Option Trading: Pricing and Volatility Strategies and Techniques

CHAPTER 5

The Solution of the Black-Scholes-Merton (BSM) Equation

It is very important that traders understand how option prices and the various greeks (derivatives of the option prices) behave. It is understandable if a trader cannot derive a pricing model, but all traders need a thorough knowledge of how option values change in response to changes in the inputs to the model. Here we give equations that describe various aspects of option behavior. We illustrate these results graphically and discuss them. Then we can use our knowledge to look further at the theory of option trading that we introduce in Chapter 4 during our derivation of the Black-Scholes-Merton (BSM) model.

Some of the discussion in this chapter might seem like overkill, and indeed many successful traders only know the characteristics of the major greeks: delta, gamma, theta, and vega. Normally this is all that is required and focusing on the higher-order greeks is distracting. An analogy that I heard recently might be useful. A good pilot normally flies a plane by looking out the window, and by watching and feeling the aircraft’s movement and orientation. He rarely glances at the instruments. However, if he needs to fly through a cloud the visual method is useless and his spatial perception will fail. Now he needs to rely on the instruments and he needs to understand what they do. This is not a good time to learn what an altimeter does. Similarly, there will come a time in a trader’s career (possibly in a job ...