Put-Call Parity With A Continuous Dividend Yield

Why a Continuous Dividend Yield Changes Parity

For a non-dividend-paying stock, European put-call parity compares a call plus a bond with a put plus the stock. When the stock pays dividends continuously, owning it also produces a stream of dividend income during the option’s life. The stock term in parity must therefore reflect the value today of obtaining the stock at expiration after accounting for that income.

This tutorial uses continuous compounding throughout. The time \(T\) is measured in years, \(r\) is the continuously compounded risk-free interest rate, and \(q\) is the constant continuous dividend yield. Thus, over \(T\) years, a dollar invested at rate \(r\) grows by the factor \(e^{rT}\), while discounting a future dollar gives the factor \(e^{-rT}\). The same time unit must be used for both rates.

The parity equation below applies to European calls and puts with the same underlying, strike \(K\), and expiration time \(T\). It assumes no arbitrage, frictionless trading, the ability to borrow and lend at the same risk-free rate \(r\), and a constant continuous dividend yield \(q\). Taxes, transaction costs, short-sale restrictions, and early exercise are not included. In particular, the equality is not the general parity formula for American options.

Why the Underlying Term Is \(S e^{-qT}\)

The quantity \(S e^{-qT}\) is the value today of a prepaid forward that delivers one share at time \(T\). To see the dividend adjustment, suppose dividends are reinvested continuously in additional shares. If an investor holds \(n(0)\) shares initially, a yield \(q\) makes the number of shares grow according to \(dn/dt=qn\). Hence \(n(T)=n(0)e^{qT}\). Starting with \(e^{-qT}\) shares therefore produces one share at time \(T\), and the initial cost of that holding is \(S e^{-qT}\). This argument is specific to the constant continuous-yield model.

Now compare two portfolios. Portfolio A consists of one call and a risk-free investment of \(K e^{-rT}\). The investment grows to \(K\) at expiration, so its combined payoff is \(\max(S_T-K,0)+K=\max(S_T,K)\). Portfolio B consists of one put and the prepaid forward. At expiration, the forward delivers a share worth \(S_T\), so the combined value is \(\max(K-S_T,0)+S_T=\max(S_T,K)\). The portfolios have equal expiration values in every state. By no arbitrage, their initial costs must be equal, which gives the parity equation.

Rearranging the same identity is useful when a different market value is unknown:

The two discounted quantities have distinct roles. The bond component is the present value of the fixed strike payment at expiration. The adjusted stock component is the present value of the share delivered at expiration, allowing for the yield earned by holding the underlying. Do not substitute \(S\) for \(S e^{-qT}\) unless \(q=0\).

Using Parity to Find a Call Price

When the put price is known, solve for the call by adding the adjusted stock value and subtracting the present value of the strike. This order also gives a useful sign check: the call price is \(P\) plus one discounted asset value minus the discounted strike. A calculation that discounts both \(S\) and \(K\) at \(r\), or discounts both at \(q\), does not follow the model.

Using a Parity Violation to Identify an Arbitrage Direction

If quoted option prices do not satisfy parity, compare the total prices of the two portfolios rather than guessing which individual option is mispriced. The parity sides represent portfolios with identical expiration values. If one side costs more, sell that portfolio and buy the cheaper one. In the idealized model, the initial price difference is received immediately and the equal future values cancel.

Common Rate and Sign Errors

A common mistake is to use the no-dividend equation \(C+K e^{-rT}=P+S\) even though \(q\ne0\). For the first worked example, that incorrect calculation would give

The correction is to account for the continuous yield in the stock term:

The wrong calculation understates the parity-implied put by approximately \(\$3.92\). It is wrong because it treats the stock as if it paid no dividends during the two-year period. Other common traps are mixing an annual rate with a time measured in months, using a simple-interest factor instead of continuous compounding, or subtracting the adjusted stock and strike terms in the wrong order when solving for an option price.

• Use \(e^{-rT}\) for the fixed strike and \(e^{-qT}\) for the underlying term.
• Measure \(T\) in years when \(r\) and \(q\) are stated as annual rates.
• Keep the same underlying, strike, and expiration on both options.
• Check the result by substituting it into both sides of parity before rounding.

This adjusted identity is aligned with the SOA Exam FM (Financial Mathematics) syllabus. It is especially useful when comparing option quotes, inferring an unknown option price, or checking whether a proposed portfolio uses the correct dividend treatment. It should not be applied unchanged to known cash dividends, a changing yield, or an option that can be exercised early; those situations require assumptions and methods appropriate to their own models.