Pricing Derivatives

GBM results in closed-form representations of asset dynamics in the risk-neutral measure $\mathbb{Q}$ are described by

$X(T) \overset{d}{=} X(t) \exp \{ (r - 2^{-1} \sigma^2) \tau + \sigma \tau^{1/2} Z \}$ where $Z$ is a normal(0,1) random variable and $\tau = T - t$ is the time to expiry.

A second result that is needed is the so-called Gaussian shift lemma

lemma: when $Z$ is a normal(0,1) random variable, $\mathbb{E} e^{tZ} g(Z) = e^{\frac{1}{2} t^2} \mathbb{E} g(Z + t)$ .

Let $\phi(x)$ be the density function of a normal(0,1) random variable. Then

$e^{tx} \phi(x) = e^{\frac{1}{2} t^2} \phi(x - t)$ and so

$\mathbb{E} e^{tZ} g(Z) = m(f : f(x) = e^{tx} \phi(x) g(x) ) = e^{2^{-1} t^2} m(f : f(x) = \phi(x - t) g(x)) = e^{2^{-1} t^2} m(f: f(x) = \phi(x) g(x + t)) = e^{2^{-1} t^2} \mathbb{E} g(Z+t)$

definition: a $n$ th-order binary is a contigent $T$ -claim such tht the payoff at a given $t_n < T$ is a $(n-1)$ th-order binary. A first-order binary is a contingent claim whose payoff at time $T$ is given by $f(X(T)) \mathbb{I}_{sX(T) > s \xi}$ where $s$ is a given binary(-1,1) variable and $\xi$ is a threshhold. If $s = 1$ then the binary is an up-binary, and if $s = -1$ the binary is a down-binary.

Asset and Bond Binaries

The simplest 1st order binaries are the asset | bond binary. The payoff function $f(x) = x | 1$ . To price these, note that $X(T) > \xi$ iff $d = \frac{\ln \frac{X(t)}{\xi} + (r - 2^{-1} \sigma^2) \tau}{\sigma \tau^{1/2}} > -Z$ .

By the FTAP, $V(t,x) = \mathbb{Q}_t e^{-r\tau} V(T,X(T))$ . In the case of the asset up binary,

$A(t,x) = \mathbb{Q}_t e^{-r\tau} X(T) \mathbb{I}(X > \xi) = x e^{- \frac{1}{2} \sigma^2 \tau} \mathbb{Q}_t e^{\sigma \tau^{1/2}Z} \mathbb{I}(-Z < d)$

By the Gaussian Shift Lemma and symmetry of the normal distribution,

$A^+_\xi(t,x) = x e^{r\tau} \mathbb{Q}_t \mathbb{I}(-(Z + \sigma \tau^{1/2})< d) = x N(d + \sigma \tau^{1/2})$

The bond up binary can be priced in a similar and even more straightforward manner:

$B^+_\xi(t,x) = \mathbb{Q}_t e^{-r\tau} \mathbb{I}(X(T) > \xi) = e^{-r\tau} \mathbb{Q}_t \mathbb{I}(-Z < d) = e^{-r\tau} N(d)$

Since the positions $1_{A^+_\xi} + 1_{A^-_\xi} \equiv_T 1_A$ by the Law of One Price, $1_{A^+_\xi} + 1_{A^-_\xi} \equiv_T 1_A$ under the assumption that $A$ is an asset with no associated flows, $1_{A^+_\xi} + 1_{A^-_\xi} \equiv_t 1_A$ and so the prices satisfy $A^+_\xi(x,t) + A^-_\xi(x,t) = x$ from which one deduces $A^-_\xi(x,t) = x - x N(d + \sigma \tau^{1/2}) = x N(-(d + \sigma \tau^{1/2}) )$ . This may be combined into one formula: $A^s_\xi(x,t) = x N(s(d + \sigma \tau^{1/2}))$ . Analogously, for the bond binary $B^+_\xi(x,t) + B^-_\xi(x,t) = B(x,t) = e^{-rt}$ and so $B^s_\xi(x,t) = e^{-rt} N(sd)$

Put and Call Options

The introduction of asset and bond binary contracts and their pricing enables calculation of the price of standard vanilla call and put options. Let $K$ be the strike price.

