we have found. We need to have an idea of what to expect. It is quite possible to be right with our volatility forecasts and still lose money. When we hedge, we become exposed to path dependency of the underlying. It matters if a stock move occurs close to the strike when we have gamma or away from a strike when we have none. If we don't hedge, we are exposed to only the terminal stock price, but we can still successfully forecast volatility and lose because of an unanticipated drift. Or we can successfully forecast the return and lose because of unanticipated volatility.
Chapter Six discusses volatility trading structures. We look at the P/L distributions of straddles, strangles, butterflies, and condors, and how to choose strikes and expirations.
In Chapter Seven we look at trading options directionally. First, we extend the BSM model to incorporate our views on both the volatility and return of the underlying. This enables us to consistently compare strikes on the basis of a number of risk measures, including average return, probability of profit, and the generalized Sharpe ratio. Chapter Eight examines the P/L distributions of common directional option structures.
The final section is about risk. Good risk control can't make money. Trading first needs edge. However, bad risk management will lead to losses.
Chapter Nine discusses trade sizing, specifically the Kelly criterion. The standard formulation is extended to allow for parameter estimation uncertainty, skewness of returns, and the incorporation of a stop level in the account.
The most dangerous risks are not related to price movement. The most dangerous risks are in the realm of the unknowable. Obviously, it is impossible to predict these, but Chapter Ten explores some historical examples. We don't know when these will happen again, but it is certain that they will. There is no excuse for blowing up due to repeat of a historical event.
It is inevitable that you will be wrong at times. The most dangerous thing is to forget this.
Summary
Find a robust source of edge that is backed by empirical evidence and convincing reasons for its existence.
Choose the appropriate option structure to monetize the edge.
Size the position appropriately.
Always be aware of how much you don't know.
CHAPTER 1 Options: A Summary
Option Pricing Models
Since all models are wrong the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned about mice when there are tigers abroad.
—Box (1976)
Some models are wrong in a trivial way. They clearly don't agree with real financial markets. For example, an option valuation model that included the return of the underlying as a pricing input is trivially wrong. This can be deduced from put-call parity. Imagine a stock that has a positive return. Naively this will raise the value of calls and lower the value of puts. But put-call parity means that if calls increase, so do the values of the puts. Including drift leads to a contradiction. That idea is trivially wrong.
Every scientific model contains simplifying assumptions. There actually isn't anything intrinsically wrong with this. There are many reasons why this is the case, because there are many types of scientific models. Scientists use simplified models that they know are wrong for several reasons.
A reason for using a wrong theory would be because the simple (but wrong) theory is all that is needed. Classical mechanics is still widely used in science even though we now know it is wrong (quantum mechanics is needed for small things and relativity is needed for large or fast things). An example from finance is assuming normally distributed returns. It is doubtful anyone ever thought returns were normal. Traders have long known about extreme price moves and Osborne (1959), Mandelbrot (1963), and others studied the non-normal distribution of returns from the 1950s. (Mandelbrot cites the work of Mitchell [1915], Olivier [1926], and Mills [1927], although this research was not well known.) The main reason early finance theorists assumed normality was because it made the equations tractable.
Sometimes scientists might reason through a stretched analogy. For example, Einstein started his theory of the heat capacity of a crystal by first assuming the crystal was an ideal gas. He knew that this was obviously not the case. But he thought that the idea might lead to something useful. He had to start somewhere, even if he knew it was the wrong place. This model was metaphorical. A metaphorical model does not attempt to describe reality and need not rely on plausible assumptions. Instead, it aims to illustrate a non-trivial mechanism, which lies outside the model.
Other models aim to mathematically describe the main features of an observation without necessarily understanding its deeper origin. The GARCH family of volatility models are phenomenological, and don't tell us why the GARCH effects exist. Because these models are designed to describe particular features, there will be many other things they totally ignore. For example, a GARCH process has nothing to say about the formation of the bid-ask spread. The GARCH model is limited, but not wrong.
The most ambitious models attempt to describe reality as it truly is. For example, the physicists who invented the idea that an atom was a nucleus around which electrons orbited thought this was actually what atoms were like. But they still had to make simplifying assumptions. For example, when formulating the theory, they had to assume that atoms were not subject to gravity. And, in only trivial situations could the equations be analytically solved. The Black-Scholes-Merton (BSM) model was meant to be of this type.
But it isn't used that way at all.
The inventors of the model envisaged that the model would be used to find a fair value for options. Traders would input the underlying price, strike, interest rate, expiration date, and volatility and the model would tell them what the option was worth. The problem was that the volatility input needed to be the volatility over the life of the option, an unknown parameter. Although it was possible for a trader to make a forecast of future volatility, the rest of the market could and did make its own forecast. The market's option price was based on this aggregated estimate. This is the implied volatility, which became the fundamental parameter. Traders largely didn't think of the model as a predictive valuation tool but just as an arbitrage-free way to convert the quickly changing option prices into a slowly changing parameter: implied volatility. For most traders, BSM is not a predictive model; it is just a simplifying tool.
This isn't to say that BSM can't be used as a pricing model to get a fair value. It absolutely can. But even traders who do this will think in volatility terms. They will compare the implied volatility to their forecast volatility, rather than use the forecast volatility to price the option and compare it to the market value. By using the model backwards, these traders still benefit from the way BSM converts the option prices into a slowly varying parameter.
We need to examine the effects of the model assumptions in light of how the model is used. Although the assumptions make the model less realistic, this isn't important. The model wasn't used because it was realistic; it was used because it was useful.
Obviously, it is possible to trade options without any valuation model. This is what most directional option traders do. We can also trade volatility without a model. Traders might sell a straddle because they think the underlying will expire closer to the strike than the value of the straddle. However, to move beyond directional trading or speculating on the value of the underlying at expiration we will need a model.
The BSM model
