for option pricing models. It has been used since 1973 and has direct ancestors dating to the work of Bachelier (1900) and Bronzin (1906). In terms of scientific theory, this age makes it a dinosaur. But just as dinosaurs were the dominant life form for about 190 million years for a reason, BSM has persisted because it is good.
We want an option pricing model for two reasons.
The first is so we can reduce the many, fast-moving option prices to a small number of slow-moving parameters. Option pricing models don't really price options. The market prices options though the normal market forces of supply and demand. Pricing models convert the market's prices into the parameters. In particular, BSM converts option prices to an implied volatility parameter. Now we can do all analysis and forecasting in terms of implied volatility, and if BSM was a perfect model, we would have a single, constant parameter.
The second reason to use a pricing model is to calculate a delta for hedging. Model-free volatility trading exists. Buying or selling a straddle (or strangle, butterfly, or condor, etc.) gives a position that is primarily exposed to realized volatility. But it will also be exposed to the drift. The most compelling reason to trade volatility is that it is more predictable than returns (drift) and the only way to remove this exposure is to hedge. To hedge we need a delta and for this we need a model. This is the most important criterion for an option trader to consider when deciding if a model is good enough. Any vaguely sensible model will reduce the many option prices to a few parameters, but a good model will let us delta hedge in a way that captures the volatility premium.
In this chapter we will examine the BSM model and see if it can meet this standard. By BSM model I mean the partial differential equation rather than the specific solution for European vanilla options. The particular boundary conditions and solution methods aren't a real concern here.
Derivations of BSM can be found in many places (see Sinclair, 2013, for an informal derivation). Here we will look at how the model is used.
Option Trading Theory
Here we will very briefly summarize the theory of option pricing and hedging. For more details refer to Sinclair (2010; 2013).
An option pricing model must include the following variables and parameters:
Underlying price and strike; this determines the moneyness of the option.
Time until expiration.
Any factors related to carry of either the option or the underlying; this includes dividends, borrow rates, storage costs, and interest rates.
Volatility or some other way to quantify future uncertainty.
A variable that is not necessary is the expected return of the underlying. Clearly, this is important to the return of an option, but it is irrelevant to the instantaneous value of the option. If we include this drift term, we will arrive at a contradiction. Imagine that we expect the underlying to rally. Naively, this means we would pay more for a call. But put-call parity means that an increase in call price leads to an increase in the price of the put with the same strike. This now seems consistent with us being bearish. Put-call parity is enough to make the return irrelevant to the current option price, but (less obviously perhaps) it is also enforced by dynamic replication.
This isn't an option-specific anomaly. There are many situations in which people agree on future price change, but this doesn't affect current price. For example, Ferrari would be justified in thinking that the long-term value of their cars is higher than their MSRP. But they can build the car and sell it at a profit right now. Their replication value as a manufacturer guarantees a profit without taking future price changes into account. Similarly, market-makers can replicate options without worrying about the underlying return. And if they do include the return, they can be arbed by someone else.
The canonical option-pricing model is BSM. Ignoring interest rates for simplicity, The BSM PDE for the price of a call, C, is
(1.1)
where S is the underlying price, σ is the volatility of the underlying, and t is the time until expiration of the option.
Or using the standard definitions where Γ is the second partial derivative of the option price with respect to the underlying and θ is the derivative of the option price with respect to time,
(1.2)
This is then solved using standard numerical or analytical techniques and with the final condition being the payoff of the particular option.
This relationship between Γ and θ is crucial for understanding how to make money with options. Imagine we are long a call and the underlying stock moves from St to St+1.The delta P/L of this option will be the average of the initial delta, Δ, and the final delta, all multiplied by the size of the move. Or
(1.3)
If this option was initially delta-hedged, the P/L over this price move would be
(1.4)
Next note that
(1.5)
so that the profit in hedging over each time interval is
(Although equation 1.6 is only asymptotically true, if we worked with an infinitesimal price change, this derivation would be exact.
This is the first term of the BSM differential equation. Literally, BSM says that these profits from rebalancing due to gamma are exactly equal to the theta of the option. Expected movement cancels time decay. The only way a directionally neutral option position will make money is if the option's implied volatility (which governs theta) is not the same as the underlying's realized volatility (which determines the rebalancing profits). This is true no matter which structure is chosen and the particulars of the hedging scheme.
If we can identify situations where this volatility mismatch occurs, the expected profit from the position will be given by
This is the fundamental equation of option trading. All the “theta decay” and “gamma scalping” profits and losses are tied up in this relationship.
Note also that this vega P/L will affect directional option trades. If we pay the wrong implied volatility level for an option, we might still make money but we would have been better off replicating the option in the underlying.
The BSM equation depends on a number
