Equity – Sisi Tang

We assume no dividend and positive risk-free interest rate.

European put-call parity

European put and call option with same maturity $T$ and strike $K$ satisfy the put-call parity:

$C_E - P_E = S_0 - Ke^{-rT},$

where $C_E$ is the price of European call option, $P_E$ is the price of the European put option, $S_0$ is the price of the underlying asset at time $t=0$ .

$C_E - P_E$ can be seen as a forward contract with maturity $T$ and strike $K$ . A short proof of European put-call parity is as follows:

$(S_T-K)^+ + (K_T-S)^+ = S_T - K$

That is to say the terminal payoff of long call and short put is equal to that of forward (with the same maturity $T$ and strike $K$ ). Hence,

$\begin{aligned}& P(0,T)E[(S_T-K)^+] + P(0,T)E[(K-S_T)^+] \\ & = P(0,T)E[S_T - K],\end{aligned}$

where $P(0,T)$ is the discount factor from $T$ to $0$ , and $E$ is the expectation under the risk neutral measure. Above equation is equivalent to the European put-call parity formula.

Never prematurely exercise American call option

If we wait until maturity, the profit of the call option is $(S_T-K)^+$ . If we exercise the option at time $t$ , then we have $-K$ cash position and a stock the worth $S_t$ at this time. Then, at time $T$ , the total portfolio value would be $S_T-Ke^{r(T-t)}$ . That is

$C_E = C_A$

But if the underlying asset pays a dividend, then it might be optimal to exercise the call option early.

American put-call parity

American put and call option satisfies the following inequality:

$S_0 - K \leq C_A-P_A \leq S_0 - Ke^{-rT}$

For the first inequality,

Suppose at time 0, we have the following portfolio: long a call option, short a put option, short underlying asset, and have $K$ cash.

At $t=0$ , the portfolio value is $C_A - P_A - S_0 + K$ .

If the long position side of the put option decides to exercise the option, we then exercise our call option at the same time, otherwise, wait until maturity to decide exercise or not. With this strategy, call and put has the same exercise time and hence can be seen as a forward with maturity undetermined. Say, the maturity is $t \in [0,T]$ .
Then, at time $t$ , the value of our portfolio is $S_t-K$ for the option part and $-S_t + Ke^{rt}$ for the asset and cash part. The total value of the portfolio is $Ke^{rt} - K > 0$ .

By the non-arbitrage principal, we have

$C_A - P_A - S_0 + K \geq 0$

For the second inequality,

$\begin{aligned} &~ C_A - P_A \\ = &~ C_E - P_E + P_E - P_A \\ = &~ S_0 - K e^{-rT} + P_E - P_A \\ \leq &~ S_0 - Ke^{-rT}\end{aligned}$

Let a stock follow a Geometric Brownian motion

$\textnormal{d}S_t = S_t(r \textnormal{d} t + \sigma \textnormal{d} W_t)$

with constant $r,\sigma > 0$ . Let

$A_T = \frac{1}{T}\int_0^T S_u \textnormal{d} u$

European Asian call option has the payoff $(A_T-K)^+$ and European vanilla call option has the payoff $(S_T-K)^+$ . Then European vanilla option has higher value.

Proof.

By (1)Jensen’s inequality, (2)Fubini, and (3)the fact that the longer the maturity is the higher the vanilla European option’s value is, we have

$\begin{aligned} &~ \mathbb{E}[e^{-rT}(A_T-K)^+] \\= &~ \mathbb{E}[e^{-rT}(\frac{1}{T}\int_0^T S_u \textnormal{d} u -K)^+] \\\leq & ~ \mathbb{E}[e^{-rT}\frac{1}{T}\int_0^T (S_u-K)^+ \textnormal{d} u] \\= &~ \frac{1}{T}\int_0^T \mathbb{E}[e^{-rT} (S_u-K)^+ ] \textnormal{d} u \\\leq &~ \frac{1}{T}\int_0^T \mathbb{E}[e^{-ru} (S_u-K)^+ ] \textnormal{d} u \\\leq &~ \frac{1}{T}\int_0^T \mathbb{E}[e^{-rT} (S_T-K)^+ ] \textnormal{d} u \\ = &~ \mathbb{E}[e^{-rT} (S_T-K)^+ ] \\ \end{aligned}$

Notice that we didn’t use the dynamics of Geometric Brownian motion. Above deduction is true as long as (3) is true.

Category: Equity

European and American Put-Call Parity

European call option v.s. Asian call option