Since

$C_K(X(T),T) = (X(T) - K)^+ = (X(T) - K) \mathbb{I}_{X(T) > K} \\= A_K^+(X(T),T) - K B_K^+(X(T),T)$

it follows that

$C_K(x,t) = A_K^+(x,t) - K B_K(x,t) = x N(d + \sigma \tau^{1/2}) - K e^{-r\tau} N(d)$ .

By put-call parity, $C_K(x,t) - P_K(x,t) = x - K e^{-r\tau}$

and so

$P_K(x,t) = -x N(-(d + \sigma \tau^{1/2})) + K e^{-r\tau} N(-d)$

Additional Standard Binary Contracts

A number of other contracts can be priced in a similar way

Q-options

These are like vanilla call/put options except the binary threshold price and strike price are not necessarily identical. The gap call has payoff $f(x) = (x - k) \mathbb{I}_{x > \xi}$ where $\xi - k > 0$ . The gap put has payoff $f(x) = (k-x) \mathbb{I}_{x < \xi}$ where $k - \xi > 0$ . The gap is the quantity $|\xi - k|$ . More generally, a Q-option hay payoff $f(x) = s(x-k) \mathbb{I}_{sx > s\xi}$ .

Capped Calls and Capped Puts

The plain vanilla call has unlimited liability for the seller. To address this, a capped call can reduce the risk. The payoff is $f(x) = \max((x-k)^+,c)$ where $c > 0$ is the cap.

lemma: $\min((x-k)^+,c) = (x-k)^+ - (x-k-c)^+$

Since $\min(a,c) + \max(a,c) = a + c$ , it follows that $\min(a,c) = a - (\max(a,c) - c) = a - (a-c)^+$ the identity follows.

Range Forward Contracts

The range forward contract is a bearish position consisting long a put and short a call at a higher strike price.

Straddle

A straddle is a a position consisting long put and long call at the same strike. This position is bullish on volatility.

Strangle

A strangle is a position consisting long put and long call at a higher strike price.

Strip and Strap

The strip and strap are long positions in puts and calls of the same strike. If more puts are purchased then calls, the position is a strip. Otherwise it is a strap.

Turbo Binary

A turbo binary has a payoff $f(x) = x^p \mathbb{I}_{sx > s\xi}$ . For the up-binary, the option price is determined as

$V^+_\xi(x,t) = \mathbb{Q}_t e^{-r\tau} x^p e^{(r- \frac{1}{2} \sigma^2) p \tau + p \sigma \tau^{1/2} Z} \mathbb{I}_{d > -Z} = x^p e^{(r- \frac{1}{2} \sigma^2) p \tau + \frac{1}{2} \sigma^2 p^2 \tau} N(d + p \sigma \tau^{1/2})$

Log-Binary

The log contract has payoff $f(x) = \ln \frac{x}{k}$ .

Dual Expiry Options

Dual expiry options are contingent claims whose payoff depends on the underlying asset price at two future dates $T_1,T_2$ such that $T_1 < T_2$ .

