hello again everyone.In a binomial model, we have the following price dynamicsAlso, let the interest rate be so the the risk-neutral probabilities are It is not difficult to see that the time-zero value of an American put with strike at $5 is $1.36. Exercise 4.2 in Shreve's book describes the following situation: "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" Faced with this exercise, I first asked myself: "Why on earth would someone do that?". My first thought was that this guy wants to, at least, hedge his position on the bank account and, if the prices happens to move in a certain favorable way, to have some profit. But then I realized that this should not be possible since this is would build an arbitrage opportunity which is excludedusing Shreve's parameters. So I would like to know if someone have a better idea about the meaning of this exercise. Anyway, I tried to sketch something and given that the agent must optimally exercise the put (in this case this means that he should exercise the put when the price of the option is the same of it's intrinsic value). So if a Tail came up he should exercise and with this money ($3) pay his loan. But if a Head comes up, then I could not figure out what to do.I would appreciate any hint or tip. I think that might be relevant to point out that this is not kind of homework. I working on all exercises alone.

its all about delta hedging the put at time zero he brrows money to buy the put and also busy a delta amt of the stock (u figure the delta from the value of the put at H and T which u would get by rolling back in the tree from maturity). rememebr he will need to fund the stock delta. repeat again for the next step

Hello again. Here is what I could do in this exercise based on your words. I will be glad if you take a look and say what you think about it.So given the scheme that I've posted in my last message in this topic, the prices for the American put option arebut, as I made a loan on time zero of $1.36 (which grows by 25% each period) and I will exercise my American put optimally (i.e when it's price equals it's intrinsic value, so that I will exercise on period 1 if a tail comes up and I will exercise on period 2 if a heads comes up).please, confirm me if you're telling me to calculate the first delta as follows:so I should short sell that fraction of stocks, invest 0.43*4 in a risk free account and continue the hedging? if so, I think that I could not hedge for example if a head came up.

QuoteOriginally posted by: fniskiHello again. Here is what I could do in this exercise based on your words. I will be glad if you take a look and say what you think about it.So given the scheme that I've posted in my last message in this topic, the prices for the American put option arebut, as I made a loan on time zero of $1.36 (which grows by 25% each period) and I will exercise my American put optimally (i.e when it's price equals it's intrinsic value, so that I will exercise on period 1 if a tail comes up and I will exercise on period 2 if a heads comes up).please, confirm me if you're telling me to calculate the first delta as follows:so I should short sell that fraction of stocks, invest 0.43*4 in a risk free account and continue the hedging? if so, I think that I could not hedge for example if a head came up.if you are long the put (which is why u borrowed the money to buy it) then you need to buy 0.43 of the stock to be delta neutral.so you have to borrow 1.36 + 0.43*4 = 3.08 now u have portfolio of put + 0.43*S and a borrowing of 3.08after 1 time step the interest on the borrowing is 0.25*3.08 but now the stock is either at 8 or 2. if the stock is @ 8 your portfolio is worth 0.4 + 0.43*8 = 3.84 so u make 0.76 but you have paid interest of 0.76if the stock is @ 2 your portfolio is worth 3 + 0.43*2 = 3.86 so again 0.78 but pay interest of 0.76hth

Quoteafter 1 time step the interest on the borrowing is 0.25*3.08 but now the stock is either at 8 or 2. if the stock is @ 8 your portfolio is worth 0.4 + 0.43*8 = 3.84 so u make 0.76 but you have paid interest of 0.76if the stock is @ 2 your portfolio is worth 3 + 0.43*2 = 3.86 so again 0.78 but pay interest of 0.76So if the stock is @2 , my portfolio is worth 3.86 and I need to pay 3.08*1.25 = 3.85 , right? so I end my operations.If the stock is 8, my portfolio is 3.84 but I need to pay 3.85. Now I tried to finish but i still can't hedge. Please, what's wrong with the following argument?Note that if the stock is @8 It is no optimal to exercise my put, so I actually can use 0.43*8 = 3.44 of my money. I then buy 1/12 shares of the stock at 8, so I spend $ 8/12 of my 3.44 available moneyand then I invest 3.44 - 8/12 = 2.773 in the free risk market.in the next period and last:I need to pay 3.85*(1.25) = 4.8125if the stock is @16, my put is worthless. I have then 16/12 = $1.33333 plus 2.773*(1.25) = $3.466 , that is $4.799 which is less then 4.8125...if the stock is @4, my put worth $1, from the stock market I have 4/12 = $0.3333, plus what I have from the risk free market = $3.466this is again $4.7999... which is less then what I need...$4.8125

I don't want to be rude but I don't get your point at allfor example, what do you mean by thisQuoteIf the stock is 8, my portfolio is 3.84 but I need to pay 3.85. Now I tried to finish but i still can't hedgewhy cant you hedge ? is it because your assets are the same value as your liabilities ?

No, you're not being rude at all. I just apologize since I'm studying all this by my own and I might have some wrong ideas in my mind.When I wrote that "I can't hedge", I meant "I could not finish the hedging procedure" meaning that I could not figure it out what actions I could do in the stock and money marketsin order to finish with a portfolio that is worth at least the same value of my liabilities. That'swhat hedge is all about right?I would like to understand what you were thinking when you wrote:"why cant you hedge ? is it because your assets are the same value as your liabilities ?"since in that message, I wrote that my portfolio is worth 3.84and but I need to pay 3.85, this way my worth assets $0.01 less then my liabilities and not the same value.

thats just a rounding error - so you made a loss of 1c. don't worry. just set up your hedge again.now with the stock at 8, you put delta is much smaller (I think its -0.0833) so you sell 0.3467 shares of the stock 8, repay 2.77 of you initial borrowings and move forward another step.your portfolio is now 0.4 (put) + 0.6667 (stock) = 1.0667stock now can either go to 16 or back to 4@ 16 the put is worth nothing, you sell your stock a d raise 1.333. also, you had to fund your portfolio of 1.0667 @ 25%, which works out to a total of 1.3333.

Thanks Dave. I realized that all the mess was mainly due rounding the numbers correctly.