Contract Multipliers And Share Quantities In Parity Trades

Why Contract Size Matters

Put-call parity is often written per share, but options are commonly traded in contracts that each cover a specified number of shares. A parity calculation can therefore be correct per share and still produce a mismatched trade if the contract multiplier is ignored. The stock position, option positions, and financing position must all refer to the same number of shares.

This tutorial uses European call and put options on the same underlying, with the same strike price $K$ and expiration time $T$. Assume the underlying pays no dividends before expiration, trading is frictionless, short sales are permitted, and borrowing and lending use the same effective annual rate $i$. Time $T$ is measured in years, so the discount factor is $v^T=(1+i)^{-T}$. Option premiums are quoted per share. These assumptions give the per-share parity relation $C-P=S_0-Kv^T$.

Definition: A contract multiplier $M$ is the number of underlying shares represented by one option contract. If a position contains $n$ contracts, its share exposure is $nM$. A zero-coupon bond with face amount $F$ pays $F$ at expiration; under the assumptions above, its value at time 0 is $Fv^T$.

For a parity portfolio, the required bond face amount is the strike payment multiplied by the number of shares in the portfolio. Thus, for $Q$ shares, the financing position has face amount $QK$, not just $K$. If the bond is quoted in units with $100 face amount, divide the required face amount by 100 to obtain the number of bond units.

From Per-Share Parity to Whole Positions

For $Q$ matched shares, the per-share equality scales to

$$ Q(C-P)=QS_0-QKv^T. $$

The left side is the value of $Q$ long calls and $Q$ short puts. The right side is the value of $Q$ shares, less the value of a zero-coupon bond with face amount $QK$. If one call contract covers $M_C$ shares and one put contract covers $M_P$ shares, the option positions must be chosen so that

$$ n_C M_C=n_P M_P=Q. $$

When contract counts must be whole numbers, choose a share quantity $Q$ that is a common multiple of the two multipliers. For example, if the multipliers are 25 and 40 shares, the smallest common share quantity is 200: eight 25-share contracts and five 40-share contracts. If both options have the same multiplier, matching one call contract with one put contract gives equal share exposure.

Key relationship: For a matched quantity of $Q$ shares, a long call and short put position corresponds to long $Q$ shares and a short zero-coupon bond with face amount $QK$. The reverse positions correspond to short $Q$ shares and a long zero-coupon bond with face amount $QK$.

Why the Share and Bond Positions Match

The quantity relationship follows from the expiration payoffs, not from a convention about how contracts are packaged. For one share, a long call and a short put have combined payoff

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

If $S_T\ge K$, the call contributes $S_T-K$ and the put expires without value. If $S_T<K$, the call expires without value and the short put contributes $-(K-S_T)=S_T-K$. In either case, the combined payoff is the same as a share worth $S_T$ accompanied by a payment of $K$ at expiration. A short zero-coupon bond with face amount $K$ supplies that payment obligation. Multiplying every payoff by $Q$ gives the result for $Q$ shares.

This argument also explains why the contract multiplier cannot be applied only to the option premium. It scales the option payoff and premium exposure together, so it must also be reflected in the stock and bond quantities used to match those options.

Worked Example: Replicating Three Option Contracts

A non-dividend-paying share is priced at $106. A two-year European call and put have strike $121, and each contract covers 100 shares. The effective annual interest rate is 10%. The call premium is $10.00 per share and the put premium is $4.00 per share. Construct the positions corresponding to three long calls and three short puts, and verify parity at initiation and expiration.

Method and assumptions. The options have the same strike and expiration, and use the common rate specified in the problem. Three contracts of each type cover $3\times100=300$ shares.

Step 1: Calculate the discount factor and parity value per share.

v^T=(1.10)^{-2}=\frac{1}{1.21}=0.826446\ldots

The present value of the $121 strike payment per share is $121/1.21=\$100.00$. Therefore, the right side of parity is $106-100=\$6.00$ per share. The option difference is $10-4=\$6.00$ per share, so the quoted premiums satisfy parity.

Step 2: Scale the option positions to 300 shares. Buy three call contracts and sell three put contracts. Each position covers 300 shares in total. The net option premium paid is $300(10-4)=\$1{,}800$.

