Actuarial Outpost > CAS Hull Ch. 20 (p. 490) - Ito's Lemma
 Register Blogs Wiki FAQ Calendar Search Today's Posts Mark Forums Read
 FlashChat Actuarial Discussion Preliminary Exams CAS/SOA Exams Cyberchat Around the World Suggestions

 D.W. Simpson and Company -- Actuary Salary Surveys Pension, Life, Health and Investment Actuarial Jobs Property and Casualty Actuarial Jobs   Registration Form

#1
02-01-2006, 12:44 PM
 Me3 Member Join Date: Jul 2004 Posts: 155
Hull Ch. 20 (p. 490) - Ito's Lemma

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?

Last edited by Me3; 02-01-2006 at 12:49 PM..
#2
02-02-2006, 10:58 AM
 rsgoldfarb Member Join Date: Jun 2003 Posts: 52

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
#3
02-02-2006, 07:29 PM
 Me3 Member Join Date: Jul 2004 Posts: 155

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).
#4
04-11-2009, 11:19 AM
 Stanley Milgram Member CAS AAA Join Date: Nov 2004 Location: Somewhere over the Rainbow Favorite beer: "Trying" to choose yoga over beer. Posts: 4,549

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
#5
04-11-2009, 03:50 PM
 Stanley Milgram Member CAS AAA Join Date: Nov 2004 Location: Somewhere over the Rainbow Favorite beer: "Trying" to choose yoga over beer. Posts: 4,549

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.

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off

All times are GMT -4. The time now is 03:43 AM.

 -- Default Style - Fluid Width ---- Default Style - Fixed Width ---- Old Default Style ---- Easy on the eyes ---- Smooth Darkness ---- Chestnut ---- Apple-ish Style ---- If Apples were blue ---- If Apples were green ---- If Apples were purple ---- Halloween 2007 ---- B&W ---- Halloween ---- AO Christmas Theme ---- Turkey Day Theme ---- AO 2007 beta ---- 4th Of July Contact Us - Actuarial Outpost - Archive - Privacy Statement - Top