Cash-Settled And Physically Settled Parity Portfolios

Why Settlement Form Matters

Put-call parity compares a call, a put, the underlying asset, and financing. The comparison is often stated as a relationship between prices today, but its foundation is the portfolios’ values at expiration. A cash-settled option pays an amount of money determined by the contract’s settlement value. A physically settled option instead exchanges the underlying asset for the strike price when exercised.

Those settlement instructions look different, but they can produce the same terminal economic value when the contracts match precisely and the underlying can be valued or traded at the same expiration price. If the reference price, asset, quantity, or settlement time differs, the equivalence may fail. The distinction is therefore important when testing whether a quoted parity relationship applies.

Definition: A European call with strike $K$ and expiration $T$ has expiration payoff $\max(S_T-K,0)$; a European put has expiration payoff $\max(K-S_T,0)$. Under cash settlement, that amount is paid in cash. Under physical settlement, exercising a call means paying $K$ to receive the specified underlying, while exercising a put means delivering the specified underlying and receiving $K$.

Compare the Portfolios at Expiration

Assume one call, one put, and one unit of the underlying, all with the same strike $K$ and expiration $T$. Also assume the call and put use the same expiration value $S_T$, and one unit of the underlying is worth $S_T$ at that time. The standard portfolios to compare are:

Call-and-cash portfolio: long one call and hold enough risk-free cash at expiration to pay $K$.
Put-and-underlying portfolio: long one put and hold one unit of the underlying.

For a physical option, “hold enough risk-free cash at expiration” means investing the present value of $K$ in a risk-free instrument that pays $K$ at $T$. The portfolios may end with different combinations of cash and the underlying, so compare their terminal wealth, not just the cash paid by the options.

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

The first expression is the value of the call-and-cash portfolio when the call is cash-settled. The second is the value of the put-and-underlying portfolio, provided the underlying can be valued at $S_T$. The same equality also describes terminal wealth under physical settlement: exercise or retain the option and asset positions as appropriate. The cash flows and deliveries differ, but the resulting wealth agrees.

Worked Example: Cash-Settled Portfolios in Three Expiration States

A hypothetical European call and put both have strike $K=\$100$, expire at the same time, and settle in cash using the same underlying’s expiration price. Compare a long call plus $\$100$ held to expiration with a long put plus one unit of the underlying when $S_T=\$80$, $\$100$, and $\$130$. Ignore transaction costs and assume the underlying can be sold for $S_T$ at expiration.

Method. Calculate each option’s cash payoff first, then add the cash or underlying value in its portfolio. This avoids treating an option payoff as the value of the entire portfolio.

When $S_T=\$80$. The call payoff is $\max(80-100,0)=\$0$, so the call-and-cash portfolio is worth $0+100=\$100$. The put payoff is $\max(100-80,0)=\$20$; the put-and-underlying portfolio is worth $20+80=\$100$.

When $S_T=\$100$. Both option payoffs are $\$0$. The call-and-cash portfolio is worth $0+100=\$100$, and the put-and-underlying portfolio is worth $0+100=\$100$.

When $S_T=\$130$. The call payoff is $\max(130-100,0)=\$30$, so the first portfolio is worth $30+100=\$130$. The put payoff is $\max(100-130,0)=\$0$, and the second portfolio is worth $0+130=\$130$.

Check. In each state, both portfolios are worth $\max(S_T,100)$: $\$100$, $\$100$, and $\$130$, respectively. The comparison depends on using the same $S_T$ for both options and for the underlying.

What Physical Settlement Changes

With physical settlement, the option payoff is not necessarily handed over as cash. Instead, exercise triggers an exchange. For a call, the holder pays $K$ and receives the underlying. For a put, the holder gives up the underlying and receives $K$. A portfolio comparison must include those exchanges and any cash already held.

If $S_T>K$, the call in the call-and-cash portfolio is exercised: the $K$ cash is paid to receive the underlying, worth $S_T$. The put in the put-and-underlying portfolio expires unexercised, leaving the investor with the underlying, also worth $S_T$. If $S_T<K$, the call expires unexercised and the portfolio retains $K$ cash. The put is exercised: the underlying is delivered in exchange for $K$, leaving $K$ cash. At $S_T=K$, exercise or non-exercise gives the same value.

