Exotic options

see chapter 24 in Hull, Options, Futures, and other Derivatives

Nonstandard American options

Standard American options can be exercised at any time up to their maturity. Restricting exercise to certain dates (Bermudan options) or disallowing during certain parts of the option’s life cycle (lock out periods) creates variations of American options. Such nonstandard American options can be valued using binomial trees where at each node a test is inserted whether early exercise is admitted and of value. ### Forward start options Valuation: … ### Compound options These are options on options:

European call on a call: $S_0e^{-qT_2}M(a_1,b_1;\sqrt{T_1/T_2})-K_2e^{-rT_2}M(a_2,b_2;\sqrt{T_1/T_2})-e^{-rT_1}K_1N(a2)$

where

The function M(a,b; ρ) is a bivariate normal distribution found on J. Hull’s website in Technical Note #5. S^* is the asset price at time T₁ for which the option price is K₁.

European put on a call: $K_2e^{-rT_2}M(-a_2,b_2;-\sqrt{T_1/T_2}) - S_0e^{-qT_2}M(-a_1,b_1;-\sqrt{T_1/T_2}) + e^{-rT_1}K_1N(-a2)$

European call on a put: $K_2e^{-rT_2}M(-a_2,-b_2;\sqrt{T_1/T_2}) - S_0e^{-qT_2}M(-a_1,-b_1;\sqrt{T_1/T_2}) - e^{-rT_1}K_1N(-a2)$

European put on a put: $S_0e^{-qT_2}M(a_1,-b_1;-\sqrt{T_1/T_2}) - K_2e^{-rT_2}M(a_2,-b_2;-\sqrt{T_1/T_2}) + e^{-rT_1}K_1N(a2)$

Chooser options

Such options can be switched to a put or a call after a certain time. Its value at time T₁ is $max(c, p)$. For European options call-put-parity provides a valuation: $max(c, p) = max(c, c+Ke^{-r(T_2-T_1)}-S_1e^{-q(T_2-T_1)}) = c + e^{-q(T_2-T_1)}max(0,Ke^{-(r-q)(T_2-T_1)}-S_1)$ It is shown that this corresponds to a package of: 1.) A call option with strike K and maturity T₂ 2.) $e^{-q(T_2-T_1)}$ put options with strike $Ke^{-(r-q)(T_2-T_1)}$ and maturity T₁

Barrier options

Either knock-in option, which begins after a barrier has been touched, or knock-out option, which ceases to exist after the event. Valuation: …

### Binary options Such options provide discontinuous payoffs after an event has occurred. Cash-or-nothing call: pays a fixed amount Q if it ends in the money (ITM), thus at time T the spot price S_T is above the strike price K. Its value depends on the probability of ending in the money: $Qe^{-rT}N(d_2)$. Cash-or-nothing put: pays a fixed amount Q if it ends ITM, thus below the strike: $Qe^{-rT}N(-d_2)$. Asset-or-nothing call: this pays the asset price if it ends ITM, thus S_T > K, otherwise nothing. Its value is $S_0e^{-qT}N(d_1)$. Asset-or-nothing put: only pays the asset price if it ends ITM, thus below the strike price. Its value is $S_0e^{-qT}N(-d_1)$.

An European call options is a combination of LONG an asset-or-nothing call and SHORT a cash-or-nothing call (Q = K). Similarly for the put option: LONG cash-or-nothing put (Q = K) and SHORT asset-or-nothing put.

Lookback options

These options provide payoffs depending on the lows and highs of S_t during their lifetime. Floating lookback call: pays the difference between S_T and the minimal observed asset price S_min. $c_{fl} = S_0e^{-qT}N(a_1) - S_0e^{-qT}\frac{\sigma^2}{2(r-q)}N(-a_1) - S_{min}e^{-rT}[N(a_2) - \frac{\sigma^2}{2(r-q)}e^{Y_1}N(-a_3)]$ where $a_1 = \frac{ln(S_0/S_{min}) + (r-q+\sigma^2/2)T}{\sigma\sqrt{T}}$ $a_2 = a_1 - \sigma\sqrt{T}$ $a_3 = \frac{ln(S_0/S_{min}) + (-r+q+\sigma^2/2)T}{\sigma\sqrt{T}}$ $Y_1 = - \frac{2(r-q-\sigma^2/2)ln(S_0/S_{min})}{\sigma^2}$

Floating lookback put: pays the difference between the maximum observed asset price S_max and S_T. $p_{fl} = S_{max}e^{-rT}[N(b_1) - \frac{\sigma^2}{2(r-q)}e^{Y_2}N(-b_3)] + S_0e^{-qT}\frac{\sigma^2}{2(r-q)}N(-b_2) - S_0e^{-qT}N(b_2)$ where $b_1 = \frac{ln(S_{max}/S_0) + (-r+q+\sigma^2/2)T}{\sigma\sqrt{T}}$ $b_2 = b_1 - \sigma\sqrt{T}$ $b_3 = \frac{ln(S_{max}/S_0) + (r-q-\sigma^2/2)T}{\sigma\sqrt{T}}$ $Y_2 = \frac{2(r-q-\sigma^2/2)ln(S_{max}/S_0)}{\sigma^2}$

### Shout options An European option with the one-time possibility to lock-in the current asset price during the lifetime of the option. At maturity the payoff depends on the maximum of S_τ or S_T.

Valuation is similar to an American option using a binomial tree. At every node in the tree the option’s value is the maximum of the value if the holder shouts or not. ### Asian options The payoff of these options depends on the average price of the underlying asset during the lifetime of the option.

Average price call: max(0, S_avg Average price put: max(0, K - S_avg) Using such options which cost less than regular options, it can be guaranteed that on average a certain price level over some time is realized.

Average strike call: max(0, S_T - S_avg) Average strike put: max(0, S_avg - S_T) These options can guarantee that the average price paid or received for an asset in frequent trading is not higher or not lower than a fixed amount.

Valuation in case of geometric average using the Black-Scholes-Merton formula and the following parameters: $\sigma_{BS} = \frac{\sigma}{\sqrt{3}}$ $q_{BS} = \frac{1}{2} (r+q+\frac{\sigma^2}{6})$

Valuation in case of the more often found arithmetic average case fits a lognormal distribution to the moments: $M_1 = \frac{e^{(r-q)T}-1}{(r-q)T}S_0$ $M_2 = \frac{2e^{[2(r-q)+\sigma^2]T} S_0^2}{(r-q+\sigma^2)(2r-2q+\sigma^2)T^2} + \frac{2S_0^2}{(r-q)T^2} [ \frac{1}{2(r-q)+\sigma^2} - \frac{e^{(r-q)T}}{r-q+\sigma^2} ]$

It follows: $F_0 = M_1$ $\sigma^2 = \frac{1}{T} ln(\frac{M_2}{M^2_1})$

which can be input into Black’s model: $c = e^{-rT}[F_0N(d_1) - KN(d_2)]$ $p = e^{-rT}[KN(-d_2) - F_0N(-d_1)]$ and $d_1 = \frac{ln(F_0/K) + \sigma^2T/2}{\sigma\sqrt{T}}$ $d_2 = d_1 - \sigma\sqrt{T}$

Options to exchange one asset for another

Valuation: …

Options involving several assets

Valuation: …

Volatility and variance swaps

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