Put-Call Parity and No-Arbitrage Pricing of Matching Portfolios

Summary

The document compares a portfolio consisting of a call minus a put with a position in the underlying asset financed by a discounted strike payment. If the options share a strike and maturity, their terminal payoff matches the financed underlying position. The question asks whether that terminal equality implies equality of their values earlier in time.

The answer uses a no-arbitrage argument: if two portfolios with identical future payoffs had different current prices, an investor could sell the more expensive portfolio and buy the cheaper one. Investing the price difference would leave a positive payoff at maturity with no remaining difference between the portfolios. This supports equal current values under the assumed financing and trading conditions. The explanation is conceptual and omits practical frictions such as transaction costs, funding differences, and restrictions on short selling.

Key ideas

• A call minus a put with matching strike and maturity has the same terminal payoff as the financed underlying position described.
• Identical future payoffs should imply identical current prices under no-arbitrage assumptions.
• If the matching portfolios have different prices, selling the expensive one and buying the cheap one creates an arbitrage opportunity.
• The argument assumes the relevant trades and financing are available without material frictions.

Tags

# Relationship between portfolios at $t=0$ based on $t=T$


# Relationship between portfolios at $t=0$ based on $t=T$












I have two portfolios $V$ and $U$ given by

$$ V(S,t) = C-P \\ U(S,t) = S-Ee^{r(t-T)} \\ $$ where $P$ and $C$ denote a put and call option with the same maturity time $T$ and strike price $E$, respectively.

The pay-off function for $V$ is a linearly increasing function that intersects the horizontal $S$-axis at $E$. This is identical to the pay-off function for $U$ at time $t=T$.

So, this means that at $t=T$ the value of $V$ and $U$ are identical. But based on this, can I say anything about the relationship of the two portfolios at $t=0$?

## Answer by Richi Wa (score 1, accepted)

https://quant.stackexchange.com/a/49013

If the value of $V(S,t)$ and of $U(S,t)$ is identical at $t=T$, then the value/price at $t=0$ should be the same too. Otherwise there is arbitrage.

Imagine $V(S,T) = U(S,T) = X_T $ for some unknown $X_T$ but $V(S,0) > U(S,0)$ then we apply sell high and buy low. We sell $V(S,0)$ and buy $U(S,0)$ and we have a gain of $x :=V(S,0)-U(S,0)$. We can put this on a bank account. At $t=T$ the portfolio is worth: $$ x(1+rT) + V(S,T) - U(S,T) = x(1+rT) + X_T - X_T = x(1+rT) >0 $$ for sure.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.