Step 3: Match the shares and strike payment. Buy 300 shares for $300(106)=\$31{,}800$. The required bond face amount is $300(121)=\$36{,}300$. Short that zero-coupon bond; its proceeds are $36{,}300/1.21=\$30{,}000$.

Step 4: Check the initiation cash flow. The replicated position costs $31{,}800-30{,}000=\$1{,}800$, exactly the net premium paid for the options.

Step 5: Check expiration. The option payoff is $300(S_T-121)$, including both possible regions of $S_T$. The stock and short bond payoffs are $300S_T$ and $-36{,}300$. Their sum is $300(S_T-121)$, matching the options.

Common mistake avoided: Using a bond face amount of $121 would cover only one share. The three contracts cover 300 shares, so the correct face amount is $36,300.

Arbitrage Positions with Different Multipliers

When parity prices do not agree, quantity matching works the same way: first find a common share exposure, then construct the corresponding stock and financing positions. The direction of the trade depends on which side is relatively inexpensive. A portfolio that is long calls and short puts has per-share value $C-P$; the corresponding stock-and-bond position has value $S_0-Kv^T$.

If $C-P<S_0-Kv^T$, the option position is cheaper. Buy the matched call-and-put package, short the shares, and buy the bond with face amount $QK$. If $C-P>S_0-Kv^T$, short the option package, buy the shares, and short the bond. In either direction, the combined expiration payoff is zero; the initial cash flow reflects the price difference.

Worked Example: Matching Calls and Puts with Unequal Multipliers

A share is priced at $101. European calls and puts have strike $100 and expire in one year. The effective annual interest rate is 5%. Call contracts cover 100 shares and put contracts cover 50 shares. The call premium is $8.20 per share and the put premium is $4.10 per share. Construct an arbitrage portfolio using the smallest whole-contract quantities that match the two option exposures.

Method and assumptions. The options have the same underlying, strike, and expiration, and the stated rate is the common financing rate. The smallest common share exposure is 100 shares: one call contract and two put contracts.

Step 1: Compare parity values per share. The call-minus-put value is $8.20-4.10=\$4.10$. The discount factor is $1/1.05=0.952381\ldots$, so the stock-and-bond value is

101-\frac{100}{1.05}=101-95.238095\ldots=\$5.761905\ldots

The option package is cheaper by $5.761905\ldots-4.10=\$1.661905\ldots$ per share. For 100 shares, the initial difference is $100(1.661905\ldots)=\$166.1905\ldots$.

Step 2: Take the positions for the cheaper side and its parity counterpart. Buy one call contract, sell two put contracts, short 100 shares, and buy a zero-coupon bond with face amount $100(100)=\$10{,}000$.

Step 3: Calculate the initiation cash flow. Buying the call costs $100(8.20)=\$820$. Selling two put contracts brings in $2(50)(4.10)=\$410$. Shorting 100 shares brings in $100(101)=\$10{,}100$. The bond costs $10{,}000/1.05=\$9{,}523.8095\ldots$. The net cash received is

-820+410+10{,}100-9{,}523.8095\ldots =\$166.1905\ldots

Step 4: Verify the expiration payoff. The call and short puts together pay $100(S_T-100)$. The short shares pay $-100S_T$, and the bond pays $+\$10{,}000$. The total is $100(S_T-100)-100S_T+10{,}000=0$.

Check. Rounding only the final cash amount to cents, the portfolio receives $166.19 initially and has a zero expiration payoff under the model assumptions.

Common mistake avoided: One put contract covers only 50 shares, not the 100 shares covered by one call contract. Selling only one put would leave an unmatched 50-share option exposure and would not produce the payoff checked above.

Worked Example: Reversing the Trade When the Option Package Is Expensive

A share is priced at $40. European calls and puts have strike $42 and expire in one year. The effective annual interest rate is 5%. Each call contract covers 25 shares, and each put contract covers 50 shares. The call premium is $2.30 per share and the put premium is $1.80 per share. Construct the smallest matched whole-contract portfolio and calculate its initial cash received.

Method and assumptions. Use the common multiple of the contract multipliers: 50 shares. The matched option positions are two call contracts and one put contract.

