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I know that Put-Call parity allows us to find the fair price of a call/put for options with the same strike price and same expiry. In the example I am working on, I have a table showing values for Spot Price $S_0$, call price $c$, put price $p$ and strike price $X$. The strike price and expiry are equal for either option.

I have calculated that the $p$ value obtained by the put-call parity equation is less than the $p$ value in the table.

Likewise, I have found that the $c$ value obtained by the put-call parity equation is greater than the $c$ value in the table

What arbitrage method would I use to exploit this mispricing for risk-free profit? Any advice is appreciated.

rainingagain
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1 Answers1

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To keep things simple assume the stock pays no dividend and the risk-free rate of interest is $r$. The put-call parity relationship is

$$C_0 -P_0 = S_0 - K e^{-rT},$$

where $C_0$ and $P_0$ are the respective fair prices for the call and put options with identical strike prices $K$ and times-to-expiration $T$.

Given market prices $c < C_0$ and $p > P_0$, we have an arbitrage opportunity where we buy the call option, sell the put option, sell short the stock, and buy a zero-coupon bond with face value $K$ maturing at time $T$ The net proceeds of setting up this portfolio is

$$\Pi= -c + p + S_0 - Ke^{-rT} > -C_0 + P_0 + S_0 - Ke^{-rT} = 0.$$

At the time of expiration, the portfolio is worth

$$\underbrace{\max(S_T-K,0)}_{\text{payoff on long call}}-\underbrace{\max(K-S_T,0)}_{\text{payoff on short put}} -\underbrace{S_T}_{\text{short stock repayment}}+ \underbrace{K}_{\text{principle from long bond}} \\= \begin{cases}S_T-K -S_T+K = 0,& S_T\geqslant K\\-(K-S_T) -S_T+K,&S_T < K \end{cases}=0$$

Thus, a riskless profit $\Pi$ has been earned.

RRL
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