Why a Replicating Portfolio’s Initial Price Is Not Its Maturity Payoff

Summary

The document clarifies the difference between a portfolio’s payoff at maturity and its price when established. A proposed structure combines European puts at two strikes with a European call at a higher strike. Evaluating the payoff formula at a particular underlying price at maturity gives the amount received in that scenario; it does not determine the portfolio’s initial cost.

Under replication, the initial value is the sum of the current prices of the component options. Those prices depend on market inputs such as volatility and interest rates, or may be supplied directly in an exercise. The question mentions a stated initial value that differs from the payoff calculated at one terminal underlying price, illustrating why the quantities should not be equated. The explanation is conceptual: it does not provide option prices, a full pricing model, or enough inputs to independently reproduce the stated initial value. The key lesson is to value each instrument today, rather than treating a single possible future payoff as its purchase price.

Key ideas

• A payoff evaluated at maturity is conditional on the underlying’s terminal price.
• The initial cost of a replicating portfolio is the sum of the component options’ current prices.
• Option prices require market inputs such as volatility and interest rates when not supplied.
• A single terminal payoff does not determine an option portfolio’s initial value.

Tags

# Misconception about replicating portfolio


# Misconception about replicating portfolio












I am solving a problem in which following payoff is provided:

With $S_0=100$ and $T=8$. Looking at the payoff it seems obvious that it is replicated with two european put options ($K=100$ and $K=150$) and an european call option ($K=200$), therefore:

$V_t=[100-S_t]^{+}+[150-S_t]^{+}+[S_t-200]^{+}$

And hence the initial value needed to replicate the derivative is $V_0=50$. However, in the solution of the problem it says that the initial value is 41.4. Where am I wrong?

## Answer by Thomasunny (score 1)

https://quant.stackexchange.com/a/61321

The 50 value you compute is the payoff of your structure if the underlying is worth 100 at maturity. The initial value is the expected payoff of your trade, then, given the decomposition in call and puts, and given the replication principle, it should be: $V_t = Put_0(100,8) + Put_0(150,8) + Call_0(200,8)$, with $Put_0(k,T)$ and $Call_0(k,T)$ the price at time $0$ of calls and puts of strikes $k$ and maturity $T$.

Either prices of calls and puts are also provided in your exercise, otherwise, you need additional information like volatilities and, as mentioned by rubikiscube09, interest rates, to discount payoffs.

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