Tailing

[kubila]

Anybody can tell me how the textbook shows that "e^−(delta*T) shares of stocks at t = 0 becomes one share at time T?"[the tailing diagram]

[jthias]

Anybody can tell me how the textbook shows that "e^−(delta*T) shares of stocks at t = 0 becomes one share at time T?"[the tailing diagram]

I'm doing this from memory, so I hope it's right.

But it seems to make sense.

Suppose a single share is $1 and delta=d is the cont. growth rate of stock.

Then at time T the stock is worth e^(dT) which will allow you to buy e^(dT) shares of stock.

so by propotion 1/e^(dT) = x/1 where x is the number of shares at stock we need at t=0 to become 1 at time T.

So we get x=e^(-dT) shares are required at t=0 to become 1 at time T.

[kubila]

I'm doing this from memory, so I hope it's right.

But it seems to make sense.

Suppose a single share is $1 and delta=d is the cont. growth rate of stock.

Then at time T the stock is worth e^(dT) which will allow you to buy e^(dT) shares of stock.

so by propotion 1/e^(dT) = x/1 where x is the number of shares at stock we need at t=0 to become 1 at time T.

So we get x=e^(-dT) shares are required at t=0 to become 1 at time T.

I am sorry to forget to say that delta here is the dividend rate (according to Guo), not the natural growth rate of the stock. Will this make any difference? Your argument makes a lot of intuitive sense though.

[jthias]

I am sorry to forget to say that delta here is the dividend rate (according to Guo), not the natural growth rate of the stock. Will this make any difference? Your argument makes a lot of intuitive sense though.

Yes, that was the part I wasn't sure about (what "d" represents).

If we gave a growth rate of 0 (r=0), it will make the calculations easier.

So price of stock discounts any dividend earnings, so P = 1*e^(-dT)

is the price of one share of stock at time T discounted back to t=0.

So at 0 growth rate with dividends being paid, e^(-dT) at time = 0 will be worth e^(-dT)*e^(dT) = 1 after dividends have been paid.

But this is a problem I have with dividends, do they get paid directly to the stockholder or do they added to the share price? I think it's the latter.

[jthias]

I just wanted to try to clarify my previous post. So if we have a share worth 1 at t=0 and continous dividend and continous growth rate, then that share will be worth 1*e^[(r+d)T] at time T.

This corresponds to e^(dT) at time T for a 0 growth rate.

By proportion 1/e^(dT) = x/1 yields x= e^(-dT) for 0 growth rate.

As for how dividends are paid, I'm still confused about this.

[jthias]

I just wanted to try to clarify my previous post. So if we have a share worth 1 at t=0 and continous dividend and continous growth rate, then that share will be worth 1*e^[(r+d)T] at time T.

This corresponds to e^(dT) at time T for a 0 growth rate.

By proportion 1/e^(dT) = x/1 yields x= e^(-dT) for 0 growth rate.

As for how dividends are paid, I'm still confused about this.

I see the mistake I was making that lead me to using a 0 growth rate for r.

I don't want to invest 1 at time 0. I want to invest s_0 at time 0 so that I get 1 at time T.

For an outright purchase, s_0 grows to s_0*e^[(r+d)T], but to have no-arbitrage pricing, a prepaid forward contract must be priced in such a way that x invested at time 0 grows to s_0*e^(rT) at time T since prepaid forward contracts buyers do not receive dividends. Note: x cannot be s_0 since it's leads to arbitrage because we are paying the same for stock at time 0 (s_0), but have two different values on the stock at time T: (s_0*e^[(r+d)T] and s_0*e^(rT)).

So by no-arbitrage pricing, we have...

s_0/[s_0*e^[(r+d)T]] = {s_0/[s_0*e^(rT)]}*x which yields x=e^(-dT) which means we must have a prepaid forward price of e^(-dT), so that it will appreciate to 1 at time T and have no arbitrage pricing.

I think it's correct now (I'll keep my fingers crossed )

[kubila]

I see the mistake I was making that lead me to using a 0 growth rate for r.

I don't want to invest 1 at time 0. I want to invest s_0 at time 0 so that I get 1 at time T.

For an outright purchase, s_0 grows to s_0*e^[(r+d)T], but to have no-arbitrage pricing, a prepaid forward contract must be priced in such a way that x invested at time 0 grows to s_0*e^(rT) at time T since prepaid forward contracts buyers do not receive dividends. Note: x cannot be s_0 since it's leads to arbitrage because we are paying the same for stock at time 0 (s_0), but have two different values on the stock at time T: (s_0*e^[(r+d)T] and s_0*e^(rT)).

So by no-arbitrage pricing, we have...

s_0/[s_0*e^[(r+d)T]] = {s_0/[s_0*e^(rT)]}*x which yields x=e^(-dT) which means we must have a prepaid forward price of e^(-dT), so that it will appreciate to 1 at time T and have no arbitrage pricing.

