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Why Stock and Bond Portfolios Cannot Super-Replicate an Exponential Payoff

Article Quant Q&A · Author: CountDOOKU
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Summary

The note considers a derivative paying an exponential function of a stock’s terminal price and asks whether a static portfolio of stock and a zero-coupon bond can guarantee at least that payoff. Such a portfolio has a payoff linear in the terminal stock price, while the derivative payoff is exponential. The proposed argument is to compare the two payoff functions across possible terminal prices.

A portfolio is a super-replicating hedge only if its payoff is at least the derivative payoff almost surely. A line may exceed the exponential over some range, but that alone is insufficient: if there is a positive probability of terminal prices where the line falls below the exponential, the hedge fails. The note gives this criterion rather than a formal proof that no coefficients can work under every possible distribution. Its conclusion depends on the stock’s attainable values and their probabilities, so the support of the terminal price distribution matters.

Key ideas

  • A static stock and bond portfolio has a payoff linear in the terminal stock price.
  • The derivative payoff grows exponentially with the terminal stock price.
  • Super-replication requires the portfolio payoff to cover the derivative payoff almost surely.
  • A region where the portfolio payoff falls short rules out super-replication if that region has positive probability.
  • The conclusion depends on the possible terminal prices and their probability distribution.

Tags

Derivatives pricingPortfolio constructionRisk management
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# Help me understand super replicating portfolio


# Help me understand super replicating portfolio












Lets consider a hypothetical stock with current price of $S_t$ at time t and it can take any positive value with a strictly positive probability.

There exists a derivative that pays $ e^{S_T}$ at maturity $T$ such that the writer pays $ e^{S_T}$ to holder, but holder pays nothing. $C_0$ is price of this derivative at time $0$.

Additionally, there is a risk-free zero-coupon bond with a face value of 1 and maturity $T$, priced at $Z_0$ at time $0$.

How can I show that the derivative cannot be super replicated if only the stock and bond are available in the market?

I am thinking in terms of the payoff structure of combination of bonds and stock (which is linear) vs payoff structure of derivative which is non linear. But depending on the units of stock or the bonds, the linear line is above the $e^{S_T}$ curve for some interval of $S_T$ right? so I can super replicate after all?

Or prove by contradiction in someway?

## Answer by Rylan (score 1)

https://quant.stackexchange.com/a/78991

I'm not familiar with super-replicating portfolios, but what I've gathered is that the idea is to find a static hedge that is greater than or equal to the asset with probability 1, ie find $a, b$ such that $aS_T + b \geq e^{S_t}$ almost surely.

I think you have generally the right idea in comparing the linear-in-$S_T$ static payoff with the exponential-in-$S_T$ derivative, and note that all we need to conclude that a portfolio is not super-replicating is some positive probabilty that that portfolio will be worth less than the derivative. In other words, when you note that the line is above the exponential for the interval, that doesn't make it a super-replicating portfolio -- having no intervals (with positive probability) where the line is below the exponential would.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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