Rolling A Parity Portfolio To An Intermediate Date

Why Revalue Parity at an Intermediate Date?

Put-call parity is often introduced by comparing option payoffs at expiration. In practice, a position may be initiated well before expiration and reviewed at an intermediate date. By then, the stock price may have changed, some dividends may already have been paid, and the discount factors to the remaining cash flows may differ from those at initiation. A correct parity check must use the information and cash flows that remain at the review date.

The central discipline is to distinguish a past cash flow from a future obligation or payment. A dividend already paid is no longer a future payment to include in the current parity equation. If the stock position received that dividend, the cash belongs in the portfolio’s cash-flow ledger; it is not an additional deduction from the stock’s current value.

Model and assumptions: Consider European calls and puts on the same non-dividend-adjusted underlying stock, with the same strike $K$ and expiration $T$. At a valuation date $t<T$, assume frictionless trading, no taxes or transaction costs, and deterministic discount factors $Z(t,u)$ for cash flows due at dates $u$. Any dividends before expiration are known fixed cash amounts. The stock price $S_t$ is its value at date $t$, after any dividend paid at that date. The parity equality below does not apply unchanged to American options or to mismatched strikes or expirations.

Intermediate-Date Put-Call Parity

Let $C_t$ and $P_t$ be the values at date $t$ of the call and put. Suppose the remaining dividends are $D_j$, payable at dates $t_j$ with $t<t_j\le T$. A dividend paid at or before $t$ is not part of this remaining-dividend list. With the stated assumptions, parity is

C_t-P_t=S_t-\sum_{t<t_j\le T}D_j Z(t,t_j)-K Z(t,T).

The discount factor $Z(t,u)$ is the date-$t$ value of one dollar payable at date $u$. The expression $\sum D_j Z(t,t_j)$ is the present value at $t$ of the remaining known dividends. If no dividends remain before expiration, the sum is zero.

Theorem: Under the model assumptions, the two portfolios Portfolio A: long one call and invest $K Z(t,T)$ in a risk-free investment that pays $K$ at $T$; and Portfolio B: long one put and one share of stock, and short claims to the remaining known dividends, have equal values at date $t$. In symbols, $C_t+KZ(t,T)=P_t+S_t-\sum D_jZ(t,t_j)$.

Why the Equality Holds

The justification starts with the expiration payoffs. At $T$, the call and put pay $\max(S_T-K,0)$ and $\max(K-S_T,0)$, respectively. For every possible $S_T$,

\max(S_T-K,0)+K=\max(K-S_T,0)+S_T.

Thus, a call plus a risk-free payment of $K$ at expiration has the same expiration value as a put plus a share of stock. But the stockholder also receives any dividends before $T$. To compare the portfolios consistently, Portfolio B includes short dividend claims: it receives each dividend as the stockholder and pays the same amount on the matching claim. These cash flows offset on each dividend date. The current value of those short claims is the negative of the present value of the remaining dividends.

If the portfolios had different values at $t$, an investor could buy the cheaper one and sell the more expensive one. Their future cash flows, including the dividend offsets, would match, leaving a positive initial difference without a future net liability. That would be an arbitrage. Therefore their date-$t$ values must be equal.

Rolling the Portfolio Through Time

“Rolling” does not mean that either portfolio’s market value must stay constant. The call, put, and stock values can change as the market changes. It means that at each review date the parity comparison is recalculated with the current stock price, the current option values, the discount factors from that date, and only the dividends still due.

Fix the review date.
Use $t$ as the date at which all portfolio values are being compared. Do not mix a price from one date with a discount factor from another.

List future cash flows.
Include each known dividend payable after the review date and through expiration. Exclude dividends that have already been paid.

Discount each cash flow from the review date.
Use $Z(t,t_j)$ for dividend $D_j$, and $Z(t,T)$ for the strike payment $K$.

Compare the portfolios.
Check whether $C_t+KZ(t,T)$ equals $P_t+S_t-\sum D_jZ(t,t_j)$. A discrepancy may indicate a pricing difference or a data, timing, or contract mismatch.

Worked Intermediate-Date Calculations

Worked Example: Finding a Call Value Before a Remaining Dividend

At an intermediate date, a stock is worth $104. One known dividend of $1.20 will be paid before expiration; its discount factor from today is 0.99. The strike is $100, the discount factor to expiration is 0.95, and the European put is worth $4.50. Find the parity-implied call value. Use the method of discounting each remaining cash flow from the intermediate date.

Step 1: Find the present value of the remaining dividend.

$$ 1.20(0.99)=1.188. $$

Step 2: Find the present value of the strike.

$$ 100(0.95)=95. $$

Step 3: Apply intermediate-date parity.

$$ C_t-P_t=104-1.188-95=7.812. $$

Since $P_t=4.50$, the call value is $C_t=4.50+7.812=12.312$, or $12.31 to the nearest cent. As a check, $C_t+KZ(t,T)=12.312+95=107.312$, and $P_t+S_t-\text{PV(dividends)}=4.50+104-1.188=107.312$. The common mistake avoided here is adding the dividend present value rather than subtracting it in the call-minus-put relation.

