Stochastic Processes and Financial Mathematics
(part two)
17.2 American and Exotic options \(\offsyl \)
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 \(T\) of an option was agreed in advance, and was deterministic. American options are options in which the holder can choose when to exercise the option. For example, an American call option with strike \(K\) and ‘final’ exercise date \(T\), gives the holder the right to buy one unit of stock \(S_t\) for a (pre-agreed, deterministic) strike price \(K\) at a time of their own choosing during \([0,T]\).
The time \(\tau \) at which the holder chooses to exercise their right to buy may depend on the current value of the stock price. As a result, the argument that we used to derive the prices (in Section 15.3) breaks down. The underlying cause is that they relate to a family of PDEs that are much more difficult to handle than the standard Black-Scholes PDEs – and for which Feynman-Kac formulas are not known to exist. In some special cases (including the case of American call options) explicit hedging strategies that are known, which allow explicit formulas for prices to be found, but in general numerical techniques are the only option.
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Remark 17.2.1 \(\msconly \) 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 by
\(\seteqnumber{0}{17.}{0}\)\begin{equation} \label {eq:op_stop_prob} \max _\tau \E ^\Q \l [e^{-r\tau }\Phi (S_\tau )\r ] \end{equation}
where the max is taken over stopping times \(\tau \). The optimal stopping problem is to identify the stopping time \(\tau \) 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, \(\ldots \)). Often such options are not traded on stock exchanges because there is insufficient demand for their individual characteristics. As a result, their prices are agreed through direct discussions between the two (or more) parties involved in the contract. We don’t attempt to make a catalogue of the theory of pricing such options. If you want to see some examples, there are plenty in Chapters 11-14 of the book ‘The Mathematics of Financial Derivatives’ by Wilmott, Howison and Dewynne.
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Remark 17.2.2 In Section 15.2 we were careful to write our contingent claims as \(X\), rather than \(\Phi (S_T)\). 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 \(X\in m\mc {F}_T\) (and not on having the form \(X=\Phi (S_T)\)).
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.