Stochastic Processes and Financial Mathematics
(part two)
17.2 American and Exotic options
We have generally used European call (and sometimes put) options as our canonical examples of financial derivatives. Whilst European call and put options are traded on many stocks, the most common type of financial derivative is actually the American call/put option.
In all our previous work, we assumed that the exercise time
The time
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Remark 17.2.1
To be precise, what American options lead too, in place of the risk-neutral valuation formula, is a so-called ‘optimal stopping problem’: their value is given bywhere the max is taken over stopping times
. The optimal stopping problem is to identify the stopping time at which the maximum occurs.Optimal stopping problems are generally quite difficult, and the mathematics needed to attack (17.1) is outside of the scope of what we can cover in this course.
Another interesting class of financial derivatives are exotic options. This is a general term used for options in which the contingent claim is either complicated to write down, or simply of an unusual form. They include
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1. digital options, which give a fixed payoff if (and only if) the stock price is above a particular threshold;
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2. barrier options, in which the exercise rights of the holder vary according to whether the stock price has crossed particular thresholds;
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3. Asian options, whose payoff depends on the average price of the underlying asset during a particular time period;
and many other variants (such as cliquet, rainbow, lookback, chooser, Bermudean,
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Remark 17.2.2 In Section 15.2 we were careful to write our contingent claims as
, rather than . The reason is that the argument we gave for the Theorem 15.2.5, which essentially stated that the Black-Scholes market was complete, relied only on the fact that (and not on having the form ).As a consequence, we do know that most types of exotic option can be hedged in the Black-Scholes model – but, in general, we don’t know how to find the replicating portfolios.