Static Arbitrage from a Log Payoff, Forward, and Discount Bond

Summary

The document poses a static arbitrage problem using a contract that pays a negative logarithm of the terminal underlying price, a forward contract, and a discount bond. Given prices for the three traded assets, it infers the underlying’s spot value from the forward price and bond discount factor, then asks how to build a zero-cost portfolio with a guaranteed gain.

The prompt does not provide a solution. It raises the key replication issue: a logarithmic payoff is nonlinear in the underlying, while the available forward and bond have linear or constant payoffs. Any arbitrage argument therefore needs to use a payoff inequality or an additional relationship, rather than assume exact replication. The stated condition that the underlying can differ from the reference level matters to whether the inequality is strict with positive probability. The document is a problem statement and partial reasoning, so it offers no worked portfolio or evidence beyond the asset-price relationships.

Key ideas

• The contract pays a negative logarithmic function of the terminal underlying price.
• The forward and discount bond prices imply a spot value through the forward pricing relationship.
• A static arbitrage can be sought by comparing the logarithmic payoff with a portfolio of linear and constant payoffs.
• The prompt leaves the portfolio construction unresolved.

Tags

# Replicating a derivative


# Replicating a derivative












Assume an underlying random variable $S_T$ which satisfies that $S_T > 0$ and that $\mathbb{P}\{S_T \neq 100 \} > 0$. Let $X_0$ be the time-0 price of a contract that pays $X_T: -2\log\left(\frac{S_T}{100}\right)$ at time T.

Let $Y_0$ be the time-0 value of a forward contract on $S_T$ with delivery price $100$ and delivery date $T$.

Let $Z_0$ but the time-0 price of a discount bond with maturity $T$.

Exactly three assets are available for you to trade: $X, Y, Z$ and Suppose that $X_0 = 0.2, Y_0 = -10, Z_0 = 0.9$. Find a static arbitrage.

Attempt at Solution:

We know that the initial Stock price $S_0 = 80$ since $Y_0 = S_0-Z_0\cdot100 = S_0 - 90 = -10 \rightarrow S_0 = 80$.

I don't think it is possible to replicate a logarithmic distribution using a zero-coupon bond and a forward contract. So how do I make sure my portfolio will have no initial cost and guarantees me to be profitable?

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.