Hull Ch. 20 (p. 490) - Ito's Lemma

[Me3]

About 60% down the page it states:



I don't see how Ito's lemma states this. The closest I find is on p. 274, in the 1st formula on the page:



The 2nd term on the right side of the equation looks similar to the right side of this equation (with G=E & S=V), but I can't figure out how they get from there to here.

Can anyone shed any light on this?

[rsgoldfarb]

1. According to Merton's model, Equity (E) is a function of the value of the firm (V) and no other stochastic processes (technically it is a call option on the value of the firm with a fixed strike price equal to the face value of the debt). If we assume that V follows Geom. Brownian Motion, then its STOCHASTIC component is sigma_v * V * dz.

2. Assuming that E also follows Geom Brownian Motion, then its stochastic component is sigma_E * E * dz.

3. Ito's Lemma tells us that since E is a function of V, then its stochastic component is dE/dv * sigma_V * V * dz.

4. Set (2) and (3) equal to each other and drop out the dz term and you get the relationship sigma_E * E = dE/dv * sigma_V * V.

Richard Goldfarb

[Me3]

Thanks a lot. I have got it now. I also see now that formula 12A.1 sort of shows your point (but I had not read it as the appendix is not on the syllabus).

[Stanley Milgram]

OK, I'm being a little dense I know, but I can't figure out Ex 22.1 in Hull (p. 507)

Equity = $3M
volatility of equity = 80%
Debt to be paid at T=1 is $10M
risk-free rate is 5% per annum.

I don't see how to get V=12.40 and volatility of V = 0.2123

[Stanley Milgram]

I have a habit of posting and then figuring it out later. Seems the public pressure of my density helps.

For those still confused, it requires Excel solver.