THE PRICING OF MULTIPLE-EXPIRY EXOTICS

An up binary option of exercise price $\xi$ on the standard option with expiry payoff $f(x)$ is a contract with expiry payoff $f(x)$ if $x>\xi$ and zero otherwise. A down binary option pays $f(x)$ if $x<\xi$ and zero otherwise. Let $s$ be the sign ”$+$” or ”$-$”. In what follows we use the sign ”$+$” and ”$-$” as sign indicators for up and down binaries, respectively. Then the expiry payoff functions for up and down binaries can be written in the form

From this there holds the following parity relation between the standard option price and the corresponding up and down binaries:

If $f(x)=x$, the standard option simply pays one unit of the asset at expiry date $T$, and the price of the standard option is $A(x,t;T)=xe^{-q(T-t)}$ for all $t<T$. And the corresponding binaries are the very asset-or-nothing binaries, here their prices are denoted by $A_{\xi}^{s}(x,t;T)$. Then from the above mentioned parity relation, we have

If $f(x)=1$, the standard option simply pays one unit of cash at expiry date $T$, and the price of the standard option is $B(x,t;T)=e^{-r(T-t)}$ for all $t<T$. And the corresponding binaries are the very cash-or-nothing binaries (or bond binaries), here their prices are denoted by $B_{\xi}^{s}(x,t;T)$. From the parity relation, we have

The concept of Q-option plays a very useful role in the pricing of dual expiry options and, in particular, simplifying the notation of the price formula. (See [2].) Consider a standard contract that pays $f(x)=s(x-K)$ at expiry date $T$; this is a kind of forward contract, where the holder must buy if $s=+$ (or sell if $s=-$) one unit of underlying asset for $K$ units of cash. Since $f(x)=s[A(x,T;T)-K\cdot B(x,T;T)]$, the price $Q(x,t;T,K)$ of this contract at time $t<T$ is given by

The binary option that pays $f(x)=s(x-K)\cdot 1(sx>s\xi)$ at expiry date $T$ is called a first order Q-option, their prices denoted by $Q_{\xi}^{s}(x,t;T,K)$. If $\xi=K$, then $Q_{K}^{s}(x,t;T,K)$ is is the very ordinary European (call if $s=+$, or put if $s=-$) option. So $Q_{\xi}^{s}(x,t;T,K)$ is called a generalized European option. These options are more general in the sense that their exercise price $\xi$ is different from their strike price $K$. The price of first order Q-options are given as follows:

The asset or nothing binary, bond binary and the first order Q-option are called the first order binaries.[2]

Proposition 2[2, 4]. The prices of asset and bond binary options are provided as follows:

Here $N(x)$ is the accumulated normal distribution function

and $d$, $d^{\prime}$ are respectively given as follows:

Definition 1. An $n$-th order binary option is a binary contract with expiry date $T_{0}$ on an underlying $(n-1)$-th order binary option. Specifically, the payoff at time $T_{0}$ has the following form

Here $F_{\xi_{1}\cdots\xi_{n-1}}^{s_{1}\cdots s_{n-1}}(x,T_{0};T_{1},\cdots,T_{n-1})$ is the price of the underlying $(n-1)$-th order binary option with expiry time $T_{1},\cdots,T_{n-1}$ at the time $T_{0}$ and either $F=A$ if the underlying binary is asset binary, $F=B$ for the underlying bond binary or $F=Q$ for the underlying $Q$-option; and $s_{0},\cdots,s_{n-1}$ are up-down indicators ($+$ or $-$) at times $T_{0},\cdots,T_{n-1}$ respectively. $\xi_{0},\cdots,\xi_{n-1}$ are their corresponding exercise prices.
The prices of these $n$-th order binary options at time $t<T_{0}$ are denoted by $F_{\xi_{0}\xi_{1}\cdots\xi_{n-1}}^{s_{0}s_{1}\cdots s_{n-1}}(x,t;T_{0},T_{1},\cdots,T_{n-1})$.
Then from the definition we have

$~Q_{\xi_{0}\xi_{1}\cdots\xi_{n-1}}^{s_{0}s_{1}\cdots s_{n-1}}(x,t;T_{0},T_{1},\cdots,T_{n-1},K)$=
$=s_{n-1}[A_{\xi_{0}\xi_{1}\cdots\xi_{n-1}}^{s_{0}s_{1}\cdots s_{n-1}}(x,t;T_{0},T_{1},\cdots,T_{n-1})-$

Note that the strike price $K$ in the higher order $Q$-binary is effective only at last time $T_{n-1}$. From the parity relation, we have

If the pricing formulae of the higher order asset binary $A_{\xi_{0}\xi_{1}\cdots\xi_{n-1}}^{s_{0}s_{1}\cdots s_{n-1}}(x,t;$ $T_{0},T_{1},\cdots,T_{n-1})$ and the higher order bond binary $B_{\xi_{0}\xi_{1}\cdots\xi_{n-1}}^{s_{0}s_{1}\cdots s_{n-1}}(x,t;T_{0},$ $T_{1},\cdots,T_{n-1})$ are provided, then the price of the higher order $Q$-binary is easily provided by (7).
Black-Scholes expressions for higher order binary options involve multidimensional normal distribution function $N(x_{0},x_{1},\cdots,x_{n-1};P)$ with zero mean vector and correlation matrix $P^{-1}$:

