10.1 Examples of Exotic Options

10.1.1 Compound Options, Option on Option

With a compound option one has the right to acquire an ordinary option at a later date. To illustrate such a compound option consider a Call-on-a-Call with the parameters:
maturity dates $ T_1 < T_2$
strike prices $ K_1, K_2.$

Such a compound option gives the owner the right at time $ T_1$ for the price $ K_1$ to buy a Call that has a maturity date $ T_2$ and a strike price $ K_2$.

The value $ V(S,t)$ of this option at time $ t$ with an actual price $ S_t=S$ can be calculated by applying the Black-Scholes formula twice:

1)
Beginning at time $ t=T_2$, calculate (explicitly or numerically) the value of the Call, which can be bought at time $ T_1$, at time $ t = T_1$. This value is $ C(S,
T_1).$

2)
The purchasing option of the Call at time $ T_1$ is only exercised when

$\displaystyle C(S, T_1) > K_1 $

Thus it holds that

$\displaystyle V(S, T_1) = \max \{C(S, T_1) - K_1,\ 0\}.$

Calculate $ V(S,t)$ for $ t < T_1$ using the Black-Scholes equation with these restrictions at $ t = T_1$ analog to the normal Call.

Remark 10.1   Since $ V(S,t)$ is only dependent on $ t$ and the price $ S_t$ at time $ t$, this value function fulfills the Black-Scholes equation so that our approach is justified.

10.1.2 Chooser Options or ``As you wish'' Options

A Chooser Option is a form of the compound option, where the buyer can decide at a later date which type of option he would like to have. To illustrate consider a regular Chooser Option with the parameters:

maturity dates $ T_1 < T_2$
strike prices $ K_1, K_2.$

This option gives one the right at time $ T_1$ for the price $ K_1$ to buy a Call or a Put (as one likes), which has a maturity $ T_2$ and a strike price $ K_2$: in the language of compound options this is referred to as a Call-on-a-Call or Put.

The value $ V(S,t)$ can be found by applying the Black-Scholes formula three times:

1)
Determine the value $ C(S,T_1)$ and $ P(S,T_1)$ of a Call and a Put with a maturity $ T_2,$ and strike price $ K_2.$
2)
Solve the Black-Scholes equation for $ t < T_1$ with the restriction

$\displaystyle V(S,T_1) = \max \{ C(S,T_1) - K_1,\ P(S,T_1) - K_1, 0 \} $

10.1.3 Barrier Options

A barrier Option changes its value in leaps as soon as the stock price reaches a given barrier, which can also be time dependent. As an example consider a simple European barrier Option which at
maturity $ T$, strike price $ K$ and
barrier $ B$

gives the holder the right to buy a stock at time $ T$ for the price $ K$ provided that

This type of Knock-out Option is worthless as soon as the price $ S_t$ reaches the barrier. A Knock-in-Option is just the opposite. It is worthless up until the barrier is reached.

For example, a European Knock-in-Call consists of the right to buy stock provided that

The value of a barrier Option is no longer dependent on a stock price at a specific point in time, but on the overall development of the stock price during the option's life span. Thus in principle it does not fulfill the Black-Scholes differential equation. The dependence however, is essentially simple enough to work with the conventional Black-Scholes application. As an example consider a Down-and-out-Call with $ K
> B$. As long as $ S_t > B$, $ V(S,t)$ fulfills the Black-Scholes equation with the restriction:

$\displaystyle V(S,T) = \max (S_T - K,\ 0) $

In the event that the price reaches the barrier $ B$, the option of course becomes worthless:

$\displaystyle V(B,t) = 0\ ,\ \quad \, 0 \le t \le T, $

is therefore an additional restriction that needs to be taken into consideration when solving the differential equation (6.22). The explicit solution is given as:

$\displaystyle V(S,t) = C(S,t) - \left( \frac{B}{S} \right)^ \alpha \, C\left(
\frac{B^ 2}{S}, t \right) $

with $ \alpha = \displaystyle
\frac{2r}{\sigma^ 2} - 1 $, where $ C(S,t)$ represents the value of a common European Call on the stock in question. The value $ \bar{V} (S,t)$ of an European Down-and-in Call can be calculated analogously. If one already knows $ V(S,t)$, one can also use the equation

$\displaystyle \bar{V} (S,t) + V(S,t) = C(S,t). $

It is fulfilled since a Down-and-in and a Down-and-out Call together have the same effect as a normal Call.

