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Why a Super-Replicating Portfolio Needs Nonnegative Bonds at Zero Spot

Article Quant Q&A · Author: dullboy
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Summary

The document explains the meaning of requiring a super-replicating portfolio to dominate a payoff at a zero underlying price. In the exercise’s setup, the portfolio holds alpha shares and beta bonds. At zero spot, the shares contribute no value, and the payoff under consideration is zero. The portfolio’s value at that state therefore comes from the bonds, whose unit value makes the dominance condition imply that beta cannot be negative.

The central idea is state-by-state payoff comparison: a super-replicating portfolio must be worth at least as much as the target payoff in every relevant final state, including the state where the underlying ends at zero. The explanation is limited to the assumed payoff and portfolio components in the exercise. It does not derive the full replicating strategy or discuss other payoff functions, bond conventions, or additional constraints that might apply in a broader model.

Key ideas

  • Dominating at zero means the portfolio value at zero spot must be at least the payoff there.
  • Shares are worth zero when the underlying spot price is zero.
  • With unit-valued bonds and a zero payoff at that state, beta must be nonnegative.
  • The argument depends on the exercise’s specified payoff and portfolio composition.

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OptionsDerivatives pricingRisk management
Full text
# Exercise 2.2 from the book "The concept and practice of Mathematical Finance"


# Exercise 2.2 from the book "The concept and practice of Mathematical Finance"












I am a newbie. Please help me understand how to resolve the exercise 2.2 from the book "The concept and practice of Mathematical Finance". The solution from the book says that our super-replicating portfolio will be $\alpha$ shares and $\beta$ bonds. It must dominate at zero. This implies that $\beta$ >= 0. First of all, what does it mean "it must dominate at zero". Secondly, why if it dominates at zero, then $\beta$ >= 0? Thanks so much for your help!

### Problem

### Solution

## Answer by q.t.f. (score 1)

https://quant.stackexchange.com/a/19178

"It must dominate at zero" means that when the final spot level is zero, the value of the super-replicating portfolio must be greater than or equal to the value of the payoff, which is zero. Since the super-replicating portfolio consists of some stock (which has zero value when the spot price is zero) and some bonds (which have value one), there must be a non-negative number of bonds.

## Answer by Bangkokian (score 1)

https://quant.stackexchange.com/a/20638

"Dominating at zero" is what it sounds like: It means that the value of the portfolio has a >0 value when the spot price (and α) is at 0.

So if α = 0 then β must be a positive (non zero) value in order to "dominate" or be >= 0.

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.