# Avellaneda and Paras

Marco Avellaneda and Antonio Paras were, together with Arnon Levy and Terry Lyons, the creators of the uncertain volatility model for option pricing. It was a great breakthrough for the rigorous, scientific side of finance theory, but the best was yet to come. This model, and many that succeeded it, was nonlinear. Nonlinearity in an option pricing model means that the value of a portfolio of contracts is not necessarily the same as the sum of the values of its constituent parts. An option will have a different value depending on what else is in the portfolio with it, and an exotic will have a different value depending on what it is statically hedged with. Avellaneda and Paras defined an exotic option's value as the highest possible marginal value for that contract when hedged with any or all available exchange-traded contracts. The result was that the method of option pricing also came with its own technique for static hedging with other options. Prior to their work the only result of an option pricing model was its value and its delta, only dynamic hedging was theoretically necessary. With this new concept, theory became a major step closer to practice. Another result of this technique was that the theoretical price of an exchange-traded option exactly matched its market price. The convoluted calibration of volatility surface models was redundant. See Avellaneda & Paras (1996).

# Brace, Gatarek and Musiela

Although the HJM interest rate model had addressed the main problem with stochastic spot rate models, and others of that ilk, it still had two major drawbacks. It required the existence of a spot rate and it assumed a continuous distribution of forward rates. Alan Brace, Dariusz Gatarek & Marek Musiela (1997) got around both of those difficulties by introducing a model which only relied on a discrete set of rates - ones that actually are traded. As with the HJM model the initial data are the forward rates so that bond prices are calibrated automatically. One specifies a number of random factors, their volatilities and correlations between them, and the requirement of no arbitrage then determines the risk-neutral drifts. Although B, G and M have their names associated with this idea many others worked on it simultaneously.

# Li

As already mentioned, the 1990s saw an explosion in the number of credit instruments available, and also in the growth of derivatives with multiple underlyings. It's not a great step to imagine contracts depending on the default of many underlyings. Examples of these are the once ubiquitous

Collateralized Debt Obligations (CDOs). But to price such complicated instruments requires a model for the interaction of many companies during the process of default. A probabilistic approach based on copulas was proposed by David Li (2000). The copula approach allows one to join together (hence the word 'copula') default models for individual companies in isolation to make a model for the probabilities of their joint default. The idea was adopted universally as a practical solution to a complicated problem. However with the recent financial crisis the concept has come in for a lot of criticism.

# Hagan, Kumar, Lesniewski and Woodward

There has always been a need for models that are both fast and match traded prices well. The interest-rate model of Pat Hagan, Deep Kumar, Andrew Lesniewski and Diana Woodward (2002), which has come to be called the SABR (stochastic, *a, jj*, p) model, is a model for a forward rate and its volatility, both of which are stochastic. This model is made tractable by exploiting an asymptotic approximation to the governing equation that is highly accurate in practice. The asymptotic analysis simplifies a problem that would otherwise have to be solved numerically. Although asymptotic analysis has been used in financial problems before, for example in modelling transaction costs, this was the first time it really entered mainstream quantitative finance.

# August 2007 quantitative finance in disrepute

In early August 2007 several hedge funds using quantitative strategies experienced losses on such a scale as to bring the field of quantitative finance into disrepute. From then, and through 2008, trading of complex derivative products in obscene amounts using simplistic mathematical models almost brought the global financial market to its knees: Lend to the less-than-totally-creditworthy for home purchase, repackage these mortgages for selling on from one bank to another, at each stage adding complexity, combine with overoptimistic ratings given to these products by the ratings agencies, with a dash of moral hazard thrown in, base it all on a crunchy base of a morally corrupt compensation scheme, and you have the recipe for the biggest financial collapse in decades. Out of this many people became very, very rich, while in many cases the man in the street lost his life savings. And financial modelling is what made this seem all so simple and safe.

# And Now a Brief Unofficial History!

Espen Gaarder Haug, as well as being an option trader, author, lecturer, researcher, gardener, soldier, and collector of option-pricing formulae, is also a historian of derivatives theory. In his excellent book *Derivatives: Model on Models* (John Wiley and Sons Ltd, 2007) he gives the 'alternative' history of derivatives, a history often ignored for various reasons. He also keeps us updated on his findings via his blog wilmott.com/blogs/collector. Here are a few of the many interesting facts Espen has unearthed.

**1688**** ****de**** *** la Vega* Possibly a reference to put-call parity. But then possibly not. De la Vega's language is not particularly precise.

* 1900s Higgins and Nelson* They appear to have some grasp of delta hedging and put-call parity.

* 1908 Bronzin* Publishes a book that includes option formula, and seems to be using risk neutrality. But the work is rapidly forgotten!

* 1915 Mitchell, 1926 Oliver and 1927 Mills* They all described the high-peak/fat-tails in empirical price data.

* 1956 Kruizenga and 1961 Reinach* They definitely describe put-call parity. Reinach explains 'conversion,' which is what we know as put-call parity, he also understands that it does not necessarily apply for American options.

* 1962 Mandelbrot* In this year Benoit Mandelbrot wrote his famous paper on the distribution of cotton price returns, observing their fat tails.

* 1970 Arnold Bernhard & Co* They describe market-neutral delta hedging of convertible bonds and warrants. And show how to numerically find an approximation to the delta.

For more details about the underground history of derivatives see Espen's excellent book (2007).

# References and Further Reading

Haug, EG 2007 *Derivatives: Models on Models,* John Wiley & Sons, Ltd.

Mandelbrot, B & Hudson, R 2004 *The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward.* Profile Books