Forward Start Calls and Puts

The forward start call is a contigent contract that delivers an ATM (at-the-money) call option at time $T_1$ that expires at time $T_2$ . That is the asset flow satisfies $1_{FSC}(t) = 1_{C(S,S(T_1),T_2)}(T_1) = (0 | 1_S - S(T_1) 1_\$)(T_2)$ . Hence

$FSC_\$(t) = \mathbb{Q}_t [e^{-r(T_1 - t)} C_\$(S,S(T_1),T_2)] = e^{-r(T_1 - t)} \mathbb{Q}_t [ S(T_1) N(d_1) - S(T_1) e^{-r(T_2-T_1)} N(d_{-1})$

where $d_s = \frac{(r + \frac{1}{2} s \sigma^2)(T_2-T_1)^{1/2}}{\sigma}$ . Since $d_s$ is deterministic, the evaluation can proceed further. Since $\mathbb{Q}_t S(T_1) = S(t) e^{r(T_1 - t)}$ ,

$FSC_\$(t) = S(t) [N(d_1) - e^{-r\tau} N(d_{-1})]$ where $\tau$ is the time-to-expiry of the FSC. A forward start put is priced in a similar manner.

Second-Order Binaries

These are contracts where the payoff depends on the price levels at time $T_1,T_2$ . That is $1_{V^{s_1,s_2}_{\xi_1,\xi_2}}(t) = F(X(T_2)) \mathbb{I}(s_1X(T_1) > s_1\xi_1, s_2 X(T_2) > s_2 \xi_2)$ where $s_1,s_2$ are binary(-1,1) variables.

As usual, asset and bond binaries are priced first. The asset price is lognormally distributed where

$X(T) = X(t) e^{(r- \frac{1}{2}\sigma^2)(T-t) + \sigma [W(T) - W(t)]}$

In this case it is important to realize that the increments $W(T_1)-W(t)$ and $W(T_2) - W(t)$ are correlated since the intervals overlap. The correlation is $\rho = \frac{T_1-t}{(T_1-t)^{1/2} (T_2-t)^{1/2}} = \frac{(T_1-t)^{1/2}}{(T_2-t)^{1/2}} = \left( \frac{\tau_1}{\tau_2} \right)^{1/2}$ where $\tau_i$ is time to reach time $T_i$ . Hence for the purposes of pricing one can use

$X(T_i) = X(t) e^{(r- \frac{1}{2}\sigma^2)(T_i-t) + \sigma (T_i - t)^{1/2} Z_i}$ where $Z_1,Z_2$ is normal(0,1; $\rho$ ).

The payoff function of the second-order asset | bond binary is $F(x) = x | 1$ . Hence by the FTAP and bivariate GSL,

[note: need to fix signs]

$1_{A}(t) = \mathbb{Q}_t e^{-r\tau_2} X(T_2) \mathbb{I}(s_1X(T_1) > s_1\xi_1, s_2 X(T_2) > s_2 \xi_2) \\ = \mathbb{Q}_t X(t) e^{\sigma \tau_2^{1/2} Z_2 - \frac{1}{2} \sigma^2 \tau_2} \mathbb{I}(s_1Z_1 > -s_1d_1, s_2 Z_2 > -s_2 d_2) \\ = X(t) \mathbb{Q}_t \mathbb{I}(s_1(Z_1+ \rho \sigma \tau_2^{1/2}) > -s_1d_1, s_2 (Z_2 + \sigma \tau_2^{1/2}) > -s_2 d_2) \\ = X(t) N(-s_1(d_1 + \sigma \tau_2^{1/2}), -s_2(d_2 + \sigma \tau_1^{1/2}); s_1 s_2 \rho)$

The bond binary is more simply priced:

$1_{B}(t) = \mathbb{Q}_t e^{-r\tau_2} \mathbb{I}(s_1X(T_1) > s_1\xi_1, s_2 X(T_2) > s_2 \xi_2) \\ = e^{-r\tau_2} \mathbb{Q}_t X(t) \mathbb{I}(s_1Z_1 > -s_1d_1, s_2 Z_2 > -s_2 d_2) \\ = e^{-r\tau_2} \mathbb{Q}_t \mathbb{I}(s_1 Z_1 > -s_1d_1, s_2 Z_2 > -s_2 d_2) \\ = e^{-r\tau_2} N(-s_1d_1, -s_2d_2; s_1 s_2 \rho)$