# Merton, again

In 1974 Robert Merton (Merton, 1974) introduced the idea of modelling the value of a company as a call option on its assets, with the company's debt being related to the strike price and the maturity of the debt being the option's expiration. Thus was born the structural approach to modelling risk of default, for if the option expired out of the money (i.e. assets had less value than the debt at maturity) then the firm would have to go bankrupt.

Credit risk became big, huge, in the 1990s. Theory and practice progressed at rapid speed during this period, urged on by some significant credit-led events, such as the Long Term Capital Management mess. One of the principals of LTCM was Merton who had worked on credit risk two decades earlier. Now the subject really took off, not just along the lines proposed by Merton but also using the Poisson process as the model for the random arrival of an event, such as bankruptcy or default. For a list of key research in this area see Schonbucher (2003).

# Boyle

Phelim Boyle related the pricing of options to the simulation of random asset paths (Figure 1.2). He showed how to find the fair value of an option by generating lots of possible future paths for an asset and then looking at the average that the option had paid off. The future important

Figure 1.2: Simulations like this can be easily used to value derivatives.

role of Monte Carlo simulations in finance was assured. See Boyle (1977).

# Vasicek

So far quantitative finance hadn't had much to say about pricing interest rate products. Some people were using equity option formulae for pricing interest rate options, but a consistent framework for interest rates had not been developed. This was addressed by Vasicek. He started by modelling a short-term interest rate as a random walk and concluded that interest rate derivatives could be valued using equations similar to the Black-Scholes partial differential equation.

Oldrich Vasicek represented the short-term interest rate by a stochastic differential equation of the form

The bond pricing equation is a parabolic partial differential equation, similar to the Black-Scholes equation. See Vasicek (1977).

# Cox, Ross and Rubinstein

Boyle had shown how to price options via simulations, an important and intuitively reasonable idea, but it was these three, John Cox, Stephen Ross and Mark Rubinstein, who gave option-pricing capability to the masses.

The Black-Scholes equation was derived using stochastic calculus and resulted in a partial differential equation. This was not likely to endear it to the thousands of students interested in a career in finance. At that time these were typically MBA students, not the mathematicians and physicists that are nowadays found on Wall Street. How could MBAs cope? An MBA was a necessary requirement for a prestigious career in finance, but an ability to count beans is not the same as an ability to understand mathematics. Fortunately Cox, Ross and Rubinstein were able to distil the fundamental concepts of option pricing into a simple algorithm requiring only addition, subtraction, multiplication and (twice) division. Even MBAs could now join in the fun. See Cox, Ross & Rubinstein (1979) and Figure 1.3.

# -81 Harrison, Kreps and Pliska

Until these three came onto the scene quantitative finance was the domain of either economists or applied mathematicians. Mike Harrison and David Kreps, in 1979, showed the relationship between option prices and advanced probability theory, originally in discrete time. Harrison and Stan Pliska in 1981 used the same ideas but in continuous time. From that moment until the mid 1990s applied mathematicians hardly got a look in. Theorem,

Figure 1.3: The branching structure of the binomial model.

proof everywhere you looked. See Harrison & Kreps (1979) and Harrison & Pliska (1981).