Replication Proof Using Stock And A Risk-Free Bond

Why Replication Proves Put-Call Parity

Put-call parity is more than an equation to memorize. Its core is a replication argument: two portfolios are assembled from different securities, yet deliver exactly the same amount at the same future date in every possible terminal stock-price state. Under no arbitrage, portfolios with identical future cash flows must have identical current values. This tutorial develops that argument using a European call, a European put, the underlying stock, and a risk-free bond.

The focus is the proof and its conditions, not an arbitrage trading strategy for a particular quote discrepancy. The result is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: A replicating portfolio is a collection of assets whose combined cash flows match those of another asset or portfolio at the relevant times and in every state being considered. If two portfolios have identical future cash flows, the no-arbitrage principle requires their current values to be equal, provided the market assumptions needed to compare and trade them hold.

Let $S_0$ denote the stock price now, $S_T$ its price at expiration $T$, $K$ the common strike, $c$ the current price of a European call, and $p$ the current price of a European put. The options are on the same stock, have the same strike and expiration, and are exercisable only at expiration. The call's expiration payoff is $\max(S_T-K,0)$; the put's is $\max(K-S_T,0)$.

Assume the stock pays no dividends before expiration. Assume a default-free risk-free bond can be bought now and pays a known amount at $T$, and that the relevant risk-free interest rate is available for borrowing and lending. Markets are frictionless for this argument: there are no transaction costs, taxes, or restrictions preventing the required purchases and sales. The options and stock can be traded in the amounts needed. The stock's future price need not have a specified probability distribution; the payoff comparison is made for every possible value of $S_T$.

Construct the Two Portfolios

The first portfolio contains one European call and a risk-free bond that pays $K$ at expiration. The second contains one European put and one share of stock. The bond is chosen to pay exactly the strike, not an arbitrary amount: that payment completes the call portfolio's payoff when the call finishes out of the money.

For an effective annual risk-free rate $i$, with $T$ measured in years under annual compounding, the present value of a bond paying $K$ at time $T$ is $K v^T$, where $v=1/(1+i)$. Thus the current value of the call-plus-bond portfolio is $c+K v^T$, and the current value of the put-plus-stock portfolio is $p+S_0$.

Choose a terminal stock price.
Compare both portfolios at the same expiration price $S_T$.

Add each portfolio's expiration payments.
The call portfolio includes the call payoff and the bond's payment $K$; the put portfolio includes the put payoff and the stock's value $S_T$.

Compare the results in all cases.
Check stock prices below $K$, equal to $K$, and above $K$. Equality in each case establishes identical terminal payoffs.

Apply no arbitrage to current values.
Identical future cash flows imply equal current portfolio prices under the stated assumptions.

Payoff-by-Payoff Justification

Write $C_T=\max(S_T-K,0)$ for the call payoff and $P_T=\max(K-S_T,0)$ for the put payoff. The call-plus-bond portfolio pays $C_T+K$, while the put-plus-stock portfolio pays $P_T+S_T$. Check the payoffs by dividing all possible terminal stock prices into three cases.

If $S_T<K$: the call expires without value, so the call-plus-bond portfolio pays $0+K=K$. The put pays $K-S_T$, so the put-plus-stock portfolio pays $(K-S_T)+S_T=K$.
If $S_T=K$: both options expire without value. The call-plus-bond portfolio pays $0+K=K$, and the put-plus-stock portfolio pays $0+K=K$.
If $S_T>K$: the put expires without value. The call-plus-bond portfolio pays $(S_T-K)+K=S_T$, and the put-plus-stock portfolio pays $0+S_T=S_T$.

The two portfolios therefore pay $K$ when $S_T\leq K$ and $S_T$ when $S_T>K$. In either portfolio, the common payoff can be written as $\max(S_T,K)$. This equality holds state by state, not merely on average.