10.1.4 Asian Options

With Asian options the value depends on the average stock price calculated over the entire life span of the option. With anaverage strike option over the time period $ 0 \le t \le T$ the payoff, for example, has at the time to maturity the form

$\displaystyle \max \left( S_t - \frac{1}{t} \int^ t_0 S_s \, ds \, ,\
0\right), t = T . $

With an American Average Strike Option this is also the payoff when the option is exercised ahead of time at some arbitrary time $ t \le T.$

To calculate the value of an Asian Option consider a general class of European Options with a payoff at time $ T$ that is dependent on $ S_T$ and $ I_T$ with

$\displaystyle I_t = \int ^ t_0 f(S_s, s) ds. $

Analogous to the Black-Scholes equation we derive an equation for the value at time $ t$ of such a path dependent Option $ V(S,I,t)$. At time $ t$ with a stock price $ S_t$ this results in

$\displaystyle I_t + dI_t$ $\displaystyle \stackrel{\mathrm{def}}{=}$ $\displaystyle I_{t+dt} = \int^ {t + dt}_0
f(S_s, s) ds$  
  $\displaystyle =$ $\displaystyle I_t + \int^ {t+dt}_t f(S_s, s) dt$  
  $\displaystyle =$ $\displaystyle I_t + f(S_t, t) dt + {\scriptstyle \mathcal{O}}(dt).$  

Thus the differential of $ I_t$ is equal to $ dI_t = f(S_t, t) dt $.

Using Itôs Lemma it follows for $ V_t = V(S_t,I_t,t)$ that:

$\displaystyle dV_t = \sigma S_t \, \frac{\partial V_t}{\partial S} dW_t + f(S_t,t) \,
\frac{\partial V_t}{\partial I} dt $

$\displaystyle + \left( \frac{1}{2} \sigma^ 2 S^ 2 \, \frac{\partial ^ 2 V_t}{\partial S}
+ \frac{\partial V_t}{\partial t} \right) dt $

Analogous to the derivation of the Black-Scholes formula continuous Delta hedging produces a risk free portfolio from an option and $ \Delta_t =
\partial V_t/\partial S$ sold stocks. Together with the restriction of no arbitrage it follows for the case of no dividends $ (b=r)$ that:

$\displaystyle \frac{\partial V_t}{\partial t} + f(S_t,t) \frac{\partial V_t}{\p...
...tial^ 2V_t}{\partial S^ 2} + rS_t
\frac{\partial V_t}{\partial S} - r\ V_t = 0 $

This is the Black-Scholes equation with an additional term $ f(S_t,t)
\frac{\partial V_t}{\partial I}.$ The boundaries in this case are

$\displaystyle V(S,I,T) = g(S,I,T) .$

For an Average Strike Call we have:

$\displaystyle g(S,I,t) = \max (S - \frac{1}{t} I,\ 0)$   und$\displaystyle \quad f(S, t) = S .$

For European options an explicit analytical solution of the differential equation exists in which complicated, specialized functions appear, the so called confluent hypergeometric functions. The numerical solution, however, is easier and faster to obtain.

The integral $ \int^ t_0 S_s ds $ in practice is calculated as the sum over all quoted prices, for example, at 30 second time intervals. Discrete time Asian Options use in place of this a substantially larger time scale. $ I_t$ changes only once a day or once a week:

$\displaystyle I_t = \sum^ {n(t)}_{j=1} S_{t_j}\ ,\ \, t_{n(t)} \le t < t_{n(t)+1} $

with $ t_{j+1} - t_j = $ 1 day or = 1 week and closing price $ S_{t_j}$.

