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Home arrow Business & Finance arrow Frequently Asked Questions in Quantitative Finance

Why do Quants like Closed-Form Solutions?

Short answer

Because they are fast to compute and easy to understand.

Example

The Black-Scholes formula are simple and closed form and often used despite people knowing that they have limitations, and despite being used for products for which they were not originally intended.

Long answer

There are various pressures on a quant when it comes to choosing a model. What he'd really like is a model that is

• robust: small changes in the random process for the underlying don't matter too much

• fast: prices and the greeks have to be quick to compute for several reasons, so that the trade gets done and you don't lose out to a competitor, and so that positions can be managed in real time as just one small part of a large portfolio

• accurate: in a scientific sense the prices ought to be good, perhaps matching historical data; this is different from robust, of course

• easy to calibrate: banks like to have models that match traded prices of simple contracts

There is some overlap in these. Fast may also mean easy to calibrate, but not necessarily. Accurate and robust might be similar, but again, not always.

From the scientific point of view the most important of these is accuracy. The least important is speed. To the scientist the question of calibration becomes one concerning the existence of arbitrage. If you are a hedge fund looking for prop trading opportunities with vanillas then calibration is precisely what you don't want to do. And robustness would be nice, but maybe the financial world is so unstable that models can never be robust.

To the practitioner he needs to be able to price quickly to get the deal done and to manage the risk. If he is in the business of selling exotic contracts then he will invariably be calibrating, so that he can say that his prices are consistent with vanillas. As long as the model isn't too inaccurate or sensitive, and he can add a sufficient profit margin, then he will be content. So to the practitioner speed and ability to calibrate to the market are the most important.

The scientist and the practitioner have conflicting interests. And the practitioner usually wins.

And what could be faster than a closed-form solution? This is why practitioners tend to favour closed forms. They also tend to be easier to understand intuitively than a numerical solution. The Black-Scholes formula are perfect for this, having a simple interpretation in terms of expectations, and using the cumulative distribution function for the Gaussian distribution.

Such is the desire for simple formula that people often use the formula for the wrong product. Suppose you want to price certain Asian options based on an arithmetic average. To do this properly in the Black-Scholes world you would do this by solving a three-dimensional partial differential equation or by Monte Carlo simulation. But if you pretend that the averaging is geometric and not arithmetic then often there are simple closed-form solutions. So use those, even though they must be wrong. The point is that they will probably be less wrong than other assumptions you are making, such as what future volatility will be.

Of course, the definition of closed form is to some extent in the eye of the beholder. If an option can be priced in terms of an infinite sum of hypergeometric functions does that count? Some Asian options can be priced that way. Or what about a closed form involving a subtle integration in the complex plane that must ultimately be done numerically? That is the Heston stochastic volatility model.

If closed form is so appreciated, is it worth spending much time seeking them out? Probably not. There are always new products being invented and new pricing models being devised, but they are unlikely to be of the simple type that can be solved explicitly. Chances are that you will either have to solve these numerically, or approximate them by something not too dissimilar. Approximations such as Black '76 are probably your best chance of finding closed-form solutions for new products these days.

References and Further Reading

Black F 1976 The pricing of commodity contracts. Journal of Financial Economics 3 167-179

Haug, EG 2003 Know your weapon, Parts 1 and 2. Wilmott magazine, May and July

Haug, EG 2006 The Complete Guide to Option Pricing Formulas.

McGraw-Hill

Lewis, A 2000 Option Valuation under Stochastic Volatility. Finance Press

 
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