Hull, Ch 5; forward contracts

[frank_exams]

The formulas are easy enough to memorize, but I'm skeptical. For the known yield/stock index formula, I'm getting F_0 = S_0 (e^{rT} - e^{qT} +1). The strategy is this (and please tell me where I went wrong):

Borrow S_0.
Buy 1 stock index at S_0 at time 0.
Enter into a short contract, forward price F_0.

The account at time T is as follows:

1. You owe S_0e^{rt} due to compounding.
2. You get F_0 in exchange for the stock index.
3. You also get dividends in the amount of S_0(e^{qt} - 1).

So, the arbitrage-free price F_0 should be S_0e^{rt} - S_0(e^{qt} - 1).

TiA, Frank

[mister nacho]

This answer was just stupid. Please ignore.

[mister nacho]

Actually, when I typed that, I should have noted that I had not clue what the hell I was talking about.

But here's what I think the real answer is, now that I've actually read over your problem and thought about it. I think what your approach fails to do is consider the impact of the compounding when you combine the dividends with the yield rate.

For example, using semi-annual compounding, look at the following example:

Time 0, invest $1000
Time .5 at 3%, $1,030
Pay a 1% Dividend based on initial balance to a friend because I'm a great guy: I now have $1,020.00
Time 1 balance = 1020 x 1.03 = $1,050.29
Pay a 1% dividend to my mom based on time = .5 balance, I now have $1,040.09.

Now do it your way: 1000x1.03x1.03 = $1,060.90
Dividends: 1000x1.01x1.01 -1 = $19.10
Total = $1,041.80

What your formula fails to do is account for the additional dividends due to the interest that will accumulate in periods past t = 0.


(Edited because I mixed up a calculation, but the final answer remains the same)

[frank_exams]

Thanks for the reply.

I just realized that they're taking into account the impractical and probably impossible: you're assumed to pay back the loan continuously as you receive dividends. Then, the formula is right.

[mister nacho]

Thanks for the reply.

I just realized that they're taking into account the impractical and probably impossible: you're assumed to pay back the loan continuously as you receive dividends. Then, the formula is right.

As I demonstrated, though, it doesn't strictly have to do with continuous compounding. It's the difference in that case of 1000(yield - div)^2 versus

1000yield^2 + 1000(div^2 -1)

It's not so much that you are "paying it back" continuously, it's that you are calculating it as if you are paying it back. Looking at it this way, it's not so impractical.

[frank_exams]

As I demonstrated, though, it doesn't strictly have to do with continuous compounding. It's the difference in that case of 1000(yield - div)^2 versus

1000yield^2 + 1000(div^2 -1)

It's not so much that you are "paying it back" continuously, it's that you are calculating it as if you are paying it back. Looking at it this way, it's not so impractical.

You post-jumped my previous reply! I just saw your 2nd response now. Yeah, that's pretty much the same thing I concluded: I'm paying back only at the end. Thanks, anyway; we'll pass this thing together.

[Hiu]

The formulas are easy enough to memorize, but I'm skeptical. For the known yield/stock index formula, I'm getting F_0 = S_0 (e^{rT} - e^{qT} +1). The strategy is this (and please tell me where I went wrong):

Borrow S_0.
Buy 1 stock index at S_0 at time 0.
Enter into a short contract, forward price F_0.

The account at time T is as follows:

1. You owe S_0e^{rt} due to compounding.
2. You get F_0 in exchange for the stock index.
3. You also get dividends in the amount of S_0(e^{qt} - 1).

So, the arbitrage-free price F_0 should be S_0e^{rt} - S_0(e^{qt} - 1).

TiA, Frank

It's an interesting question. But I think the key is the bold sentence. This is just the nominal amount of the dividends w/o considering the reinvestment, I mean. You'd remember that all the dividends should be reinvested at the risk-free rate. In your approach, the future price was overstated due to the ignore of the dividend reinvestment. So I agree with the book.

[Hiu]

Actually, when I typed that, I should have noted that I had not clue what the hell I was talking about.

But here's what I think the real answer is, now that I've actually read over your problem and thought about it. I think what your approach fails to do is consider the impact of the compounding when you combine the dividends with the yield rate.

For example, using semi-annual compounding, look at the following example:

Time 0, invest $1000
Time .5 at 3%, $1,030
Pay a 1% Dividend based on initial balance to a friend because I'm a great guy: I now have $1,020.00
Time 1 balance = 1020 x 1.03 = $1,050.29 (Typo, should be $1050.60)
Pay a 1% dividend to my mom based on time = .5 balance, I now have $1,040.09. (Typo, should be $1040.60, and the future price is $40.60)

Now do it your way: 1000x1.03x1.03 = $1,060.90
Dividends: 1000x1.01x1.01 -1 = $19.10
Total = $1,041.80

What your formula fails to do is account for the additional dividends due to the interest that will accumulate in periods past t = 0.


(Edited because I mixed up a calculation, but the final answer remains the same)

Under the semi-annual discrete circumstance, the loan at time T is 1000*1.03*1.03=1060.90

The stock together with the accumulated dividends is 1000+1000*0.01*1.03+1000*0.01=1020.30

So the future price is 1060.90-1020.30=40.60, consistent with the above explanation.