Worked Example: Following Physical Exercise and Delivery

A hypothetical European call and put each cover one unit of the same asset, have strike $K=\$75$, and expire at the same time. One portfolio holds the call and $\$75$ payable at expiration; the other holds the put and one unit of the asset. The options are physically settled. Compare terminal wealth when $S_T=\$60$ and $S_T=\$90$. Assume the asset can be valued at those prices at expiration.

Method. Record what is delivered or received on exercise, then value any remaining cash and asset. Do not add a cash-settled intrinsic-value payment to a physical option’s delivery.

When $S_T=\$60$. The call is out of the money, so it expires unexercised. Its portfolio retains $\$75$ cash and is worth $\$75$. The put is in the money: its holder delivers the asset worth $\$60$ and receives $\$75$. The put-and-asset portfolio therefore ends with $\$75$ cash and is worth $\$75$.

When $S_T=\$90$. The call is in the money. Its holder pays $\$75$ from the cash holding and receives the asset worth $\$90$, leaving an asset worth $\$90$. The put expires unexercised, and its portfolio retains the asset worth $\$90$.

Check. The two portfolios have equal terminal wealth in both cases: $\max(60,75)=\$75$ and $\max(90,75)=\$90$. Their physical deliveries differ by state, but those deliveries produce the same wealth under the stated asset-valuation assumption.

Deriving the Price Relationship

Now assume European options on a non-dividend-paying underlying, with matching strike $K$, expiration $T$, and settlement value $S_T$. Assume frictionless trading, no default risk in the risk-free investment, and the ability to buy or sell the underlying at its quoted price. Let $i$ be the effective annual risk-free rate for the term, and let $v^T=(1+i)^{-T}$ be the discount factor to time 0. A risk-free investment of $Kv^T$ today pays $K$ at expiration.

For cash-settled options, the terminal cash value of the call-and-cash portfolio is $\max(S_T-K,0)+K$. The terminal wealth of the put-and-underlying portfolio, after valuing or selling the underlying at $S_T$, is $\max(K-S_T,0)+S_T$. These values are equal in every possible expiration state. Therefore, under no arbitrage, the two portfolios must have the same price today.

$$ C_0+Kv^T=P_0+S_0 $$

Rearranging gives the no-dividend European put-call parity relation. The same price relationship applies to physically settled contracts when their exercise exchanges produce the same terminal wealth under the assumptions above.

$$ C_0-P_0=S_0-Kv^T $$

The reason equal terminal values imply equal prices is the law of one price: if two portfolios have identical future values in every state but different current prices, an investor could buy the cheaper portfolio and sell the more expensive one. The initial difference would be locked in while the future obligations and receipts offset. This argument requires that the portfolios really do have identical terminal values, not merely similar option names or strikes.

Worked Example: Applying Parity to Matching Settlement Contracts

A hypothetical non-dividend-paying asset is priced at $S_0=\$110$. European cash-settled call and put quotes have the same strike $K=\$120$, expiration in two years, and reference the same expiration price. The annual effective risk-free rate is $i=5\%$, and the call price is $C_0=\$15$. Under the no-arbitrage assumptions, find the put price. Explain why the result also applies if both options instead use matching physical settlement.

Method. Discount the strike to the present, then use $C_0-P_0=S_0-Kv^T$. The rate is annual effective, so the two-year discount factor uses two annual periods.

Step 1: Find the discount factor.

v^T=\frac{1}{(1+i)^T} =\frac{1}{(1.05)^2} =\frac{1}{1.1025} =0.907029478\ldots

Step 2: Find the present value of the strike.

Kv^T=120(0.907029478\ldots) =108.843537\ldots

Step 3: Solve for the put price.

15-P_0=110-108.843537\ldots =1.156463\ldots

Thus $P_0=15-1.156463\ldots=\$13.843537\ldots$, or $\$13.84$ to the nearest cent. The calculation uses the two-year discount factor, not a one-year factor. For matching physical options, each exercise exchange gives the same terminal wealth as the corresponding cash-settled payoff when the same asset and expiration value are used, so the parity price is unchanged.

Check. $C_0-P_0\approx 15-13.843537=1.156463$, while $S_0-Kv^T\approx110-108.843537=1.156463$. The two sides agree.

