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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?

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  • $\begingroup$ No, but isn't the point here to hedge dXt/dSt, and the derivative of the log of X is 1/X? Wihch creates a ratio/quotient, that maybe can be hedged? $\endgroup$
    – demully
    Oct 7, 2020 at 21:22
  • $\begingroup$ I might have found the answer by going long on 50X and 1 Y. You get that the combination of those 2 has a global minimum at S_T = 100 and its strictly greater than 0 for S_T > 0 except at S_T = 100 where it is 0. $\endgroup$
    – FatFeynman
    Oct 7, 2020 at 22:03
  • $\begingroup$ Hi, I think your replication strategy is correct. $\endgroup$ Oct 8, 2020 at 14:07

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