Why Continuous Dividend Yield Changes Parity
Put–call parity links the prices of a European call, a European put, the underlying share, and a risk-free investment. When a share pays dividends, holding it today is not equivalent to receiving a share only at expiration: the holder of the share receives dividends along the way, while the holder of a European option does not. A continuous dividend yield adjusts the share term to account for those distributions.
This relationship is useful for checking option prices and for calculating a price implied by the other market inputs. The key is to match the contracts exactly and discount the exercise price and share value using the correct rates and time period.
\(C + Ke^{-\delta T}=P+S_0e^{-qT}\).
Here \(Ke^{-\delta T}\) is the present value of \(K\), and \(S_0e^{-qT}\) is the prepaid forward value of a share to be delivered at expiration. The equation applies to matching European options; it is not, without adjustment, a parity identity for American options or for a share with discrete dividends that are not represented by the assumed continuous yield.
What the Discounted Terms Represent
With a continuously compounded rate \(\delta\), one dollar payable in \(T\) years has present value \(e^{-\delta T}\). Thus \(Ke^{-\delta T}\) is the amount invested at the risk-free rate today that will grow to \(K\) at expiration. The time \(T\) must be in years when \(\delta\) and \(q\) are annual rates.
The dividend-adjusted share term has a different interpretation. A prepaid forward is an agreement paid for today that delivers one share at time \(T\), without requiring a payment for the share at delivery. Under a constant continuous dividend yield \(q\), its value is \(S_0e^{-qT}\). This is the share value adjusted for the dividends that a share held directly would pay before expiration. In parity, the prepaid forward is not interchangeable with the current share price \(S_0\).
| Term | Meaning today | What it provides at expiration |
|---|---|---|
| \(Ke^{-\delta T}\) | Present value of the fixed exercise-price payment | \(K\) dollars |
| \(S_0e^{-qT}\) | Prepaid forward value of one share | One share, with no payment at delivery |
The continuous rates are applied over the same horizon. A rate quoted per year should not be multiplied directly by a number of months: first express the time in years, then evaluate the exponentials.
Why the Relationship Holds
The equation follows by comparing two portfolios with the same expiration payoff. This argument requires the matching European contracts and the standard assumptions in the definition: the share follows the specified continuous-yield convention, borrowing and lending at the risk-free rate are available, and there are no transaction costs or restrictions that prevent replication.
Buy one call and invest \(Ke^{-\delta T}\) in a risk-free asset. The investment grows to \(K\) at expiration, so the portfolio value then is \(\max(S_T-K,0)+K=\max(S_T,K)\).
Buy one put and a prepaid forward that delivers one share at expiration. Its expiration value is \(\max(K-S_T,0)+S_T=\max(S_T,K)\).
The two portfolios have equal expiration values in every state. Under no arbitrage, their prices today must therefore be equal: \(C+Ke^{-\delta T}=P+S_0e^{-qT}\).
The identities in the first two steps hold for either \(S_T\geq K\) or \(S_T<K\). For example, if \(S_T\geq K\), the first portfolio is worth \(S_T-K+K=S_T\), while the second is worth \(0+S_T=S_T\). If \(S_T<K\), they are worth \(0+K=K\) and \(K-S_T+S_T=K\), respectively. Since their terminal values agree in both cases, the no-arbitrage price equality gives parity.
Rearranging Parity to Find an Implied Price
The equation can be rearranged according to which quantity is unknown. For example, the put price implied by a call price is
The corresponding call-price form is
To infer a share price from matching calls and puts, isolate \(S_0\):
These equations are not separate valuation rules; each is an algebraic rearrangement of the same parity identity. Keep the sign of each term when rearranging. In particular, the put price equals the call price plus the present value of the exercise price minus the dividend-adjusted share term.
Worked Example: Finding a Put Price with a Continuous Yield
A hypothetical share has \(S_0=\$75\), a continuous dividend yield \(q=0.015\), and a continuously compounded risk-free rate \(\delta=0.035\). Matching European options expire in \(T=0.5\) years, have \(K=\$78\), and the call price is \(C=\$5.20\). Find the put price to the nearest cent and check it in the parity equation.
Method and assumptions. Use the European parity equation because the call and put have the same share, exercise price, and expiration. Rates are annual, so the half-year horizon is \(0.5\) in each exponent.
First calculate the discounted terms:
Now solve for the put:
Rounded to the nearest cent, \(P=\$7.41\). To check, use the unrounded value:
The two sides agree. Common mistake avoided: using \(S_0\) in place of \(S_0e^{-qT}\) omits the continuous dividend-yield adjustment and produces the wrong put price.
Worked Example: Calculating a Synthetic Call Price
A hypothetical share is priced at \(S_0=\$120\), with continuous dividend yield \(q=0.020\). The continuously compounded risk-free rate is \(\delta=0.050\). A European put with \(K=\$125\) and \(T=1.5\) years costs \(P=\$9.40\). Find the price of the matching European call to the nearest cent.
Method and assumptions. Use \(C=P+S_0e^{-qT}-Ke^{-\delta T}\). The options match in share, exercise price, and expiration, and the two annual rates are constant over the 1.5-year term.
Calculate the adjusted share term:
Calculate the present value of the exercise price:
Substitute both values into parity:
The synthetic call price is \(\$9.89\) to the nearest cent. As a check, \(C+Ke^{-\delta T}\approx9.88552+115.96794=125.85346\), and \(P+S_0e^{-qT}=9.40+116.45346=125.85346\). Common mistake avoided: subtracting the adjusted share term instead of the discounted exercise price reverses the parity signs.
Worked Example: Inferring a Synthetic Share Price
Matching European options on a hypothetical share have \(C=\$14.80\), \(P=\$8.10\), and \(K=\$90\). They expire in \(T=2\) years. The continuous dividend yield is \(q=0.030\), and the continuously compounded risk-free rate is \(\delta=0.040\). Use parity to infer the share price today, to the nearest cent.
Method and assumptions. Rearrange parity to \(S_0=e^{qT}(C-P+Ke^{-\delta T})\). This infers the share price consistent with the two option prices under the stated constant-rate model.
First find the present value of the exercise price:
The call-minus-put difference is \(C-P=14.80-8.10=\$6.70\). Therefore, the implied dividend-adjusted share value is
Undo the dividend-yield adjustment:
The inferred share price is \(\$95.33\). Check by discounting this price for the continuous yield: \(95.33219e^{-0.060}\approx \$89.78047\). Then \(P+S_0e^{-qT}\approx8.10+89.78047=\$97.88047\), while \(C+Ke^{-\delta T}\approx14.80+83.08047=\$97.88047\). Common mistake avoided: stopping at \(\$89.78047\) treats the dividend-adjusted prepaid forward value as if it were the current share price.
Common Traps and Limits of the Formula
Parity is sensitive to contract matching and to the meaning of each rate. Before using it, confirm that both options are European, written on the same underlying share, and have the same \(K\) and \(T\). A different expiration or exercise price means the two portfolios in the derivation do not have the same payoff.
- Omitting the yield. With \(q>0\), the share term is \(S_0e^{-qT}\), not \(S_0\). The standard no-dividend form is recovered when \(q=0\).
- Discounting at the wrong rate. The exercise price is discounted using \(\delta\); the dividend yield \(q\) adjusts the share term. Swapping these rates changes the equation.
- Using inconsistent time units. If rates are annual, \(T=9\) months must be entered as \(0.75\) years, not as \(9\).
- Applying the continuous-yield model to discrete dividends without adjustment. A known cash dividend on a specific date is not automatically the same as a constant proportional yield. The dividend model must match the security’s actual cash flows.
- Applying European parity directly to American options. Early exercise can change the relationship. American options require bounds and additional reasoning rather than this equality.
In practice, prices may differ slightly from a theoretical equality because of bid–ask spreads, transaction costs, taxes, financing constraints, or model assumptions that do not hold exactly. Under the stated ideal assumptions, however, a discrepancy means the two equal-payoff portfolios would have different prices and the parity relationship would fail.
Check Your Understanding
For each problem, use annual rates and time in years. Round dollar answers to the nearest cent unless stated otherwise.
- A hypothetical share has \(S_0=\$100\), \(q=0.010\), and \(\delta=0.040\). Matching European options have \(K=\$102\), \(T=1\), and \(C=\$8.00\). Find the parity-implied put price. (Answer: \(\$7.00\))
- A matching European put costs \(\$6.50\). The share price is \(S_0=\$64\), the exercise price is \(K=\$66\), \(q=0.025\), \(\delta=0.045\), and \(T=2\). Find the synthetic call price. (Answer: \(\$7.06\))
- Matching options have \(C=\$11.00\), \(P=\$7.00\), and \(K=\$100\). They expire in \(1.5\) years; \(q=0.020\) and \(\delta=0.030\). Infer the share price today. (Answer: \(\$102.63\))
- A share costs \(\$80\). Matching options have \(C=\$8.90\), \(P=\$7.10\), and \(K=\$82\); \(T=1.25\) years and \(\delta=0.040\). Assuming a constant continuous dividend yield, infer \(q\) to four decimal places. (Answer: \(q\approx0.0020\), or about \(0.20\%\) per year)
- A matching European put costs \(\$12.00\). The share price is \(\$150\), \(K=\$155\), \(q=0.040\), and \(\delta=0.060\). Expiration is in nine months. Find the synthetic call price, taking care to convert the time to years. (Answer: \(\$9.39\))