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2.1.4 Exotic options $ \diamondsuit $

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People generally refer to an exotic option when the contract is not a plain vanilla put or call that are traded on an open exchange and is instead traded over-the-counter (OTC).

* Binary or digital options
may be the simplest form of exotic contracts: they only differ from the vanilla options by the terminal payoff L(S)   that can be any positive function of the asset price S. Some binaries can be obtained from the superposition of vanilla options: straddles, bullish / bearish vertical spreads and butterfly spreads are the subject of exercises 2.05-2.07. Others have payoffs that remind well known functions, such as the cash-or-nothing call reproducing the Heavyside H(x)

$\displaystyle \Lambda_\mathrm{cash-or-nothing} =b \mathcal{H}(S-K)$ (2.1.4#eq.1)

and the supershare reminding the Dirac delta function

$\displaystyle \Lambda_\mathrm{supershare} =\frac{1}{d}[\mathcal{H}(S-K) -\mathcal{H}(S-K-d)]$ (2.1.4#eq.2)

with terminal payoff diagrams illustrated in (2.1.4#fig.1).
Figure 2.1.4#fig.1: Example of binary / digital options with a general terminal payoff $ \Lambda(S)$ : a cash-or-nothing call (left) and a super-share (right).
\includegraphics[width=6cm]{figs/payCashNothing.eps}        \includegraphics[width=6cm]{figs/paySupershare.eps}

This generalization does not present any formal difficulty, but the discontinuities in terminal payoff L(S)   do however seriously stretch the non-arbitrage arguments that will be used to derive the Black-Scholes equation in chapter 3. Indeed, a large amount of cash would be required to hedge small changes in the price of the underlying as the option jumps from zero to a finite value, only to fall back to zero shortly afterwards.
* Compound options
can be understood as options on options. In the simplest case, this involves only put and call options and leads to four types compound options. For example, the call-on-put carries the right at time T1   to purchase for a price K1   a put option P2(K2,T2).   The coming chapters will show how to calculate the value of a put option before it expires; denoting this as VP2(S,T1),   the payoff at expiry T1   gets

$\displaystyle \Lambda_\mathrm{call-on-put} =V(S,T_1)=\mathrm{max}(V_\mathrm{P_2}(S,T_1)-K_1,0).$ (2.1.4#eq.3)

Since only one random variable governs the underlying asset price S and its derivatives, the value of a compound option can be calculated by solving first for the value of the option that may be bought or not, e.g. P2(K2,T2);   inserting this solution into (2.1.4#eq.3), the value of the compound option is then obtained as usual.
* Chooser options
are an extension of compound options, giving its holder the right at time T1   to purchase for an amount K1   either a call C2(K2,T2) or a put P2(K2,T2)   Going through all the possibilities, the same reasoning shows that at expiry T1  

$\displaystyle \Lambda_\mathrm{chooser} =V(S,T_1) =\mathrm{max}(V_\mathrm{P_2}(S,T_1)-K_1, V_\mathrm{C_2}(S,T_1)-K_1)$ (2.1.4#eq.4)

* Barrier options
are characterized by a a condition set on the existence of the option. When triggered, the right to exercise the option either appears (in) or disappears (out) if the asset price is above (up) or below (down) a prescribed barrier B: Barrier options can be further complicated by making the knockout boundary a function of time B(t) or by having a rebate if the barrier is activated. In the latter case, the holder of the option receives a specified amount if the barrier is reached.
* Asian options
have a payoff that depends on the price history of the underlying via some kind of average. Different definitions use a continuous S(t)   or a discrete sampling of the price history {S(t1),S(t2)... S(tN)}   and involve an arithmetic

$\displaystyle \bar{S}=\frac{1}{\Delta t}\int_{t-\Delta t}^{t} S(\tau)d\tau\hspace{1cm} \bar{S}=\frac{1}{N}\sum_{j=1}^N S(t_j)$ (2.1.4#eq.5)

or geometric average

$\displaystyle \bar{S}=\exp\left[\frac{1}{\Delta t}\int_{t-\Delta t}^{t} \log S(\tau)d\tau \right]\hspace{1cm} \bar{S}=\left[\prod_{j=1}^N S(t_j)\right]^{1/N}$ (2.1.4#eq.6)

to define the strike price on the expiry date T. An average strike call, for example, is structurally similar to a vanilla call, with a payoff equal to the difference between the asset price at expiry and its average if the difference is positive and zero otherwise

$\displaystyle \Lambda_\mathrm{average-strike-call}=\mathrm{max}(S-\bar{S},0).$ (2.1.4#eq.7)

Such a product can be used to average out the price of an underlying without the need for continuous re-hedging.
* Lookback options
are similar in spirit as Asian options, except that the strike price is a suitable definition of the maximum or minimum of the underlying price history

$\displaystyle \Lambda_\mathrm{lookback-call}=\mathrm{max}(S-\min_{0\le\tau\le t} S(\tau),0).$ (2.1.4#eq.8)

Such options can result in extremely advantageous payoffs and can therefore be very expensive: think of an option that allows the holder to buy the underlying at a low and sell it at a high.
* Russian options
are an example of perpetual options with an American exercise style: at any time, Russian options pay out the maximum realized asset price up to that date.
A variety of option can moreover be constructed by combining several exotic features and the list presented here far from exhaustive.

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