Replicating Portfolio

[Phillip]

Hello David,
Just have a question about the spreadsheet 2011.T5.b.4. Please could you elaborate a bit more about the formula =G23/(G22/1000-G18/1000) on cell J12 and J13? Thank you.
Phillip

 

[David Harper CFA FRM]

Hi Phillip,

I copied the relevant section below, from Tuckman Chapter 9. It's really at the heart of the whole idea; i.e., how much do we buy of the 1.0 year bond and short of the 0.5 year bond, such that our portfolio payoff is the same as a call option with a payoff of either $3.0 (up-state) or $0 (down-state). So, there are two formulas, both for the payoff in six months (at T = 0.5), one for the up-state with a payoff of $3 and one for a down state with a payoff of zero. But both portfolios hold long F(0.5) amount of the six month and short F(1.0) of the one-year bond:

F(0.5) + .97324*F(1.0) = $0 (formula 9.3)
F(0.5) + .97800*F(1.0) = $3 (formula 9.4)

That is two equations with two unknowns, if we subtract the first from the second, we can eliminate the F(0.5), which is the operation in J12:
F(1.0)*[.97800 - .97324] = $3, and F(1.0) = $3/[.97800 - .97324]; my $629.34 in the XLS is the same as the $630.2521 in the text, and -612.50 is the same as -613.3866, because Tuckman rounds off his inputs but the XLS does not (i.e., $973.24 in Tuckman is $973.23601xxx in the XLS). J13 applies formula 9.3 directly to solve for F(0.5) since it can use the F(1.0) = $629.34

I hope that helps,

Tuckman: "To price the option by arbitrage, construct a portfolio on date 0 of underlying securities, namely six-month and one-year zero coupon bonds, that will be worth $0 in the up state on date 1 and $3 in the down state. To solve this problem, let F(0.5) and F(1.0) be the face values of six-month and one-year zeros in the replicating portfolio, respectively. Then, these values must satisfy the following two equations:

F(0.5) + .97324*F(1.0) = 0 (formula 9.3)
F(0.5) + .97800*F(1.0) = $3 (formula 9.4)

Equation (9.3) may be interpreted as follows. In the up state, the value of the replicating portfolio’s now maturing six-month zero is its face value. The value of the once one-year zeros, now six-month zeros, is .97324 per dollar face value. Hence, the left-hand side of equation (9.3) denotes the value of the replicating portfolio in the up state. This value must equal $0, the value of the option in the up state. Similarly, equation (9.4) requires that the value of the replicating portfolio in the down state equal the value of the option in the down state.

Solving equations (9.3) and (9.4), F(.5) =–$613.3866 and F(1.0) =$630.2521. In words, on date 0 the option can be replicated by buying about $630.25 face value of one-year zeros and simultaneously shorting about $613.39 face amount of six-month zeros. Since this is the case, the law of one price requires that the price of the option equal the price of the replicating portfolio. But this portfolio’s price is known and is equal to .... $0.58" --Tuckman Chapter 9, page 175

 

[tpchan]

Why is the replicating portfolio long the 1.0 year bond and short the six month bond?

Thank you

 

[tpchan]

I got it. Sorry for overlooking the texts extracted from Tuckman in your reply

 

[Delo]

Hi,
This is related to Tuckman Ch 7 - Term Structure.




How do we come to the conclusion in the first place that "The replicating portfolio will be long the 1.0 year bond and short the six month bond"?

Anything after that is mere calculation / mechanics - easy to solve the simultaneous equations.
I read through Tuckman but I don't think that is clearly explained.