September 2012 Question 7

[s16455441]

Hello, I was hoping for some help understanding the last part of question 6 ii)

S0.25 = S0 exp(σ Z0.25 −0.5σ2(0.25) + 0.25r) (equation 1)

S0.5 = S0.25 exp(2σ(Z0.5 − Z0.25 ) − 0.5(2σ)2 (0.25) + 0.25r) (equation 2)
= S0 exp(2σ(Z0.5 − Z0.25 ) + σ Z0.25 − 0.5σ2(1.25) + 0.5r) (equation 3)

I understand equation (1) and equation (2), however I cannot work out how they get to equation (3).

please may someone help explain how they get here?

I think it is something to do with equation (1) plus equation (2), but I can't see how the maths works or if my thinking is correct.

Additionally I have a question on exam technique here. I thought this question had little marks for the work I put into solving.

My starting point was setting f = ln(St) and solving the SDE, integrating df etc. Do the examiners expect us to learn the formula for St by heart which would mean I didn't need to solve the SDE?

Many thanks in advance for your help.

 

[John Potter]

S0.25 = S0 exp(σ Z0.25 −0.5σ2(0.25) + 0.25r) (equation 1)

S0.5 = S0.25 exp(2σ(Z0.5 − Z0.25 ) − 0.5(2σ)2 (0.25) + 0.25r) (equation 2)

and sub in?

= S0 exp(σ Z0.25 −0.5σ2(0.25) + 0.25r) exp(2σ(Z0.5 − Z0.25 ) − 0.5(2σ)2 (0.25) + 0.25r)

= S0 exp(σ Z0.25 −0.5σ2(0.25) + 0.25r + 2σ(Z0.5 − Z0.25 ) − 0.5(2σ)2 (0.25) + 0.25r)

and now let's colour-code it...

= S0 exp(σ Z0.25 −0.5σ2(0.25) + 0.25r + 2σ(Z0.5 − Z0.25 ) − 0.5(2σ)2 (0.25) + 0.25r)

= S0 exp(2σ(Z0.5 − Z0.25 ) + σ Z0.25 − 0.5σ2(1.25) + 0.5r)

Yes, exam technique is important, especially when you consider the seeming disparity of marks per minute to which you (correctly) allude. The secret is to "save until the end of an exam" those questions that are clearly going to take us more than 2 minutes per mark - this would most likely be a question that is difficult to even understand and hence going to take time to decipher (or possibly even lose time before giving up!) but it can also crop up in questions that we know how to do but just aren't worth the marks. Getting good at knowing when to "save a question until the end" comes with past paper practice - which is good, right? It rewards those of us who are prepared to work hard In terms of the specifics of what to learn in the geometric Brownian motion journey, I would probably learn how to derive the solution as well as learning the final distribution of St. There are many past paper questions where magicking this out of nowhere (or your head!) would have been helpful, yes.

In terms of what the Examiners "expect" students to know, this implies a level of planning and consideration that might not even be there. We should use past papers as a guide to what might or might not be expected that we learn going into an exam but I'd say it's worth being emotionally prepared for a break in the implied constancy of this. In other words, don't fall off your chair (bad exam technique, in any case) if you're suddenly asked to remember a formula that no past paper ever had suggested was worth memorising.

John

 

[s16455441]

Thank you John - I understand this now.