Settlement Assumptions and Common Traps

The parity argument is only as good as the contract match. Before applying it, verify that the call and put have the same strike, expiration, underlying, unit or multiplier, and settlement reference. Also check that the cash payment and physical delivery occur on the same economic date. If one option uses a closing value and another uses an average over several days, they do not generally have the same $S_T$.

A delay between the option’s stated expiration and actual cash payment or asset delivery can also matter. A cash amount paid later is not identical in value to that amount paid at expiration; discounting and any exposure during the delay may affect the comparison. Likewise, if a cash-settled contract references an index but the physical contract delivers a different asset, a difference between the index settlement value and the deliverable’s market value can create a settlement basis.

For an underlying that pays dividends, the simple no-dividend equation $C_0-P_0=S_0-Kv^T$ must be adjusted for the value of the dividends or other distributions over the option’s term. The key settlement lesson remains the same: compare portfolios using consistent underlying cash flows and values. Settlement form alone does not invalidate parity, but unmatched contract terms or underlying cash flows can.

Common mistake: Treating physical exercise as an additional cash payoff of $\max(S_T-K,0)$, while also counting the asset delivered or received. That double-counts the option’s value. For a physical call, record payment of $K$ and receipt of the underlying; for a physical put, record delivery of the underlying and receipt of $K$. Then value what remains.

For example, with a physical call, $S_T=\$90$, and $K=\$75$, the call holder pays $\$75$ and receives an asset worth $\$90$. The resulting position is worth $\$90$, not $\$90+\$15=\$105$. The $\$15$ intrinsic value describes the net economic benefit of exercising relative to buying the asset for $\$90$ in the market; it is not an extra payment on top of the delivered asset.

Worked Example: A Settlement-Reference Mismatch

A hypothetical cash-settled call and put use a reference value $R_T$, with strike $K=\$100$. At expiration, $R_T=\$104$, while a different asset proposed for the put-and-underlying portfolio is worth $S_T=\$100$. Compare the cash-settled call plus $\$100$ with the cash-settled put plus that different asset. Assume each option’s cash payoff is calculated using $R_T$.

Method. Use the contract’s specified reference value for both option payoffs, but value the proposed asset at its own market value. This makes the basis mismatch explicit.

The call payoff is $\max(104-100,0)=\$4$. The call-and-cash portfolio is therefore worth $4+100=\$104$. The put payoff is $\max(100-104,0)=\$0$. The put-and-asset portfolio is worth $0+100=\$100$, because the proposed asset is worth $S_T=\$100$, not $R_T=\$104$.

Check. The first portfolio exceeds the second by $\$104-\$100=\$4$, exactly the difference between the reference value and the asset value. The standard equality does not apply to these mismatched portfolios. This is not a settlement-form effect by itself; it is a failure to match the underlying value used in the option payoff with the value of the asset in the parity portfolio.

Check Your Understanding

For each problem, state the relevant settlement values and compare complete portfolios rather than option payoffs alone.

A cash-settled call and put have $K=\$75$, and both use $S_T=\$68$. Compare a call plus $\$75$ cash with a put plus one unit of the underlying, which can be sold for $S_T$. (Answer: Both portfolios are worth $\$75$.)
A cash-settled call and put have $K=\$80$ and $S_T=\$92$. Find the terminal values of the call-plus-$\$80$ portfolio and the put-plus-underlying portfolio. (Answer: Both are worth $\$92$.)
A physically settled call and put each cover one asset, with $K=\$50$ and $S_T=\$63$. Describe what happens to each portfolio: long call plus $\$50$ cash, and long put plus one asset. (Answer: The call is exercised and the cash is exchanged for an asset worth $\$63$; the put expires and its portfolio retains an asset worth $\$63$. Both portfolios are worth $\$63$.)
A non-dividend-paying asset has $S_0=\$84$. Matching European options have $K=\$90$, $C_0=\$11$, and the discount factor to expiration is $v^T=0.96$. Find $P_0$. (Answer: $P_0=\$13.40$.)
A cash-settled call and put use a reference value $R_T=\$107$ and $K=\$100$. A proposed put-and-underlying portfolio uses a different asset worth $S_T=\$103$. Compare its value with the call plus $\$100$ cash. (Answer: The call-and-cash portfolio is worth $\$107$; the put-and-asset portfolio is worth $\$103$; the difference is $\$4$.)

What You'll Learn