Models for Options and Contingent Claims

While financial options have existed for 2000 years (Greeks dealt in options on olive production), their use increased greatly after Black-Scholes-Merton introduced a mathematical model for valuing options in 1972. Importantly, this model also provided a mechanism for hedging the risk associated with holding an option. The ability to hedge meant that market-makers dealing in options became equally willing to buy or sell options, regardless of whether they were bullish or bearish on the stock (or commodity or bond) from which the option derived its value. Rather than betting on market direction, they could make money on the spread between the bid and ask price for an option. There are many natural financial uses for options, and with willing middle-men in the form of market makers, option market volume increased exponentially. Soon there developed new kinds of options (and contingent claims and structured products) – designed to satisfy virtually any financial need or to express any view on market direction.

The classic call and put options became known as “plain vanilla” options and a zoo of “exotic” options sprang into existence. More complex options required more complex models and people with analytical¬†and computational modeling skills developed in physics or mathematics were in demand to develop and implement models — and to assess the risk associated with the use of those models. It was in that context that I entered the financial world in 1995, after over 20 years as a plasma physicist. The notebooks below describe some of the models and modeling concepts on which I worked for banks who dealt in options.

17-6 Spread Options 02-14-11

Spread options are contingent claims on the difference in the prices of two or more underliers. Such contingencies arise in quite natural ways and have relevance in many financial contexts. In this notebook the classic call option on the price difference is discussed and double quadrature model for its valuation is developed. A closed form is obtained for the special case of a Margrabe exchange option. Models are also given for Bachelier’s approximation and Kirk’s approximation.

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17-7 Compound Options - 08-20-16

A compound option is an option on an option. Valuation models are developed for the simple call-on-call compound options. Two numerical models are developed for the COC, as well as a semi-closed model. The derivation of the latter requires a rather long and complicated calculation that is facilitated by the use of symbolic manipulation. Integrals and identities involving the bivariate probability distribution are derived along the way.

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17-8 Worst of Call Option - Three Ways 07-30-16

Three different models are derived for a call option on the worst performing of two equity underliers. First, a brute-force double numerical quadrature model is implemented. Second, a single numerical quadrature model over the minimum of two underliers is developed – but that requires deriving the risk-neutral distribution function for that minimum. Finally, a closed form expression in terms of bivariate normal distribution functions is derived for this option. This derivation follows a quite involved early calculation by Stulz. Several integrals and identities have to be derived and the details of the derivation process demonstrate the utility of Mathematica in such contexts.

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17-40 Moment matching for Asian Options 02-22-17

An approximation method called moment matching is frequently used to develop fast evaluation option models for Asian options (average option prices). I illustrate this approximation for continuously and discretely sampled averages.

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