Here $y^{\mathsf{T}}=(y_{0},y_{1},\cdots,y_{n-1}).$
Let define the matrix $A=(a_{ij})_{i,j=0,1,\cdots,n-1}$ related to the expiry dates $T_{0},T_{1},\cdots,$ $T_{n-1}$ as follows:

$a_{00}=(T_{1}-t)/(T_{1}-T_{0})$,

$a_{n-1,n-1}=(T_{n-1}-t)/(T_{n-1}-T_{n-2})$,

$a_{ii}=(T_{i}-t)/(T_{i}-T_{i-1})+(T_{i}-t)/(T_{i+1}-T_{i}),~~1\leq i\leq n-2$,

$a_{i,i+1}=a_{i+1,i}=-\sqrt{(T_{i}-t)(T_{i+1}-t)}/(T_{i+1}-T_{i}),~~0\leq i\leq n-2$,

and another elements are all zero. Then we have

Let $s_{i}=+$ or $-$ (that is, sign indicators and $i=0,1,\cdots,n-1$) and define a new matrix by

Then we have

Theorem 1 The price of $n$-th order asset binary and bond binary are provided as follows :

$A_{\xi_{0}\xi_{1}\cdots\xi_{n-1}}^{s_{0}s_{1}\cdots s_{n-1}}(x,t;T_{0},T_{1},\cdots,T_{n-1})=$

$B_{\xi_{0}\xi_{1}\cdots\xi_{n-1}}^{s_{0}s_{1}\cdots s_{n-1}}(x,t;T_{0},T_{1},\cdots,T_{n-1})=$

Here

  $d_{i}=\frac{\ln\frac{x}{\xi_{i}}+(r-q+\frac{\sigma^{2}}{2})(T_{i}-t)}{\sigma\sqrt{T_{i}-t}},~~~~$

Proof: The cases of $n=1$ and $n=2$ were proved by Buchen in [2] using probability theory and can be easily proved using the formula (3) too. In the case of $n>2$ we will give a sketch of the proof by induction.
We assume that theorem 1 holds in the case of $n-1$. From the definition 1, $A_{\xi_{0}\xi_{1}\cdots\xi_{n-1}}^{s_{0}s_{1}\cdots s_{n-1}}(x,t;T_{0},T_{1},\cdots,T_{n-1})$ satisfies (1) and

Therefore by the formula (3), If we let

then $A_{\xi_{0}\xi_{1}\cdots\xi_{n-1}}^{s_{0}s_{1}\cdots s_{n-1}}(x,t;T_{0},T_{1},\cdots,T_{n-1})$ is provided as follows:

$A_{\xi_{0}\xi_{1}\cdots\xi_{n-1}}^{s_{0}s_{1}\cdots s_{n-1}}(x,t;T_{0},T_{1},\cdots,T_{n-1})=$

$=x\cdot e^{-q(T_{0}-t)}\int_{0}^{\infty}G(z)A_{\xi_{1}\cdots\xi_{n-1}}^{s_{1}\cdots s_{n-1}}(z,T_{0};T_{1},\cdots,T_{n-1})\cdot 1(s_{0}z>s_{0}\xi_{0})dz$.

Here $A_{\xi_{1}\cdots\xi_{n-1}}^{s_{1}\cdots s_{n-1}}(z,T_{0};T_{1},\cdots,T_{n-1})$ is the price of the underlying $(n-1)$-th order asset binary option. By induction-assumption, the formula (9) holds for $A_{\xi_{1}\cdots\xi_{n-1}}^{s_{1}\cdots s_{n-1}}(z,T_{0};T_{1},\cdots,T_{n-1})$. Thus we have

$A_{\xi_{1}\cdots\xi_{n-1}}^{s_{1}\cdots s_{n-1}}(z,T_{0};T_{1},\cdots,T_{n-1})=$

Substitute this equality into the above singular integral representation and calculate the integral, then we have (9) for the case of $n>2$. The proof for (10) is similar. (Proof End)

The formulae (7), (9) and (10) give the following price of higher order $Q$-option :

$Q_{\xi_{0}\xi_{1}\cdots\xi_{n-1}}^{s_{0}s_{1}\cdots s_{n-1}}(x,t;T_{0},T_{1},\cdots,T_{n-1},K)=$

$=s_{n-1}[x\cdot e^{-q(T_{n-1}-t)}N(s_{0}d_{0},s_{1}d_{1},\cdots,s_{n-1}d_{n-1};A_{s_{0}s_{1}\cdots s_{n-1}})-$

In this section, we applied the results of previous section to the pricing of some multiple expiry exotics.

The Static Replication Theorem If the payoff of an option at expiry time $T_{0}$ is a linear combination of prices at time $T_{0}$ of higher order binaries, then its price at all time $t<T_{0}$ is the combination of the corresponding prices at time $t$ of the higher order binaries.

The static replication theorem can be proved from the uniqueness of solution to the initial value problem of Black-Scholes equation. And we need one lemma that guarantees the monotonousness of the option price on the underlying asset price and provides some estimates about the gradient of the price on the underlying asset price.

Lemma 1 Assume that $f(x)$ is continuous and piecewise differentiable. If $V(x,t)$ is the solution of (1) and (2), then we have

In particular, if $E=\{z:\frac{\partial f}{\partial x}(z)<\max_{x}\frac{\partial f}{\partial x}\},~F=\{z:\frac{\partial f}{\partial x}(z)>\min_{x}\frac{\partial f}{\partial x}\},|E|>0$ and $|F|>0$, then

Using the formula (3) and the assumption of lemma 1, we can easily prove the required results.