The Jump-Diffusion Pricing Formula

The flaws are not singular, but multiple. At least 9 flaws are easily identified. These are

1. Focus on European exercise only;
2. No allowance for dividends;
3. Calculation for calls only, and not for puts;
4. Assumption that option profits are not taxed;
5. No transaction fees are applied;
6. Interest rates do not change over time;
7. Trading is continuous, and no price gaps occur;
8. Price movement is normally distributed; 
9. Volatility does not change over the life of an option.
    

All these flaws bring into question the basic reliability of the BSM model. Even so, most subsequent models were built on and expanded beyond the BSM itself, so that even when a single flaw was adjusted for, the others remained. It would be impossible to construct a fully reliable model, because an infinite number of variables apply for dividends, taxes, transaction fees, interest rates, distribution of outcomes, and most of all, volatility.

Among the later adjustments was the Jump-Diffusion Model, developed by Robert C. Merton in 1976. Merton observed that stock returns tend to exhibit fat tails and not normal distribution. The BSM did not allow for these fat tails. By adding in a jump component to the calculation, Merton approximated what is called Brownian motion, which captures the nature of a random walk. It describes the random movement of stocks or particles or other forms of matter, and that movement is not predictable. Moreton’s new model was intended to account for unexpected changes in the underlying price caused by developments of new information.
 

This study of discontinuous price movement, or jumps, often is based on a momentum-based trading strategy. This involves buying positions that were profitable during the past 3 months (or a full year), based on a belief that the trends will be likely to repeat. In comparison, losing positions under this strategy are not likely to reverse and become winners. Momentum is believed to work both for winners and losers.

Employing a jump approach assumes, correctly, that volatility is stochastic and not predictable. The assumption in BSM that volatility remains unchanged for the life of the option is simply unrealistic, and that is the greatest challenges to any traders using BSM to determine whether an option is reasonably or unreasonably priced. When random jumps and exceptionally large jumps occur (and they do regularly), the assumptions of BSM are demonstrably flawed, to the point of complete inaccuracy. Although Merton’s jump-diffusion model is designed to modify assumptions for these events, they cannot be anticipated with any reliability.

Options traders are constantly seeking clear answers to the dilemma of unpredictable volatility. However, there are no answers. It is the risk unavoidably connected with all trading, and especially for options. It might be the case that options are priced assuming traders and investors are risk-neutral, so that BSM could work or at least be applied equally to all traders. But BSM has three specific assumptions that make this impossible. They include (1) no jumps occur in pricing of the option; (2) the underlying price volatility is not related to other variables affecting the underlying price (for example, current news, earnings, dividends announcements, legal actions, etc.); and (3) risk-free interest rates are universally applied and will not change.

These assumptions, all false, are necessary on order to be able to develop a model without the influence of infinite variables. However, anyone relying on this modeling assumptions must realize that it destroys reliability in BSM or any other model. Jump-diffusion at least provides a partial remedy. As new information becomes known, this model is supposed to account for it. However, the full range and scope of these developments can never be fully known by traders. As one new variable is introduced and taken into the calculation, what prevents another, or two or three others, from occurring at the same time?

The truth is that options trading, even using jump-diffusion in place of BSM, is not a solution as much as a form of fine-tuning and attempting to offset known flaws and variables as they arise. However, forms of jump risk also evolve. BSM was developed assuming only one form of systematic risk was at play, that of risks in the underlying asset’s pricing and volatility. But volatility and jump risk may have greater influence over option pricing than anything involving developments in the underlying price.

Merton’s model assumed that the jump risk was specific to each company and could be diversified to reduce it. But when markets behave with great volatility and rapid price changes, Merton’s assumption is questionable. It may be that his formula replaced one set of flawed assumptions with another set, and that no formula would ever be able to eliminate flows well enough to provide a fully reliable result.

Most traders adopt a belief that risk is always best managed through diversification, and to a degree this is true. However, it does not extend in every case to pricing of options or even of the underlying. Diversification does not always offset even known risks, and when modeling is employed to define option pricing, new uncertainties are introduced, no matter which models are used.

No pricing model is going to accurately adjust the flaws introduced by BSM or jump-diffusion. The complexity of the entire process is a struggle with an unknown set of variables and risks, including jump risk, and no matter how detailed the analysis, modeling itself cannot be used to eliminate the need for common sense in trading. Options traders usually have good instincts about level of risk, even without employing pricing models. The question becomes a most important one: Is a trader wise enough to listen to these instincts, or are they impulsive enough to ignore their own common sense?

Michael C. Thomsett is a widely published author with over 80 business and investing books, including the best-selling Getting Started in Options, coming out in its 10th edition later this year. He also wrote the recently released The Mathematics of Options. Thomsett is a frequent speaker at trade shows and blogs on his website at Thomsett Publishing as well as on Seeking Alpha, LinkedIn, Twitter and Facebook.