Created: October 26, 2021
Modified: October 26, 2021
Modified: October 26, 2021
Black-Scholes
This page is from my personal notes, and has not been specifically reviewed for public consumption. It might be incomplete, wrong, outdated, or stupid. Caveat lector.- A model of option prices that assumes:
- The existence of a risk-free asset paying some interest rate, for example, US Treasury bonds.
- That the price of the underlying stock follows a geometric Brownian motion (a Brownian motion in log space) with known drift and volatility parameters.
- That the asset does not pay dividends.
- It also assumes that the market is frictionless (no transaction fees) and that it is possible to buy and sell arbitrary fractional quantities of both the stock and the risk-free asset at any moment.
- Under these assumptions, the price of a European put or call option is completely determined, in the sense that any other price would create an arbitrage opportunity.
- The basic idea (?) is that we can continuously buy and sell the underlying stock to maintain a perfectly hedged position, one whose value has no relationship to the value of the underlying stock. This is called 'delta hedging'.
- The modeled option prices depend on the asset price, asset volatility, strike price, risk-free interest rate, and time to expiry. All of these are directly observable except for volatility. Given the option price and the other observed quantities, one can back out an implied volatility. The model asserts that this should be constant across strike prices and expiry dates for a given asset. In practice, it is not. Options that are very far in- or out- of the money (very far from the strike price) empirically tend to have higher prices (thus, higher implied volatility) than the model suggests. This pattern is known as the volatility smile. It does not always appear; in particular, it would not appear in a market where everyone was naively using Black-Scholes to price options. In American markets it began to appear after the crash of 1987.
- Estimates of the volatility smile can be used as corrections to Black-Scholes models.