8.2 No-arbitrage relations that hold in every model

Put–call parity#

The present value of receiving $S_T$ at $T$ is $Se^{-qT}$ (buy $e^{-qT}$ units now and reinvest dividends). The present value of receiving $K$ at $T$ is $Ke^{-rT}$. So the forward payoff $S_T-K$ is worth

using the forward price $F$ derived in M07 · 7.1.

Proof

At $T$, $(S_T-K)^+-(K-S_T)^+=S_T-K$ for every $S_T$: if $S_T\ge K$ the left side is $S_T-K-0$, otherwise it is $0-(K-S_T)$. So a long call plus short put has the same payoff as a long forward at $K$. By the law of one price their values today agree, and the forward is worth (8.3).∎

If (8.4) fails, say $c-p > Se^{-qT}-Ke^{-rT}$, sell the call, buy the put, buy $e^{-qT}$ units of the underlying and borrow $Ke^{-rT}$. The initial cash flow $c-p-Se^{-qT}+Ke^{-rT}$ is positive and the terminal value is $-(S_T-K)^++(K-S_T)^++S_T-K=0$. This trade is called a conversion; the reverse is a reversal.

When a futures contract expires on the same day as the options (true for Nifty monthly options and futures), substitute $F$ from the futures market into (8.3):

This form needs no estimate of dividends. It is the form Indian desks use, and it lets you read the option-implied forward $F^{\text{impl}}=K+e^{rT}(c-p)$ from any strike.

Bounds on option prices#

Proof

$c\ge0$ because the payoff is non-negative (a negative price is an arbitrage: buy and receive cash, with no liability). Since $(S_T-K)^+\ge S_T-K$, the call dominates the forward, so $c\ge Se^{-qT}-Ke^{-rT}$. Since $(S_T-K)^+\le S_T$, the call is dominated by $e^{-qT}$ units of the underlying, so $c\le Se^{-qT}$. For the put, $(K-S_T)^+\ge K-S_T$ gives the lower bound and $(K-S_T)^+\le K$ the upper bound. Each inequality is a payoff domination; if the price inequality failed, buying the dominating portfolio and selling the dominated one would be an arbitrage.∎

The same argument gives monotonicity and convexity in the strike (Problem 8.3): $c$ is non-increasing and convex in $K$, with $-e^{-rT}\le \partial c/\partial K\le0$.

American options and early exercise#

Proof

The American holder can follow the European exercise policy, so the set of attainable payoffs contains the European one.∎

Proof

At any $t<T$, with $\tau=T-t$,

$$C_t \ \ge\ c_t \ \ge\ S_t - Ke^{-r\tau} \ \ge\ S_t-K ,$$

using Proposition 8.2.3, then the lower bound (8.6) with $q=0$, then $e^{-r\tau}\le1$. The left side is what the holder gets by selling the option; the right side is what she gets by exercising. Selling is at least as good, and strictly better if $r>0$. So no rational holder exercises early, and the American call's value equals that of the European call with the same exercise policy.∎

The proof has two economic parts. Exercising early pays $K$ earlier than necessary, which forgoes interest $K(1-e^{-r\tau})$. It also throws away the insurance in the call: if $S$ falls below $K$ by $T$, the unexercised holder loses nothing more. Neither argument works for a put: early exercise of a deep ITM put receives $K$ early, and when $S$ is near zero the remaining insurance is worth little. So American puts are worth strictly more than European ones in general (8.4). Dividends break the call result too: exercising just before an ex-dividend date captures the dividend (Problem 8.6).

Parity on Nifty prices#