Step 1: Compare values per share. The option difference is $2.30-1.80=\$0.50$. The discounted strike is $42/1.05=\$40.00$, so $S_0-Kv^T=40-40=\$0.00$. The option package is overpriced by $0.50 per share.

Step 2: Take the reverse positions. Short two calls, buy one put, buy 50 shares, and short a zero-coupon bond with face amount $50(42)=\$2{,}100$.

Step 3: Calculate the initial cash received. Selling two calls brings in $2(25)(2.30)=\$115$. Buying the put costs $50(1.80)=\$90$. Buying the shares costs $50(40)=\$2{,}000$. Shorting the bond brings in $2{,}100/1.05=\$2{,}000$. The net amount received is $115-90-2{,}000+2{,}000=\$25$.

Step 4: Check expiration in both regions. If $S_T\ge42$, the short calls and long shares contribute $-50(S_T-42)+50S_T=2{,}100$, which is offset by the bond payment of $-2{,}100$. If $S_T<42$, the long put and long shares contribute $50(42-S_T)+50S_T=2{,}100$, again offset by $-2{,}100$. The expiration payoff is zero in either case.

Check. The $25 initial receipt equals $50$ shares multiplied by the $0.50 per-share parity difference. The contract quantities, share position, and bond face amount all refer to the same 50 shares.

Quantity Checks and Common Traps

Before treating a parity portfolio as matched, check each position in units rather than relying on the number of contracts. A call contract and a put contract do not necessarily represent the same number of shares. Likewise, a bond position should be recorded by its face amount and maturity payment, not just by its market price at initiation.

Convert contracts to shares.
Multiply each option contract count by its own multiplier. Confirm that the call and put exposures are equal.

Set the common share quantity.
Call this quantity $Q$. Use whole contract counts when required, choosing a common multiple of the multipliers.

Scale the underlying and financing.
Use $Q$ shares and a bond face amount $QK$. Discount that face amount to calculate its initiation value.

Check both dates.
Calculate the initial cash flow from premiums, shares, and bond proceeds. Then verify that the total payoff at expiration is zero for both $S_T\ge K$ and $S_T<K$.

A frequent error is to multiply the option premium by the contract multiplier but leave the stock position or bond face amount at a per-share quantity. The correction is to scale every part of the portfolio to the same $Q$. Another error is to treat a bond’s present value as its expiration payment: a bond with face amount $QK$ is purchased or sold at $QKv^T$ today, but its maturity cash flow is $QK$.

Key takeaway: A parity trade is matched only when the call exposure, put exposure, stock quantity, and strike-payment obligation all represent the same number of shares. Contract multipliers determine the option counts; the common share quantity determines the stock position and bond face amount.

Check Your Understanding

For each problem, assume European options on a non-dividend-paying underlying, the same strike and expiration for calls and puts, and the stated effective annual rate as the common financing rate.

A call contract covers 40 shares and a put contract covers 20 shares. What smallest whole-contract combination gives equal call and put exposure? If the strike is $55, what bond face amount matches that exposure? (Answer: 1 call contract and 2 put contracts; 40 shares; $2,200 face amount.)
Three matched option contracts each cover 100 shares. The strike is $90, expiration is in two years, and $i=5\%$. Find the stock quantity and the present value of the matching bond face amount. (Answer: 300 shares; bond face amount $27,000; present value $24,489.80.)
Call contracts cover 25 shares and put contracts cover 40 shares. Find the smallest whole-contract combination with equal exposure. If $K=\$32$, find the matching bond face amount. (Answer: 8 call contracts and 5 put contracts; 200 shares; $6,400 face amount.)
A share costs $70. A one-year call and put have $K=\$72$, and both multipliers are 100. With $i=4\%$, $C=\$5.50$, and $P=\$4.00$, determine the parity direction and the matched positions for an arbitrage portfolio. Find its initial cash receipt. (Answer: $C-P=\$1.50>S_0-Kv^T=\$0.7692$; short one call, buy one put, buy 100 shares, and short a $7,200-face bond; initial receipt $73.08.)
A call multiplier is 100 shares and a put multiplier is 75 shares. What smallest whole-contract combination matches the options? If $K=\$120$, state the share quantity and bond face amount. (Answer: 3 call contracts and 4 put contracts; 300 shares; $36,000 face amount.)

What You'll Learn