The following question is taken from Steven Shreve Volume 1, Chapter 1, Exercise $1.6$ (Hedging a long position-one period)

Consider a one period binomial stock model with $S_0=4$, $S_1(H)=8$ and $S_1(T)=2$. The interest rate is $25$%. Consider a bank that has a long position in the European call written on the stock price. The call expires at time one and has strike price $K=5$. It is determined that the time-zero price of this call to be $V_0=1.20$. At time zero, the bank owns this option, which ties up capital $V_0=1.20.$ The bank wants to earn the interest rate $25\%$ on this capital until time one (i.e. without investing any more money, and regardless of how the coin tossing turns outs, the bank wants to have $$\frac{5}{4}\cdot 1.20 = 1.50$$ at time one, after collecting the payoff from the option (if any) at time one). Specify how the bank's trader should invest in the stock and money markets to accomplish this.

My attempt:

Let $X_0, X_1$ be portfolio values at time zero and one respectively. From the question, we have $$X_1(T)= X_1(H) = 1.50.$$ From the wealth equation, we have $$X_1(H) = \Delta_0S_1(H) + 3 +(1+r)(X_0-\Delta_0S_0 - 1.20),$$ $$X_1(T) = \Delta_0S_1(T) + 0+ (1+r)(X_0-\Delta_0S_0- 1.20).$$ Substituting appropriate values, we have $$1.50 = 8\Delta_0 + 3 +\frac{5}{4}(X_0-4\Delta_0- 1.20),$$ $$1.50 = 2\Delta_0 + 0 +\frac{5}{4}(X_0-4\Delta_0- 1.20).$$ Solving the 2 equations above for $\Delta_0$ and $X_0$ leads to $$\Delta_0 = -\frac{1}{2}, X_0= 1.20.$$ I do not know how to proceed from here. Is my attempt above correct?


In both of your wealth equations, there is no need to subtract 1.2 from your investment in money market because you already own the option.

Solve it you will get $\Delta=-\frac{1}{2}$ and $X_0=0$. Now, it means you need to short sell $-\frac{1}{2}$ shares, get $2\\\$$, and put $2\\\$$ into your investment account to guarantee that at $t=1$, you will get $1.5 \\\$ $ (you only get $1.5 \\\$ $ from that account because part of it is needed to cancel with the option payoff).


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