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

Because the terminal cash flows are identical, the no-arbitrage principle gives equal current values. The bond's present value is $K v^T$, so the replication proof yields:

$$ c+K v^T=p+S_0 $$

This is European put-call parity for a non-dividend-paying stock under the stated assumptions. The proof works because the bond supplies the strike amount when the call is not exercised; when the call is exercised, its payoff plus that bond payment equals the stock's terminal value. In the other portfolio, the put fills any shortfall of the stock below the strike.

Worked Replication Examples

Worked Example: Verifying Both Portfolios Across Terminal Prices

A hypothetical non-dividend-paying stock has a European call and put with strike $K=\$110$, both expiring in one year. The effective annual risk-free rate is $i=10\%$. The call price is $c=\$15$, and the stock price is $S_0=\$100$. Use replication to find the put price, then verify the common expiration payoff when $S_T=\$80$, $\$110$, and $\$140$.

Method and assumptions. Use a one-year default-free bond that pays $\$110$ at expiration. The options are European, match in strike and expiration, and the stock pays no dividends. The terminal prices are state checks; no probability distribution is required.

Step 1: Price the bond. With $v=1/(1+i)=1/1.10$, its present value is:

110v^1=\frac{110}{1.10}=\$100

Step 2: Equate the portfolio values. The call-plus-bond portfolio costs $15+100=\$115$. The put-plus-stock portfolio must have the same value, so $p+100=115$, giving $p=\$15$.

Step 3: Check the expiration payoffs. At $S_T=\$80$, the call-plus-bond portfolio pays $\max(80-110,0)+110=0+110=\$110$. The put-plus-stock portfolio pays $\max(110-80,0)+80=30+80=\$110$.

At $S_T=\$110$, the first portfolio pays $\max(110-110,0)+110=0+110=\$110$, and the second pays $\max(110-110,0)+110=0+110=\$110$.

At $S_T=\$140$, the first portfolio pays $\max(140-110,0)+110=30+110=\$140$, and the second pays $\max(110-140,0)+140=0+140=\$140$.

Check. Both portfolios cost $\$115$ and pay $\$110$, $\$110$, or $\$140$ in the three tested states. The mistake avoided is treating the bond as if it paid the call's payoff; it pays the fixed strike, $\$110$.

Worked Example: Replication With a Two-Year Bond

A hypothetical stock pays no dividends. A European put with strike $K=\$121$ and expiration in two years costs $\$6$. The stock costs $S_0=\$112$. The effective annual risk-free rate is $i=10\%$. Find the price of a European call with the same strike and expiration, then compare the portfolios' payoffs at $S_T=\$90$ and $S_T=\$150$.

Method and assumptions. Replicate the call-plus-bond portfolio with a put and one share. Use annual effective compounding for two years, and assume the bond is default-free and pays $\$121$ at expiration.

Step 1: Calculate the present value of the bond payment.

v=\frac{1}{1.10}, \qquad 121v^2=\frac{121}{(1.10)^2} =\frac{121}{1.21} =\$100

Step 2: Calculate the value of the put-plus-stock portfolio. Its current value is $p+S_0=6+112=\$118$. Equal current values require $c+100=118$, so $c=\$18$.

Step 3: Check at $S_T=\$90$. The call-plus-bond portfolio pays $\max(90-121,0)+121=0+121=\$121$. The put-plus-stock portfolio pays $\max(121-90,0)+90=31+90=\$121$.

Step 4: Check at $S_T=\$150$. The call-plus-bond portfolio pays $\max(150-121,0)+121=29+121=\$150$. The put-plus-stock portfolio pays $\max(121-150,0)+150=0+150=\$150$.

Check. The common current value is $\$118$, and the two portfolios match in both terminal states. The common mistake avoided is discounting the two-year bond payment for only one year; the correct factor is $(1.10)^{-2}$.

Worked Example: Recovering a Call Price From the Replication

A hypothetical non-dividend-paying stock is priced at $S_0=\$104$. A European put with strike $K=\$108.16$ and two-year expiration costs $\$7$. The effective annual risk-free rate is $i=4\%$. Find the price of the matching European call and verify the portfolio values.

