math.sqrt(T));
const d2 = d1 - sigma * Math.sqrt(T);
const call = S * Math.exp(-q * T) * N(d1) - K * Math.exp(-r * T) * N(d2);
const put = K * Math.exp(-r * T) * N(-d2) - S * Math.exp(-q * T) * N(-d1);
return {call, put, d1, d2};
}
// Calculate current option prices
prices = blackScholes(S0, K, T, r, sigma, q);
callPrice = prices.call;
putPrice = prices.put;Factors affecting option prices
This interactive tool demonstrates how the six key factors affect European option prices using the Black-Scholes model. Adjust the sliders to see how changes in each parameter impact both call and put option values in real-time.
Black-Scholes Parameters
Current Prices
Quick Overview
Watch how all factors affect option prices as you adjust the sliders above.
Impact of Each Factor
The graphs below show how each factor affects option prices while holding all other parameters constant. The red dot indicates the current price for your selected parameters.
Effect of Stock Price (S₀)
As the stock price increases, call option values increase while put option values decrease. The sensitivity of an option’s price to a change in the underlying stock price is known as Delta (\(\Delta\)).
Effect of Strike Price (K)
As the strike price increases, call option values decrease, while put option values increase. A higher strike makes a call option less likely to finish in-the-money and a put option more likely to finish in-the-money.
Effect of Time to Expiration (T)
More time until expiration generally increases the value for both calls and puts, as there is more time for the stock price to move favorably. This sensitivity of an option’s price to the passage of time is known as Theta (\(\Theta\)).
Effect of Volatility (σ)
Higher volatility increases the value of both call and put options. This sensitivity to volatility is known as Vega (\(\nu\)). Greater uncertainty increases the likelihood of large price swings, making a profitable outcome more possible.
Effect of Risk-Free Rate (r)
A higher risk-free rate increases a call’s value and decreases a put’s value. This sensitivity to interest rates is known as Rho (\(\rho\)). This is because the present value of the strike price (which will be paid or received in the future) is lower when rates are higher.
Effect of Dividend Yield (q)
A higher dividend yield decreases call values and increases put values. This is because dividends reduce the stock price on the ex-dividend date. The sensitivity to dividend yield is sometimes referred to as Psi (\(\Psi\)).
Summary
The six factors affect European option prices as follows:
| Factor | Call Option | Put Option |
|---|---|---|
| Stock Price (S₀) ↑ | Price ↑ | Price ↓ |
| Strike Price (K) ↑ | Price ↓ | Price ↑ |
| Time to Expiration (T) ↑ | Price ↑ | Price ↑ |
| Volatility (σ) ↑ | Price ↑ | Price ↑ |
| Risk-Free Rate (r) ↑ | Price ↑ | Price ↓ |
| Dividend Yield (q) ↑ | Price ↓ | Price ↑ |
Note: These relationships hold for European options. American options may exhibit slightly different behaviors, particularly around early exercise for dividend-paying stocks, and time-to-expiration effects can reverse for equities with large discrete dividends near expiration.