I think it's correct now (I'll keep my fingers crossed )

Hi Jthias, thanks a lot for your trying hard to answer my questions. I really appreciate that. I have been tried to understand your logic, but still have confusions which I hope you can help clarify. First, Is the "x" number of shares? Second, recall that Guo asks this question:" how many shares of stocks do I need to hold at t=0 in order to get one SHARE of stock at maturity date T?" So I am a little confused about your " I want to invest s_0 at time 0 so that I get 1 at time T." Maybe you normalized it? Then, I dont understand this :"For an outright purchase, s_0 grows to s_0*e^[(r+d)T]". I think the stock price is random, you probably can determine its expected value but you never know for sure its exact value in the future.

Given that, I do have my own opinion on this question. I think Guo's question can be reinterpreted this way: what's the price at t=0 in terms of stocks for one share of stock at t=T? There are two ways to answer this. First, one can use risk-neutral pricing method to derive that the price of a stock S(T) at t=0 is nothing but S(0)e^(-dt) where d is the continuous dividend rate. This seems awkward but it is true. A security that has a payoff of S(T) has a price given by S(0)e^(-dt). When there's no dividend, the price is simply S(o).

The second way of thinking is like this: with S(0)e^(-dt) amount of money I can deposit in the bank and simultaneously long a stock forward. At maturity I get S(0)e^(rt-dt), which is nothing but the fairly priced forward price F. Since I am in a forward contract, I pays F and get ONE SHARE of stock. This is exactly how one can use e^(-dt) shares of stocks at t=0 to get one share of stock at T.

[jthias]

I apologize. I was tired when I made that post. I knew the post was wrong after I finished correcting some of it, (for example, I wrongly concluded that, “…we must have a prepaid forward price of e^(-dT), so that it will appreciate to 1…”), but I couldn’t fix it because I didn’t understand my mistakes. You were asking how e^(-dT) shares becomes one share. I was considering the stock in terms of its value but not with respect to its purchasing power. I won’t mention prepaid forwards, or forwards, or no-arbitrage pricing. It just adds more to the confusion. I’ll just stick to an outright purchase at time 0.

“I think the stock price is random, you probably can determine its expected value but you never know for sure its exact value in the future.” I agree with you that stock prices are unpredictable. I shouldn’t choose “r” as the stock’s growth rate because it could grow at a faster or slower rate than the risk-free rate.

This is my opinion now of what is happening. I should play it safe by assuming the value of the stock at time T is s_T rather than trying to assign a growth rate to it (like the risk-free rate: another mistake I made). If you buy a single share of stock (with no dividends) at time 0 for an amount s_0, then the value of the stock at time T is s_T, but in terms of purchasing power at time T, you would only be able to purchase a single share of stock worth s_T.

If dividends are paid then we have a different situation. Suppose dividends are paid continuously at a rate of d, then the value of the stock at time T is s_T*e^(dT), but our purchasing power has increased because of the dividends, so we can afford to buy s_T*e^(dT)/s_T = e^(dT) shares of stock at time T.

So initially, 1 share worth s_0 grows to e^(dT) shares worth s_T*e^(dT).

So by proportion 1/ e^(dT) = x/1 implies x=/ e^(-dT) shares at t=0 will buy one share at time T.

I agree with both interpretations that you have. They both make perfect sense.

[kubila]

I apologize. I was tired when I made that post. I knew the post was wrong after I finished correcting some of it, (for example, I wrongly concluded that, “…we must have a prepaid forward price of e^(-dT), so that it will appreciate to 1…”), but I couldn’t fix it because I didn’t understand my mistakes. You were asking how e^(-dT) shares becomes one share. I was considering the stock in terms of its value but not with respect to its purchasing power. I won’t mention prepaid forwards, or forwards, or no-arbitrage pricing. It just adds more to the confusion. I’ll just stick to an outright purchase at time 0.

“I think the stock price is random, you probably can determine its expected value but you never know for sure its exact value in the future.” I agree with you that stock prices are unpredictable. I shouldn’t choose “r” as the stock’s growth rate because it could grow at a faster or slower rate than the risk-free rate.

This is my opinion now of what is happening. I should play it safe by assuming the value of the stock at time T is s_T rather than trying to assign a growth rate to it (like the risk-free rate: another mistake I made). If you buy a single share of stock (with no dividends) at time 0 for an amount s_0, then the value of the stock at time T is s_T, but in terms of purchasing power at time T, you would only be able to purchase a single share of stock worth s_T.

If dividends are paid then we have a different situation. Suppose dividends are paid continuously at a rate of d, then the value of the stock at time T is s_T*e^(dT), but our purchasing power has increased because of the dividends, so we can afford to buy s_T*e^(dT)/s_T = e^(dT) shares of stock at time T.

So initially, 1 share worth s_0 grows to e^(dT) shares worth s_T*e^(dT).

So by proportion 1/ e^(dT) = x/1 implies x=/ e^(-dT) shares at t=0 will buy one share at time T.

I agree with both interpretations that you have. They both make perfect sense.

This is exactly right. I really appreciate your efforts and your honesty attitude toward your mistakes. That's a great personality!

[jthias]

Thanks for being forgiving about my mistakes