Black-Scholes
Part of the the investment sequenceBlack-Scholes Model
A derivative is a contract whose value is derived by the underlying assets or benchmarks. Consider a derivative whose value is a mathematical function ($f$) of some underlying asset price (let $x$ by the log-price of the underlying asset).
Over small amounts of time, we can approximate the derivative's price moves using a Taylor expansion:
$$ df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dx + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} dx^2 + \cdots $$Now, assume the underlying log stock returns follows Brownian motion with drift ($\mu$):
$$ dx = \left( \mu x dt + \sigma x dB \right) $$First, we can substitute this into the Taylor expansion. Then, we can note that as $dt \rightarrow 0$, all the higher-order $dt$ terms tend towards zero much faster. If we assume markets are perfectly liquid, then we can instantly trade when the situation changes, which allows us to ignore these second-order terms. This gives us
$$ df = \left( \frac{\partial f}{\partial t} + \mu x \frac{\partial f}{\partial x} \right) dt + \frac{\sigma^2 x^2}{2} \frac{\partial^2 f}{\partial x^2} dB^2 + \sigma x \frac{\partial f}{\partial x} dB $$By the quadratic variation of Brownian motion, $dB^2 = dt$ Stochastic Processes II, which allows us to achieve Itô's lemma Itô's lemma but using geometric Brownian motion rather than Brownian motion Itō Calculus. MIT OpenCourseWare:
$$ df = \left( \frac{\partial f}{\partial t} + \mu x \frac{\partial f}{\partial x} + \frac{\sigma^2 x^2}{2} \frac{\partial^2 f}{\partial x^2} \right) dt + \sigma x \frac{\partial f}{\partial x} dB $$Rearranging these terms yields
$$ df - \mu x \frac{\partial f}{\partial x} dt - \sigma x \frac{\partial f}{\partial x} dB = \left( \frac{\partial f}{\partial t} + \frac{\sigma^2 x^2}{2} \frac{\partial^2 f}{\partial x^2} \right) dt $$
At this point, some insight is needed. Given small changes in $t$ and $x$, the following two portfolios are equivalent:
- Buy one derivative and invest the rest at the risk free rate.
- Buy $\frac{\partial f}{\partial x}$ stocks and invest the rest at the risk free rate.
From this we can conclude that if you buy the derivative and then short $\frac{\partial f}{\partial x}$ units of stock, you achieve a portfolio that is effectively risk free.
This insight implies the left hand of the formula side must equal
$$ r \left( x \frac{\partial f}{\partial x} - f \right) dt $$Making this substitution yields:
$$ r \left( f - x \frac{\partial f}{\partial x} \right) dt = \left( \frac{\partial f}{\partial t} + \frac{\sigma^2 x^2}{2} \frac{\partial^2 f}{\partial x^2} \right) dt $$Rearranging these terms yields:
$$ 0 = \left( \frac{\partial f}{\partial t} + \frac{\sigma^2 x^2}{2} \frac{\partial^2 f}{\partial x^2} + r x \frac{\partial f}{\partial x} - r f \right) dt $$This finally gives us the Black–Scholes differential equation Black–Scholes equation:
$$ 0 = \frac{\partial f}{\partial t} + \frac{\sigma^2 x^2}{2} \frac{\partial^2 f}{\partial x^2} + r x \frac{\partial f}{\partial x} - r f $$If we phrase this in the language of the Greeks Greeks (finance), the three derivatives are theta, gamma, and delta, respectively.
Calls, Puts, and Forwards
Just to be clear, we can't just solve the above formula for $f$ with simple algebra because there is a "circular dependency" of sorts going on where the partial derivatives depend on our solution for $f$. In other words, we have to use tools from differential equations and consider the overall structure of whatever derivative we're considering. The most popular examples here are options.
A call option gives the buyer the right to buy an asset at a specified price ("strike price"). A European call option gives them this right on a specific date ("expiry date"), while an American call option gives them this right at any time before some date - note that American options are, therefore, more valuable than European ones. If a call option is exercised, the profit is simply the price of the underlying asset ("spot price") minus the strike price.
Put options likewise give the buyer the right to sell an asset at a specified price. They work similarly.
The Black-Scholes formula Black–Scholes model is a closed form solution that gives the theoretical price a European call/put option should be. Here it is for the special case for an option of a stock that pays no dividends
[ $t$ is the time until maturity, $k$ is the strike price, and $N$ is the normal CDF ]
American options are more complicated and there is no known closed form solution, but a variety of approximations exist Black–Scholes equation Option style Binomial options pricing model. That being said it is typically irrational to exercise an American option early Merton, so the to options of the two styles should trade for similar prices.
One interesting thing to note that doesn't require any fancy math is that the Black-Scholes differential equation doesn't include the stock's bias term ($\mu$) at all. This is similar to what happened when we discussed put-call parity, where we also saw that the actual returns completely disappeared from the formula.
From this comes a startling conclusion: the price that options on a stock trade at depend on the stock's expected variance, but not on its expected return. A very similar conclusion holds for futures, which can, after all, be replicated by buying (selling) a call option and selling (buying) a put option.
todo Merton Hull