Such a discrete time Asian Option is largely consistent with a common option with discrete dividend payments at time periods $ t_1, t_2, \ldots .$ From the assumption of no arbitrage follows a continuity restriction at $ t_j$:

$\displaystyle V(S,I,t_j -) = V (S,I + S,\ t_j^+) $

To determine the value of the option one begins as usual at the time to maturity where the value of the option is known:

1)
$ T=t_n$

$\displaystyle V(S,I,T) = \max (S - \frac{1}{T} I_{T},\ 0) $

$\displaystyle I_{T} = \sum^ n_{j=1} S_{t_j} $

Solve the Black-Scholes equation backwards to time $ t_{n-1}$ and obtain

$\displaystyle V(S, I + S,\ t_{n-1}^+ ) $

2)
Calculate using the continuity restriction the new terminal value
$ V(S, I, t_{n-1}^- ) $. Solve the Black-Scholes equation backwards to time $ t_{n-2}$ and obtain

$\displaystyle V(S, I + S,\ t_{n-2}^+) $

etc.

10.1.5 Lookback Options

The value of a lookback Option depends on the maximum or minimum of the stock price over the entire life span of the option, for example, a lookback put over the time period $ 0 \le t \le T$ has at maturity the payoff

$\displaystyle \max ( M_{T} - S_{T} ,\ 0)$   with  $\displaystyle \displaystyle \qquad M_t = \max_{0 \le s \le t} S_s.$

To calculate the value of such an option first consider a path dependent option with

$\displaystyle I_t (n) = \int^ t_0 S_s ^ n ds \, ,\ $   i.e.$\displaystyle \quad f(S,t) = S^n. $

With $ \, M_t(n) = (I_t (n) )^ {\frac{1}{n}} $ it holds that:

$\displaystyle M_t = \lim_{n\rightarrow\infty} M_t (n). $

Form the differential equation for $ I_t(n)$ and $ n \rightarrow
\infty$ it follows that the value $ V_t=V(S_t,M_t,t)$ of an European lookback put fulfills the following equation:

$\displaystyle \frac{\partial V_t}{\partial t} + \frac{1}{2} \sigma^ 2 S_t^ 2 \f...
...l
^ 2V_t}{\partial S^ 2} + r\ S_t \frac{\partial V_t}{\partial S} - r\ V_t = 0
$

This is the normal Black-Scholes equation. $ M$ only appears as an argument of $ V$ and in the boundary conditions:

$\displaystyle V(S,M,T) = \max (M - S,\ 0) $

The solution is for a remaining time period of $ \tau = T -t,\ \,
\alpha = 2r/\sigma^ 2$:

$\displaystyle V(S,M,t)$ $\displaystyle =$ $\displaystyle S\left( \Phi (y_1) \cdot (1 + \frac{1}{\alpha} ) -
1\right)$  
    $\displaystyle + M\ e^ {-r\tau} \left( \Phi (y_3) - \frac{1}{\alpha} \left(
\frac{M}{S}\right) ^ {\alpha - 1} \Phi (y_2) \right)$  


with$\displaystyle \qquad y_1$ $\displaystyle =$ $\displaystyle \frac{1}{\sigma\sqrt{\tau}}
\{ \ln \frac{S}{M} + (r + \frac{1}{2} \sigma^ 2) \tau \}$  
       
$\displaystyle y_2$ $\displaystyle =$ $\displaystyle \frac{1}{\sigma\sqrt{\tau}}
\{ \ln \frac{S}{M} - (r + \frac{1}{2} \sigma^ 2) \tau \}$  
       
$\displaystyle y_3$ $\displaystyle =$ $\displaystyle \frac{1}{\sigma\sqrt{\tau}}
\{ \ln \frac{M}{S} - (r + \frac{1}{2} \sigma^ 2) \tau \}$