April 2001 - Ques 6 - Pricing Kernels

Hi All,

From the Core Reading, I thought that the formula for the state price deflator consisted of both q from the risk neutral measure and p from the real world measure. For the April 2001 #6 question, they have q as 0.4 and p as 0.6, which would be under the real world measure. Why wasn't q derived from the risk neutral measure?

Thanks.

 

Any help here? Please.

 

I don't think it's very useful looking at this question, as Subject 109 dealt with pricing kernels differently from the way Subject CT8 deals with state-price deflators (SPDs). However, it might be useful to practise our SPD calcs.

In CT8, you're right, we would first work out the RN probs:

q = (1-0.9)/(1.15-0.9)=0.4; 1-q = 0.6

Then the SPDs are given by:

A(up) = exp(-r) x q/p = 0.94 x 0.4/0.6 = 0.6267
A(down) = exp(-r) x (1-q)/(1-p) = 0.94 x 0.6/0.4 = 1.41

These are the same as K(good) and K(bad) given in the Examiners' Report, so we can see they're the same as our SPDs.

The formula given for the pricing kernel is just an alternative way of deriving SPDs without needing to go via the RN probs, but it isn't covered in CT8, so I wouldn't worry about it.

 

Thanks Mike

 

Oh... Uh I have one more question. Shouldn't the value of u = 1.15/0.94, and d = 0.9/0.94?

Thanks again.

 

Yes, u and d are as you say. If you put these into the equation for q you will get the same as me. Each of your 4 bits in the q equation will be the same as mine, but divided by 0.94, so the 0.94's in the numerator and denominator will cancel out.

So you can either get q using returns (As you want to do) or values (as I've done) - the two approaches are equivalent.

 

Mike, in your calculation you use the risk free rate as 0, since q = e^r - d/.... but for the SPD you have e^-r = 0.94.

Also Mike for the initial question I had used the fact that the risk free rate r should be equal to the return on a zero coupon bond. i.e r = (1/0.94 - 1) = 6.3829...%

This gave me q = (e^0.0638...- 0.90)/(1.15 - 0.90) = 0.4078251783

And by following the algebra I ended up with V(0) = 0.07652143505

Help

 

No. I'm using q = (e^r - d) / (u-d) with the proper risk-free rate.

Multiplying top and bottom by S(0) gives:

q = [S(0)e^r - S(0)d] / [S(0)u - S(0)d]

S(0)=0.94
Cash grows from 0.94 to 1.00, so S(0)e^r = 1.00
S(0)d = 0.90 and S(0)u = 1.15 as stated in the question.

The RFR you're calculating is an annual effecvtive rate, but you're using it as though it were a force of interest.

Also, your q formula is missing a 0.94 before the exponential. You need this because the 1.15 and 0.90 aren't up and down factors, they are up and down share values, given the share price starts at 0.94

Either set r=ln(1.00/0.94)=6.1875%, or plug your value into:

q = [0.94(1+i) - 0.90] / [1.15 - 0.90]

 

Awesome!!