# Tutorial 4 Q4

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I dont understand how we arrived at selling 3 calls, borrowing 102.5 and and buying one share. It makes sense why we have to find the value of the replicating portfolio because that enables us to show that there exists an arbitrage opportunity. I also know that we have to sell the more expensive and buy the cheaper. How did we determine how many call options to sell?

answered May 25 by (820 points)

This follows from the principal of a delta hedge.

The call has a delta, or lookthrough exposure to the underlying asset, of $$\frac{1}{3}$$. So in order for the calls to perfectly hedge the risk of one unit of the underlying asset we have to purchase three.

This can be seen by comparing the difference of the terminal underlying asset price, to the call terminal value. Units to be held:

$$\frac{140 - 110}{10 - 0} = 3 \;\mathrm{units}$$.

We fund the calls from the bank account so that the cost now is zero which ensures the arbitrage conditions are met.

commented May 25 by (820 points)

See here for the numerical solution to the exact bank account values: