Chapter 26: Exotic Options

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What Makes an Option "Exotic"?

Exotic options have more complex payoff structures than standard (vanilla) calls and puts. They are mostly traded OTC and are tailored to specific hedging or investment needs.

Packages

A package is a portfolio of standard vanilla options. Examples: bull spreads, butterfly spreads, straddles, strangles. Some packages have standard names in certain markets (e.g., "risk reversal" = long OTM call + short OTM put).

Perpetual American Options

Options with no expiration date ( $\to \infty$ ). Closed-form solutions exist for perpetual American calls and puts. For a perpetual American call on a non-dividend stock:

$\left(\frac{S}{H}\right)^{\gamma} \cdot (H - K)$

where $H$ is the critical exercise boundary and $γ\gamma$ depends on $r$ , $σ\sigma$ .

Nonstandard American Options

Variations on the American exercise feature:

Bermudan options: Exercisable on specific dates (between European and American)
Canary options: Exercisable on certain dates before a period of "lock-out" after which they become Bermudan
Early exercise restricted to specific windows

Gap Options

A gap call pays $S_T - G$ when $S_T > K$ (gap between payoff trigger $K$ and payoff calculation $S_T - G$ ). A gap put pays $G - S_T$ when $S_T < K$ .

The price is:

$c = S_0 N(d_1) - G e^{-rT} N(d_2)$

where $d_1$ and $d_2$ use $K$ as the strike for the exercise criterion, but $G$ for the payoff.

Forward Start Options

Options that are paid for now but start at a future date. At the start date, the strike is typically set at-the-money. Used in employee stock option plans with staggered vesting. Valuation: the option is worth the same as an at-the-money option starting immediately, discounted for the forward start period.

Cliquet Options

A cliquet (ratchet) option is a series of forward start options where the strike for each period is reset to the asset price at the start of that period. The payoff accumulates over sub-periods with a cap and floor applied to each period's return.

Compound Options

An option on an option — gives the right to buy/sell another option. Four types:

Call on a call
Call on a put
Put on a call
Put on a put

Uses: bidding on projects (option to acquire a follow-on option), hedging future option positions.

Chooser Options

The holder can decide at a future date whether the option is a call or a put with the same strike and remaining maturity. This has the same value as:

$Chooser=c(S0,K,T)+p(S0,Ke−r(T−tc),tc)\text{Chooser} = c(S_0, K, T) + p(S_0, K e^{-r(T-t_c)}, t_c)$

where $t_c$ is the choice date (via put–call parity).

Barrier Options

The payoff depends on whether the underlying reaches a certain barrier level during the option's life.

Type	Description
Knock-out	Option ceases to exist if barrier is hit
Knock-in	Option comes into existence only if barrier is hit
Down-and-out call	Regular call unless $S$ falls below $H$
Up-and-out put	Regular put unless $S$ rises above $H$
Down-and-in	Becomes a vanilla option if $S$ falls below $H$

Barrier options are cheaper than vanilla options (because of the knock-out risk) and provide continuous monitoring (or discrete monitoring at fixing times).

Key relationship: $Knock-in+Knock-out=Vanilla\text{Knock-in} + \text{Knock-out} = \text{Vanilla}$

Binary (Digital) Options

Pay a fixed amount if a condition is met:

Cash-or-nothing call: Pays $Q$ if $S_T > K$
Asset-or-nothing call: Pays $S_T$ if $S_T > K$

Price for cash-or-nothing call: $\cdot e^{-rT} \cdot N(d_2)$

Lookback Options

Payoff depends on the minimum or maximum asset price during the option's life:

Floating lookback call: Payoff = $S_T - S_{\min}$
Floating lookback put: Payoff = $S_{\max} - S_T$
Fixed lookback call: Payoff = $max(S_{\max} - K, 0)$

Lookback options are expensive because they guarantee a holder-optimal exercise price.

Asian Options

Payoff depends on the average price over a period, not the spot price at a single point:

Average price call: $max⁡(Savg−K,0)\max(S_{\text{avg}} - K, 0)$
Average strike call: $max⁡(ST−Savg,0)\max(S_T - S_{\text{avg}}, 0)$

Asian options are cheaper than vanilla because averaging reduces volatility. They reduce manipulation risk at expiration (harder to manipulate an average than a spot price).

Pricing is complex because the average of lognormal variables is not lognormal. A common approximation treats the average as lognormal with adjusted mean and variance (Turnbull–Wakeman approximation).

Exchange Options

An option to exchange one asset for another. Margrabe's formula (1978):

$c = S_0^{(1)} N(d_1) - S_0^{(2)} N(d_2)$

$d1=ln⁡(S0(1)/S0(2))+σ2T/2σT,d2=d1−σTd_1 = \frac{\ln(S_0^{(1)} / S_0^{(2)}) + \sigma^2 T/2}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}$

where $σ2=σ12+σ22−2ρσ1σ2\sigma^2 = \sigma_1^2 + \sigma_2^2 - 2\rho\sigma_1\sigma_2$ .

Volatility and Variance Swaps

Variance swap: Pays realized variance minus the strike variance
Volatility swap: Pays realized volatility minus the strike volatility

Variance swaps can be statically replicated using a portfolio of European options — no dynamic hedging needed. The fair strike for a variance swap is:

$Kvar=2T[∫0F0P(K)K2dK+∫F0∞C(K)K2dK]K_{\text{var}} = \frac{2}{T}\left[\int_0^{F_0} \frac{P(K)}{K^2} dK + \int_{F_0}^{\infty} \frac{C(K)}{K^2} dK\right]$

where $P (K)$ and $C (K)$ are OTM put and call prices, and $F_0$ is the forward price.