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I have a question on the following exercise from S. Shreve: Stochastic Calculus for Finance, I:

Exercise 4.2. In Example 4.2.1, we computed the time-zero value of the American put with strike price $5$ to be $1.36$. Consider an agent who borrows $1.36$ at time zero and buys the put. Explain how this agent can generate sufficient funds to pay off his loan (which grows by $25 \%$ each period) by trading in the stock and money markets and optimally exercising the put.

The model from Example 4.2.1 he refers to is the following: \begin{align*} S_0 & = 4 \\ S_1(H) & = 8, S_1(T) = 2 \\ S_2(HH) & = 16, S_2(HT) = S_2(TH) = 4, S_2(TT) = 1 \end{align*} with $r = 1/4$ and risk-neutral probabilities $\hat p = \hat q = 1/2$.

So now my question, I am not sure how to solve this, I guess the agent wants in some way hedge that he can always pay the credit by utilising the option. First how should I think about it, that he should pay back his credit at time step $2$, or earlier if possible by exercising the option, or should he stay liquide until the end? And what means optimal, hedging with minimal invest?

Okay, I solved it by considering two scenarios, first using the option at time step $1$, and then at time step $2$. By using it at time step one I found that he has to invest additional $0.16$, buy $1/2$ from the share, and accordingly borrow $0.16 - 1/2 \cdot 4$ from the bank/monkey market. Then at the first time step, as the option was exercised and the value of the portfolio equals $(1 + r)1.36$ he could just invest everything riskless, i.e. readjusting his portfolio by not buying any shares of stock, and investing in the riskless asset $(1+r)1.36$, in this way at time step $2$ he could pay $(1+r)^2 1.36$.

In the second scenario, i.e. exercising the option at time step $2$, I found that he has to invest additional $1.36$ and buy no share at the initial step, and then readjust in the next step his portfolio as to buy $1/12$ of the share if it goes up and $1.06$ if it goes down, and by exersing the option, if it goes up after paying his debt $(1+r)^2 1.36$ his portfolio has the value $1.3$, meaning he still has money, or $-0.3$ if it goes down, meaning he still has some debt (this point I do not understand fully?...)

So can someone help me in understand and solving this exercise (if my approach is wrong...)?

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I think the question is about dynamic replication in a binomial context except it is worded a bit odd. Normally a tree would be given and one needs to find x(t) and y(t), where x(t) is amount of stock at time t and y(t) is amount of cash at time t. Then you do your replication argument and you have that the option price P(0) = X(0) + Y(0).

The way this problem is worded you already have the price at time zero. In order to be able to pay off the loan of 1.36 at every time you would still have to go over the replication.

So at t= 0 you need X(0) stocks such that no matter what (either up or down) your portfolio consisting of stocks, cash and put is flat. Note that you will be long the put, you will be long the stock (cause the put has negative delta) and therefore you need to fund both the put (1.36 as well as X(0)*4 for your stock position).

Solving gives X(0) = 0.4333 and y(0) = 1.36 + 0.433*4 = 3.0933. You can check that this position will always be flat. if stock goes down then you lose on the long stock but make cash on your put (which you need to exercise). etc. If stock goes up you need to readjust your X(1) and rebalance Y(1) to replicate the put payoff again.

If you need more help let me know, but you should be fine from here on. Just solve the linear algebra.

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$$\Delta_1(H)=\frac{V_2(HH)-V_2(HT)}{S_2(HH)-S_2(HT)}=-\frac{1}{12}$$ and $$\Delta_1(T)=\frac{V_2(TH)-V_2(TT)}{S_2(TH)-S_2(TT)}=-1$$ and $$\Delta_0=\frac{V_1(H)-V_1(T)}{S_1(H)-S_1(T)}=-0.433$$ The optimal exercise time is $$\tau(HH)=\infty $$ $$\tau(HT)=2 $$ $$\tau(TH)=1 $$ $$\tau(TT)=1 $$

AS a result, you should borrows $1.36$ at time zero and buys the put option. At the same time, to hedge the long position, you need to borrow again and buy $0.433$ shares of stock at time zero.

Step 1: If the result of toss is tail and the stock price goes down to $2$, you should exercise the put and get $3$ to pay off your debt.

Step 1 : If the result of toss is head and the stock price goes up to $8$, you should should borrow to buy $\frac{1}{12}$ shares of stock.

Step 2: If the result of toss is head and the stock price goes up to $16$, should let the put expire because the value of the portfolio is $0$.

Step 2: If the result of toss is tail and the stock price goes down to $4$, you should exercise the put to get $1$.

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