# 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?

• 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? Oct 7, 2020 at 21:22
• 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. Oct 7, 2020 at 22:03
• Hi, I think your replication strategy is correct. Oct 8, 2020 at 14:07