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29-March-2006
Alex Castaldo reviews Option Pricing and Spikes in Volatility, a paper by Paola Zerilli

This paper by a graduate student at Boston College should be looked at as the latest in a large (and growing) series that tries to develop mathematical models of option pricing in the Black Scholes tradition but with additional advanced features. I thank Dr. Zachar for sending it to me but I would caution that this research is so new and advanced that only the people active in this field (of which I am not one) can tell us if this particular paper represents a real advance or not. AFAIK the Duffie-Pan-Singleton and Eraker papers are considered the state of the art at the present time.

Let us take a giant step backwards and start at the beginning.

Diffusion processes and jump processes

In the mathematical theory of stochastic processes (which has a lot of applications other than options) a basic distinction is made between processes that change randomly but continuously and those that are discontinuous. Brownian motion is the protoypical example of a "diffusion", that is a continuous process.

Drag a pen across the paper while making random shaking motions with your hand and you have drawn the path of a diffusion. On the other hand the Poisson Counting Process is the prototype of a pure jump process. It remains at zero until a poisson event occurs, at which time it jumps discontinuously) to the value 1, it stays there for a random (exponentially distributed) time, then jumps to 2, and so on. Of course mixed processes containing both a diffusion and a jump component are possible; such processes are continuous most of the time but once in a while make a jump as well.

Twenty second summary of option pricing models

If we assume that stock prices follow some well defined random process, we can derive formulas to value options.

CV-NJ Black and Scholes assumed that stocks follow a particular diffusion, called GBM. This diffusion has a constant volatility and (like all diffusions) is continuous, so we can call it the CV-NJ case. Constant volatility, no jumps.

CV-JP A few years after Black-Scholes, Merton came up with a model in which the stock price can make random jumps at random times. The volatility is still assumed to be constant. This was the first model of the "CV-JP" class, constant volatility, jumps in price. Because real-life one-day moves in stock prices are larger than is consistent with a long term diffusion, this model achieves greater realism in the prices of very short term options (a few days to maturity); the prices of long term options are not affected, because the jumps come out in the wash in the long run.

SV-NJ The constant volatility assumption of Black and Scholes is clearly contradicted by the data. (The volatility of US stock prices in 2005 was clearly much lower than in 2002). To remedy this, Stochastic Volatility models were created. The volatility itself is assumed to change randomly, according to another diffusion. So there are two stochastic equations in these models: one for stock prices and one for volatility. Both are continuous, so these models are SV-NJ: stochastic volatility, no jumps. The best known and most often used SV-NJ model is the Heston model; this model is more accurate than the previous in pricing long term options. However the "smile" produced by this model does not quite match the "smile" observed in reality. So back to the drawing board.

SV-JP By combining the possibility of jumps in price with a randomly changing volatility, it would seem possible to match long term and short term option prices well. Unfortunately these models (Stochastic volatility, jumps in price) were not successful in matching the observed option prices either.

SV-JPJV The next (and latest) step are models that allow the volatility to change according to a mixed diffusion/jump process while the stock price follows another process with diffusion and jumps. These models are called Stochastic Volatility with Jumps in Price and Jump in Vol (SV-JPJV). Duffie Pan Singleton is an example. These models are quite complex:

1. there are many choices to be made in designing these models: are the jumps in vol and price correlated or independent, is there are a risk premium associated with the jumps or not, etc. etc.
2. the models are quite difficult to solve and estimate, often requiring very advanced math and econometrics; it is a very rarefied area of research with only a handful of participants, of which Ms. June Pan of MIT is perhaps the best. Even with all this work, the SV-JPJV models, while an improvement over previous ones, do not spit out the exact prices of options traded on the CBOE either!

This paper's critique of (and improvement on?) Duffie-Pan-Singleton

The interesting point to me in this paper is the observation that Duffie-Pan-Singleton, the king of the option models at this time, may be flawed. The DPS model allows volatility to jump up, as occurred for example after the market break of October 1987. But we often see the Implied Vol jump up one day (a day of falling stock prices) and then jump back down the next day. The author develops another SV-JPJV model in which volatility can jump both up and down. To her credit she is able to solve and estimate this model and show some interesting results in reproducing the implied volatility spikes.

But the question I have is does it really work this way? Does God decide one day: "I am going to increase the underlying vol for the US market from now on. I am allowed to do this from time to time." Note that no human being knows what this volatility is, but the implied vol is the market's attempt to estimate this latent vol. Then the next day God says "Ha! Ha! Ha! I have changed my mind, I am going to change the underlying vol for the US market again, this time down, back to the previous value". The market senses this and ratchets the IV back down. It certainly might work this way, but it does not seem reasonable to me.

Does it not make more sense that the one day spike in IV is caused by panic amongst human beings and that God's underlying vol is unchanged. I am willing to believe that the real vol changes gradually in response to the state of the economy, the earnings season, etc. and that once in a while it jumps randomly for some reason (maybe we reset our understanding of the economy's vol from time to time, based on a particularly significant announcement). But a vol that is always jumping up and down seems like a "deus ex machina" to generate IV spikes. If there is an underlying vol it must be determined by some significant economic factors and not just be arbitrary on any day.

Of course it is difficult or impossible to model this since it would mean that IV is at times a mistaken estimate (i.e. a non-optimal estimate) of the real underlying vol. The process of making mistakes would itself have to be modeled mathematically in some way. Good luck.

This line of option research may be barking up the wrong tree

In addition to the great complexity of these models, there may be another reason why this line of research may not be the best to pursue. Recall that what these people are trying to do is to write down a mathematical description of stock price behavior, from which equations giving option prices will mechanically follow. In other words options are purely derivative securities whose price is determined by the stock price process. These researchers believe that if only we could know what the real process driving the S&P 500 is, we could derive the prices of listed options to the penny.

An alternative point of view is that options are not completely derivative. They are subject to the laws of supply and demand, just like apples and every other commodity. For example there could be hedging pressures that arise and affect option prices. There could be temporary factors, squeezes of one sort or another, that affect option prices. Researchers like Bates, Posteshman and others have pursued such ideas. Dynamic hedging is a form of arbitrage that tends to counteract these pressures, but every form of arbitrage has limits, so there is some room for demand and supply to have an effect on the prices of options. This line of option research seems more promising to me.