The Black-Scholes Formula is Wrong! - Part 3/12 - Formula Derivation

Table of Contents

Part 1 - Overview and Background
Part 2 - Axioms
Part 3 - Formula Derivation
Part 4 - The Put/Call Parity Formula
Part 5 - The Volatility Smile
Part 6 - The Contradiction
Part 7 - Resolving the Contradiction
Part 8 - The Kelly Criterion
Part 9 - How FSK Trades Options
Part 10 - Only Fools and Hedge Funds Write Covered Calls
Part 11 - Other Options
Part 12 - Summary

Armed with all the axioms, we can now price an option. The Black-Scholes formula uses a delta-hedging approach. If we own a call option, we will simultaneously have a short stock position of 0-100 shares. Similarly, if we own a put option, we will simultaneously have a long stock position of 0-100 shares; for short calls and short puts, we have a long stock or short stock hedge position, respectively.

A call and a put are actually nearly equivalent. If I own a call, I can short sell 100 shares of stock and now my payoff is as if I owned a put. With American-style options, they are not 100% equivalent due to the early exercise right.

Consider the example where we buy a call and delta hedge it with short stock. We paid money to buy the call option, but we will collect interest on our short stock position and we will make money as we adjust our hedge. As we adjust our hedge, we will be selling more shares short as the stock goes up and buying shares as the stock goes down. As we adjust our hedge, we are buying low and selling high; the more volatility there is, the more we will profit as we adjust our hedge. This practice is usually known as "gamma scalping".

A deep in-the-money put is like a promise to pay the strike at expiration. Money now is better than money later, so if you have a deep-in-the-money American-style put, you should early exercise it. Similarly, if you have a deep in-them-money American-style call *AND* the stock pays a dividend, it pays to early exercise the option to collect the dividend.

A common mistake in many options textbooks is "An American-style call option on a stock that pays no dividend is equivalent to a European option. You never early-exercise an American-style call option on a stock that pays no dividend." This is true in the abstract theoretical model. It is false in practice, because you pay short interest fees on your hedge. If you own a call option that is sufficiently deep-in-the-money, it pays to early-exercise it. First, you avoid the short interest fees. Second, you free up capital that can be more profitably used elsewhere.

Our goal is that, at any given instant, we are indifferent to an epsilon percent increase or an epsilon percent decrease in the stock price. At the next instant, we will make a trade and adjust our hedge to our new target. If our volatility prediction was accurate, then our actual realized profit on our hedge will equal the premium we paid for the option and break even.

You can construct a tree, based at the current stock price, that shows the future possible paths of the stock price. The volatility assumption and expected growth rate tell you how to build the tree. In practice, 100 levels is enough to price an option. For European options, you can use the calculus trick and take the limit as the step size goes to zero and get an exact price. You can't get an exact formula for American options, because of the difficulty in pricing the early exercise right. The "Should I exercise early?" decision is evaluated at each node of the tree, by comparing the option price at future nodes to the value of immediately exercising.

By setting up the tree correctly, you can make a recombining binomial tree. This way, for an n level tree, there are only O(n^2) nodes to evaluate, instead of O(2^n) nodes. This makes the calculation feasible. In other words, suppose you move up at T=0 and down at T=1. Suppose you move down at T=0 and up at T=1. These have the same value, if you set up the tree correctly. The volatility tells you the width of the tree and the expected gain tells you how much the price should increase on average. An up move and a down move are each given a 50% chance.

What is the correct hedge? At expiration, it is obvious. If the option is in-the-money, the correct hedge is 100 shares. If the option is out-of-the-money, the correct hedge is 0 shares. There is a singularity when the strike equals the current stock price, but that's mostly irrelevant. Then, we work backwards in time to figure out the hedge at times prior to expiration. Once you know your expected future hedge, you know your expected future hedging profit, and you can price the option. At each node of the tree, we can calculate what our gain or loss on our hedge will be in the future, and that's the price of the option at that time. The option price is the expected gain from hedging over the life of the option. Money in the future is worth less than money in the present. The future cashflow must be discounted back to the present, and the risk-free interest rate is usually used for this purpose.

Conversely, if we were short the option, we would have collected some money as we sold the option and gradually lost that money as we adjusted our hedge. When we are short an option, we are buying high and selling low as we adjust our hedge.

Notice that the actual expected gain in the stock is practically irrelevant. It drops out of the equation due to our hedging policy. That's the reason professional options traders assume that the expected gain in the stock equals the risk-free interest rate. There's another reason, called the "put/call parity formula", which I mention and debunk in the next post.

There are two benefits from hedging. The first benefit is that it reduces your variance. If the variance of your trading system is lower, then it's easier to tell if you have a positive expectation trading system. The second benefit of hedging is that a hedged position receives much more favorable margin treatment than an unhedged position. This makes sense, because the hedged position is much less risky compared to an unhedged option position. One drawback of hedging is that you pay hedging costs. You pay short interest fees on a short stock position, and you pay transaction costs every time you trade the underlying stock to adjust your hedge.

You can think of the call option price as a function CP(S, X, t, v, r), where

S is the current stock price
X is the strike price
t is the time to expiration
v is the volatility assumption
r is the interest rate assumption and assumption for the expected growth rate of the underlying

Professional options traders always use the same value for "interest rate assumption" and "expected future growth rate of the underlying" and "rate at which we discount future cashflow to the present". The reason they do this is the put/call parity formula.

Options traders assign all of the partial derivatives of this function a name, as a shorthand. They are named after Greek letters.

dCP/dS is called delta
d(delata)/dS = d^2(CP)/dS^2 is called gamma, the origin of the name "gamma scalping"
dCP/dt is called theta
dv/dt is called vega
dr/dt is called rho

Delta is the sensitivity of the option price to changes in the underlying stock price. Professional options traders use the value of delta as their hedge.

Gamma is the second derivative of the option price relative to changes in the underlying stock price. Gamma tells you how much you would gain or lose if the stock gapped up or down a huge amount in one day, before you can adjust your hedge. When you profit from adjusting your hedge, you are taking advantage of the concavity of the option price function; this is the reason that is called "gamma scalping". "Gamma scalping" itself is not profitable; you are merely reducing your variance, minus your transaction costs.

Theta tells you how much the time-value of the option is eroding each day, assuming the underlying price does not change.

Vega tells you the sensitivity to changes in the volatility assumption. Notice that "vega" is not actually a Greek letter, although it sounds Greek to me.

Rho tells you the sensitivity to changes in the interest rate assumption. Professional options traders always use the same value for interest rate assumption and assumed growth rate in the underlying stock.

When thinking about these "Greeks", you should just think in terms of partial derivatives.

In my next post, I write about the put/call parity formula.