Implied Dividend Yield from Put-Call Parity

Worked Solution

How to Think About It: Put-call parity is the single most useful no-arbitrage relationship in equity options. It holds model-free (no volatility assumption needed) and lets you extract market-implied information about dividends, forward prices, and interest rates. In practice, when you see call and put mid-prices at the same strike and maturity, you can immediately back out the forward price -- and from that, the implied dividend yield. This is how equity derivatives desks calibrate their dividend assumptions daily.

Quick Estimate: With $C - P = 8.00 - 6.50 = 1.50$, and $K = S_0 = 100$, parity says $C - P = S_0 e^{-qT} - K e^{-rT}$. Since the strike and spot are both 100, the difference $C - P$ is driven entirely by the gap between the dividend discount and the interest rate discount. With $r = 0.02$ and $T = 1$, $Ke^{-rT} \approx 98.02$. So $S_0 e^{-q} \approx 1.50 + 98.02 = 99.52$, giving $e^{-q} \approx 0.9952$, i.e., $q \approx 0.0048$ or about 48 bps. Very small dividend yield -- makes sense for a growth stock.

Approach: Apply put-call parity with continuous dividends and solve for $q$.

Formal Solution:

Put-call parity with continuous dividend yield:

$C - P = S_0 e^{-qT} - K e^{-rT}$

Plugging in:

$8.00 - 6.50 = 100 \, e^{-q \cdot 1} - 100 \, e^{-0.02 \cdot 1}$

$1.50 = 100 \, e^{-q} - 100 \, e^{-0.02}$

$1.50 = 100 \left(e^{-q} - e^{-0.02}\right)$

$e^{-q} = 0.015 + e^{-0.02} = 0.015 + 0.98020 = 0.99520$

$q = -\ln(0.99520) \approx 0.00481$

So the implied continuous dividend yield is $q \approx 0.48\%$.

Part 2 -- Discrete Dividend Effect:

If the stock pays a single discrete cash dividend $D$ at time $t_d$, the correct parity becomes:

$C - P = S_0 - D \, e^{-r t_d} - K e^{-rT}$

The present value of the discrete dividend reduces the forward price by $D e^{-r t_d}$, which is a lump amount rather than a continuous yield. Converting this to an equivalent continuous yield depends on the timing $t_d$: a dividend paid early in the year has a larger impact on the equivalent $q$ than one paid late, because the continuous yield model assumes the "leakage" is spread evenly. In general, for a fixed total dividend amount, the implied $q$ from the naive continuous model will overestimate or underestimate the true forward depending on where in the year the dividend falls relative to the "center of mass" assumed by the continuous model.

Directionally: if the discrete dividend is large and paid early, the effective reduction in the forward price is larger than what a small continuous $q$ would produce, so the naive $q$ would need to be higher. If the dividend is paid late, the present-value effect is smaller.

Part 3 -- Backing Out Implied PV of Dividends:

When dividends are unknown, use the model-free version of parity:

$C - P = S_0 - \text{PV}(\text{Divs}) - K e^{-rT}$

Rearranging:

$\text{PV}(\text{Divs}) = S_0 - K e^{-rT} - (C - P)$

This gives the market-implied present value of all dividends between now and expiry. You observe $S_0$, $K$, $r$, $T$, $C$, and $P$ -- everything on the right side is known. In our example:

$\text{PV}(\text{Divs}) = 100 - 100 \, e^{-0.02} - 1.50 = 100 - 98.02 - 1.50 = 0.48$

This is exactly how practitioners extract implied dividends from listed options across multiple maturities.

Answer: The implied continuous dividend yield is $q \approx 0.48\%$. With a discrete dividend, parity becomes $C - P = S_0 - D e^{-r t_d} - K e^{-rT}$, and the equivalent continuous $q$ shifts depending on dividend timing. To back out implied dividends model-free: $\text{PV}(\text{Divs}) = S_0 - K e^{-rT} - (C - P)$.