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September 2019 Q8

D

Darragh Kelly

Ton up Member
Hi,

I've been looking at method 1 of the examiners solution to this question. So they differentiate B(t)=F(t)-E(t) w.r.t vega. But they say that the vega of a bond is unknown. However the black-scholes framework is being applied, can we not use the fact that del,F(t)/del,sigma = 0 (as all greeks except for delta are 0 for the share price which in this case is the total firm value F(t)). So then we have del,B(t)/del,sigma = -del,E(t)/del,sigma, and then as E(t) is equivalent to the value of the call option and the vega of a call = S(t)*N'(d1)*(T-t)^0.5 and just replace del,E(t)/del,sigma = F(t)*N'(d1)*(T-t)^0.5. So then finally as minus del,E(t)/del,sigma = del,B(t)/del,sigma, the vega of the bond is just - F(t)*N'(d1)*(T-t)^0.5? I don't get the same answer as the acted solution which looks at it differently...

Thanks,

Darragh
 
That's a really neat way of approaching the question - nice one.
Here are your calculations:
del,B(t)/del,sigma = - F(t)*N'(d1)*(T-t)^0.5 =-100*NORM.S.DIST(1.3015,FALSE)*SQRT(10) = -54
This figure can be interpreted as being the approximate change in the value of the B if sigma were to increment by 1. The solution in the examiner's report quotes vega in terms of %^(-1), and this is the approximate change in B if sigma were to increment by 0.01. This makes their answer about 100 times smaller than yours. It think that's where the key difference lies.
 
Hi Steve,

Thanks for your detailed reply.

Yeah ok so is my approach acceptable do you think from an examiners point of view? Why did the examiner quote %^(-1)?

And finally last question - the acted solution is perfect to follow, but when you say the value of the firm is fixed do you mean at an instant in time the value of a firm does not change, so then we change the vol parameter to see how the value of the equity changes and hence the value of the bond will have to change in opposite direction for fixed value of firm to hold? Surely over time firms increase in value due to share price increasing?

Thanks very much,

Darragh
 
Correct answers expressed in different units are still correct. For example, 1 = 100% = 10,000%%, although some notation might feel more natural than others!
Yes, the value of the company's assets are considered to be fixed when the vega is calculated because that's how partial derivatives work; all other parameters are assumed constant. Actually the total assets of the company is assumed to follow geometric Brownian motion, so it definitely changes over time.
Thanks
 
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