Method and assumptions. Use the call-plus-bond portfolio and the put-plus-stock portfolio. Both options have the same strike and expiration, and the bond pays the strike at the end of two years. Assume default-free borrowing and lending at the stated rate.

Step 1: Find the bond's present value.

v=\frac{1}{1.04}, \qquad 108.16v^2=\frac{108.16}{(1.04)^2} =\frac{108.16}{1.0816} =\$100

Step 2: Find the put-plus-stock portfolio value. The current value is $7+104=\$111$. Therefore the call-plus-bond portfolio must also cost $\$111$, so $c+100=111$ and $c=\$11$.

Step 3: Check terminal payoffs. If $S_T=\$90$, the call-plus-bond portfolio pays $0+108.16=\$108.16$; the put-plus-stock portfolio pays $(108.16-90)+90=18.16+90=\$108.16$. If $S_T=\$130$, the call-plus-bond portfolio pays $(130-108.16)+108.16=21.84+108.16=\$130$; the put-plus-stock portfolio pays $0+130=\$130$.

Check. Both initial portfolio values are $\$111$, and the payoffs agree in both cases. The mistake avoided is adding the stock price to the call price without accounting for the bond's present value.

When the Proof Applies—and Common Traps

The equality depends on the portfolios truly matching their cash flows. The proof as given uses European options, since each option's payoff is determined at the common expiration date. It also assumes the stock pays no dividends before expiration. A dividend paid to a stockholder in the put-plus-stock portfolio is a cash flow that is absent from the call-plus-bond portfolio, so the two portfolios would not have matching cash flows at all dates without an adjustment.

Likewise, if the bond cannot be bought at the stated risk-free rate, if it may default, or if borrowing and lending rates differ, the specific present-value comparison may not hold as written. An American option can be exercised before expiration, so matching only expiration payoffs is not enough to prove equality of its current value with the European-option expression.

Common exam trap: Checking only one terminal stock price is not a replication proof. Matching at $S_T>K$ alone does not show that the portfolios match when $S_T<K$. Verify the payoff in both regions, including $S_T=K$, and confirm that the bond pays exactly $K$ at expiration.

The value of the argument is its generality: once payoffs match for every terminal stock price, the comparison does not depend on forecasting the stock, estimating probabilities, or selecting a particular future scenario. It is a cash-flow identity followed by a no-arbitrage valuation principle. Keep those two steps distinct: payoff replication establishes the matching future cash flows, while no arbitrage turns that match into an equality of current values.

Check Your Understanding

For each problem, assume frictionless markets, a default-free risk-free bond, and a non-dividend-paying stock unless the question says otherwise. Round dollar answers to the nearest cent.

A stock costs $\$75$. A one-year European call with strike $\$80$ costs $\$9.25$. The effective annual risk-free rate is $4\%$. Find the present value of the bond paying the strike, the matching put price, and the common portfolio value. (Answer: bond value $\$76.92$; put price $\$11.17$; common value $\$86.17$.)
A two-year European put costs $\$14.50$, the stock costs $\$118$, and the strike is $\$125$. At an effective annual rate of $5\%$, calculate the matching call price using the replication argument. (Answer: $\$19.12$.)
A call-plus-bond portfolio and a put-plus-stock portfolio use strike $\$96$. State their common payoff at expiration when the stock price is $\$72$, $\$96$, and $\$130$. (Answer: $\$96$, $\$96$, and $\$130$, respectively.)
A matching European call costs $\$17$, a put costs $\$12$, and the stock costs $\$150$. The options expire in two years, and the effective annual risk-free rate is $5\%$. Use replication to recover the strike. (Answer: $\$159.86$.)
Suppose the put in the replication portfolio is American while the call is European, with all other conditions unchanged. Does matching only the expiration payoffs prove the same current-value equality? (Answer: No; the American put can be exercised before expiration, so expiration-payoff equality alone does not establish equality of current values.)

What You'll Learn