Worked Example: Rechecking Parity After a Dividend Is Paid

At date 0, a stock is worth $120. A known $3 dividend is payable at date 0.25, and the options expire at date 0.75. The discount factors are $Z(0,0.25)=0.99$ and $Z(0,0.75)=0.96$. The strike is $100, the call is worth $30, and the put is worth $8.97. At date 0.25, immediately after the dividend is paid, the stock is worth $117, the put is worth $6, and the discount factor to expiration is 0.97. Find the call value at date 0.25 and check parity at both dates.

At date 0: the present value of the dividend is $3(0.99)=2.97$, and the present value of the strike is $100(0.96)=96$. The left side of the portfolio-value equality is

$$ C_0+100Z(0,0.75)=30+96=126. $$

The other side is $P_0+S_0-\text{PV(dividend)}=8.97+120-2.97=126$. The two sides agree.

At date 0.25: the $3 dividend has been paid. It is not a remaining dividend, so the future-dividend sum is zero. The present value of the strike is $100(0.97)=97$. Therefore

C_{0.25}-6=117-97=20, \qquad C_{0.25}=26.

The portfolio values are $26+97=123$ and $6+117=123$, so parity still agrees. A past dividend must not be deducted again at date 0.25. The $117 stock value is already the post-dividend stock value; retaining the paid dividend in the future-dividend sum would count it twice.

Worked Example: Two Remaining Dividends and a Quoted Call

At a review date, a stock is worth $88. Two fixed dividends remain: $1.50 with discount factor 0.995, and $1.00 with discount factor 0.982. A European call and put have strike $90 and the same expiration; the discount factor to expiration is 0.94. The put is quoted at $7.25, while the call is quoted at $8.30. Calculate the parity-implied call and determine the direction and amount of the quote discrepancy. Use the method of comparing both portfolio values.

Step 1: Discount the remaining dividends.

1.50(0.995)=1.4925,\qquad 1.00(0.982)=0.982,

so their total present value is $1.4925+0.982=2.4745$. The present value of the strike is $90(0.94)=84.60$.

Step 2: Find the parity-implied call.

C_t-P_t=88-2.4745-84.60=0.9255, \qquad C_t=7.25+0.9255=8.1755.

Thus the parity-implied call is $8.18 to the nearest cent. The quoted call of $8.30 is $8.30-8.1755=0.1245$, or about $0.12 above the parity value.

Step 3: Check using portfolio values. At the quoted prices, Portfolio A is worth $8.30+84.60=92.90$. Portfolio B is worth $7.25+88-2.4745=92.7755$. Portfolio A is more expensive by $92.90-92.7755=0.1245$, consistent with the call quote being above parity. The comparison is made at the same date and includes both remaining dividends.

Common Timing and Financing Traps

Carrying a paid dividend forward into the formula. Only dividends payable after the valuation date belong in the remaining-dividend sum. A dividend already paid is part of the past cash-flow record.
Using the wrong discount date. Each dividend is discounted from the current review date to its own payment date. The strike is discounted from the review date to expiration.
Forgetting the strike-side investment. The matched portfolio contains an investment that pays $K$ at expiration, with current value $KZ(t,T)$. The current amount is generally not $K$.
Subtracting dividends twice. In $C_t-P_t=S_t-\text{PV(remaining dividends)}-KZ(t,T)$, dividends are subtracted once. Do not also adjust $S_t$ for those same future payments.
Treating parity as an unchanged portfolio value. Parity equates two portfolio values at a given date. It does not assert that either portfolio’s value remains fixed as the stock price, options, or time to expiration change.
Ignoring contract matching. The call and put must have the same underlying, strike, expiration, and relevant contract terms. Otherwise their expiration payoff identity does not establish this equality.

Key takeaway: At an intermediate date, revalue the strike payment and each still-future dividend from that date. Then compare $C_t+KZ(t,T)$ with $P_t+S_t-\sum D_jZ(t,t_j)$. A dividend already paid is not part of the remaining-dividend adjustment.

Check Your Understanding

For each problem, use the review date as the valuation date and include only dividends that remain payable after that date.

At an intermediate date, $S_t=$95$, $K=$100$, $Z(t,T)=0.97$, and a $2 dividend remains with discount factor 0.99. If $P_t=$10$, find the parity-implied call value. (Answer: $6.02)
A $2.50 dividend was paid before the review date. Now $S_t=$110$, $K=$105$, $Z(t,T)=0.96$, and $P_t=$4.25$. Find the call value and state whether the paid dividend is deducted. (Answer: Call value is $13.45; the paid dividend is not deducted.)
At the review date, $S_t=$140$, $K=$135$, and $Z(t,T)=0.93$. Remaining dividends are $2 with discount factor 0.98 and $1.50 with discount factor 0.95. If $P_t=$6.40$, find the parity-implied call value. (Answer: $17.47 to the nearest cent.)
At a review date, $S_t=$76$, $K=$80$, $Z(t,T)=0.96$, and a remaining $1.20 dividend has discount factor 0.99. The put is $8.10 and the call quote is $6.60. Find the parity-implied call and say whether the quote is above or below it. (Answer: Parity-implied call is $6.11 to the nearest cent; the quote is about $0.49 above it.)
At an intermediate date, $S_t=$125$, $K=$120$, $Z(t,T)=0.95$, and a $2 dividend remains with discount factor 0.99. A different $1 dividend was already paid. If $P_t=$5.50$, find the call value and explain whether the paid $1 dividend enters. (Answer: Call value is $14.52; the paid $1 dividend does not enter.)

What You'll Learn