Put-Call Parity

Study guides: Exam FAM, Exam INV-201

Put-call parity is the no-arbitrage identity linking a European call $C$ and a European put $P$ on the same underlying, with the same strike $K$ and expiry $T$ : a long call plus a short put pays $S_T - K$ in every state, so it must cost what a forward purchase of the asset at $K$ costs today, the present value of the asset less the present value of the strike.

$C - P = F^P_{0,T}(S) - K e^{-rT}$

$C - P = S_0 e^{-\delta T} - K e^{-rT}$

$F^P_{0,T}(S)$ is the prepaid forward price: $S_0$ for a non-dividend stock, $S_0 e^{-\delta T}$ with a continuous dividend yield $\delta$ , and $S_0 - PV_{0,T}(\text{Div})$ with discrete dividends; $r$ is the continuously compounded risk-free rate
It is model-free: it follows from the payoffs alone ( $\max(S_T-K,0) - \max(K-S_T,0) = S_T - K$ ), so it holds under the binomial model, the Black-Scholes Model and any other arbitrage-free model, and under any volatility
Rearranged it builds synthetic positions: long call + short put = synthetic forward; stock + put = call + bond (the protective put behind a guaranteed minimum maturity benefit)
It holds only for European options. For American options on a non-dividend stock it weakens to the bounds $S_0 - K \leq C - P \leq S_0 - K e^{-rT}$ , part of the price bounds INV-201 pairs with parity

Pricing a Put from a Quoted Call An index is at $S_0 = 50$ and pays dividends continuously at $\delta = 0.02$ . The continuously compounded risk-free rate is $r = 0.05$ . A 6-month European call with strike $55$ costs $2.80$ . Find the price of the 6-month European put with strike $55$ .
Answer
 $\begin{align*} P &= C - S_0 e^{-\delta T} + K e^{-rT} \\ &= 2.80 - 50e^{-0.01} + 55e^{-0.025} \\ &= 2.80 - 49.5025 + 53.6420 \\ &= 6.94 \end{align*}$ 
The put is worth $6.94$ : it is in the money ( $K > S_0$ ), so it costs more than the out-of-the-money call.

Exploiting a Parity Violation A non-dividend-paying stock trades at $100$ , $r = 0.04$ , and 1-year European options with strike $100$ are quoted at $C = 10.50$ and $P = 6.00$ . Show an arbitrage and its profit per share.
Answer Parity requires
 $\begin{align*} C - P &= S_0 - K e^{-rT} \\ &= 100 - 100e^{-0.04} \\ &= 3.9211 \end{align*}$ 
but the quotes give $C - P = 4.50$ , so the call is rich relative to the put. Sell the call, buy the put, buy the stock, and borrow $96.0789$ (the present value of $K$ ). Today's cash flow is
 $\begin{align*} \Pi_0 &= 10.50 - 6.00 - 100 + 96.0789 \\ &= 0.5789 \end{align*}$ 
At $T$ the stock plus put minus call is worth exactly $K = 100$ whatever $S_T$ is, which repays the loan. The riskless profit is $0.58$ per share, the gap $4.50 - 3.9211$ .

Parity with Discrete Dividends A stock trades at $80$ and will pay dividends of $1$ at $t = 0.25$ and $t = 0.75$ . With $r = 0.06$ , a 1-year European put with strike $75$ costs $3.20$ . Find the price of the 1-year European call with strike $75$ .
Answer
 $\begin{align*} PV(\text{Div}) &= e^{-0.06(0.25)} + e^{-0.06(0.75)} \\ &= 1.9411 \\ C &= P + (S_0 - PV(\text{Div})) - K e^{-rT} \\ &= 3.20 + 78.0589 - 75e^{-0.06} \\ &= 3.20 + 78.0589 - 70.6323 \\ &= 10.63 \end{align*}$ 
The call is worth $10.63$ . Ignoring the dividends would give $12.57$ : the call holder does not receive them, so they come off the prepaid forward.

How it appears on the exam

FAM (objective "apply put-call parity"): given three of $C$ , $P$ , $S_0$ , $K$ , $r$ , $\delta$ , solve for the fourth, often a put price from a Black-Scholes call or a binomial call computed earlier in the question.

FAM: parity used backwards to find an implied dividend yield, an implied risk-free rate, or the strike at which $C = P$ (the forward price, $K = S_0 e^{(r-\delta)T}$ ).

INV-201 ("understand put-call parity and price bounds"): identify which quotes violate parity or the bounds, and describe the arbitrage trades and their cash flows today and at expiry, as a written answer.

INV-201: parity for delta and the other Greeks, $\Delta_C - \Delta_P = e^{-\delta T}$ and equal gammas and vegas, used to convert a hedge in calls into one in puts.

Exam traps

Dropping the dividend term. With a dividend yield the stock enters as $S_0 e^{-\delta T}$ , not $S_0$ ; with cash dividends subtract their present value. A put price that comes out too low by about $S_0(1-e^{-\delta T})$ is this slip.

Discounting the strike at the wrong rate basis. $Ke^{-rT}$ needs a continuously compounded $r$ ; given an annual effective rate, use $K(1+i)^{-T}$ . Mixing them moves the answer by cents and lands on a wrong choice.

Applying parity to American options. Early exercise breaks the equality; only the inequalities hold, and with dividends even those change.

Getting the arbitrage direction backwards. If $C - P$ exceeds $F^P - Ke^{-rT}$ , sell the call, buy the put and buy the stock with borrowed money; reversing every leg locks in a loss of the same size.