Help needed.

[olly]

If anyone could explain to me the following I would be greatly appreciative.

P.104 Baxter & Rennie (Pricing foreign exchange securities) deals with pricing a call option on an exhange rate, C_t. It tells us that as the forward price F is the expectation of C_T under the martingale measure Q, the value of C_T can be written in the form:

F * exp( sb*Z - 0.5*sb^2) where sb denotes sigma bar which is the volatility of of log(C_T) i.e. (sigma^2)*T and Z is a normal(0,1) random variable under Q.

This expression change (or at least a similar technique) for C_T comes up later in the book so I would be happier if I understood it. It seems to quote it as formal result but without showing how. Perhaps its very simple if I think about it but my brain is a little fried at the moment!

 

[examstudent]

Hi
Im crap at writing in maths symbols on computer, but ill try anyway

Look at top P102 denotes Ct = Co * exp * (sigma Wbt + (r – u - 0.5 * SIGMA ^2 ) t )
Where t,0 referes to time t,0 repectively and Wbt refers to Q brownian motion at time t

Similarly We can write C (T) = Co * exp (sigma WbT + (r – u – 0.5 * SIGMA ^2 ) T )
Where T refers to time T (expiry) and WbT refers to Q Brownian motion at time T

P103 tells us forward price F = Eq (C (T) ) = Co * exp ( r – u ) T

Hence C(T) = F * exp (sigma WbT – 0.5 * SIGMA ^2 ) from removing the Co * exp (r – u ) T

But a Wb T random variable (has mean zero and variance Sigma ^2 * T )
And is hence identical to (sigma * ( T ^0.5 ) * Z ) random variable where Z is N (0,1)


Hence C(T) = F * exp ( sigma * ( T^0.5) * Z - 0.5 * SIGMA ^2 * T )

Which should be identical to expression on P104 which is parameterised as sigma with bars on them…



NB the infamous ST6 core reading proves BS formula by replacing the Brownian increment with a Z N(0,1) * square root of time , as does literally every other form of proof in other books etc.

I hope my notation haven’t confused or mislead but probably work through a similar kind of analysis to satisfy yourself…or maybe someone could provide you with a neater version

With regards to similar techniques coming on later in the book, I suspect that you mean the spot value of the underlying at expiry T (usually an exponential function) is rewritten in terms of its forward price F and hence yields a black forward formula type approach…

 

[Gareth]

olly, i think you are reading to deeply here.

We have that:

and from page 102:

Thus:

which is the result they state.

[edit] for writing maths in posts u can use http://rogercortesi.com/eqn/index.php

 

[examstudent]

wow gareth

i always wondered hwo equations could be typed on screen

thnsk for the link

 

[Gareth]

if you know latex then it's pretty easy to do online equations. latex beats word's equation editor any day imo, but it takes a while to learn.

shame we dont have the facility for latex equations built into this forum software, like wilmott.com do.

 

[olly]

Thank you ES and Gareth.

It was, as I pretty much suspected, right there under my nose. Thanks for the help though.
With no disrespect intended to ES here, Gareth - you seem to be well versed in stochastic calculus and this subject in general, judging by some of the other threads. Have you studied this subject before either in an actuarial exam context or as part of some other programme?

 

[Gareth]

i've never studied it before doing 103. but i find the subject very interesting, and have read many papers and books on the subject. i think we will find that arbitrage pricing methods will become a big part of actuarial mathematics in the future - e.g. in general insurance bonds are now traded on insurance risk. we also have mortality bonds being traded in life.

i find the maths side of st6 quite easy, but it's the wordy questions that worry me.

i'm doing it this april with CT8, so hopefully it will go ok.

 

[olly]

Well, you seem to have picked it up pretty well!, if CT8 is the equivalent of 109 there's a nice bit of synchronicty there I seem to remember. I'm taking st6 for the 3rd time, this time armed with the hull and b&r books - wish they'd recommended them the first time around.
As for your query re: x5.4 part 3, i agree but figured you probably didn't need me to say so for your own peace of mind.

 

[examstudent]

hi all

olly-no offence taken about the “disrespect”- although me and gareth gave you the same derivation (albeit mine was with dodgy symbols and his was with neat equations) - gaerths post on bounded variation/stochastic vs newtonian claculus confirms him as the "actuarial heavyweight of stochastic calculus"!!! LOL…..

gareth - given the recent news on the institute website about a quantitative finance initiative, you re surely right about arbitrage free pricing being a key place in the actuarial arena in the future - but a big question must be how these methodologies apply in these mortality/insurance bond markets where black scholes type assumptions arent so valid?

But do you reckon actuarial involvement in these securities will be limited to appraisal from a risk perspective/macro type outlook as opposed to pure pricing/modelling, or a balanced hybrid of these approaches, because as you know from wilmott forums, pricing these securities requires pretty heavy quant methodologies which are well well well beyond ST6? Would the actuarial profession really be willing raise the minimum maths requirement to MSc/phd level just so as to allow actuaries a pricing role in these exotic securities?

-gareth - the wordy questions in ST6 arent as bad as some of the other 300 type wordy questions as there are less points you can make – and it may well have more maths in it now that the additional textbooks have been mentioned…

With regard to the textbooks which do you guys prefer….hull or B & R ..
i haven’t fully read all of both books (as arent planning to retake this in april but maybe in september) but I really like the style of b & r chapters 1-4 as it is very intuitive, fully emphasising the martingale approach to pricing (the core reading has sloppy notation on this!)
Hull though seems to be a disjointed practical manual and doesn’t derive half the results!!!grrrr
Gareth –what other books do you find a nice read on this subject – wilmott’s books are comprehensive, but seems more computationally orientated (numerical methods/PDEs)

Good luck to both of you for st6, and i hope both this option you took out on st6 ends up in the money for you both (i.e a pass!!!!!!! )…..

 

[olly]

I can't really say I prefer one or the other ES although I think Hull is probably more help. As you say, b&r walks you through it in a nicely intuitive fashion although I laughed a little when it suggested at the start that no particular body of knowledge was assumed from the start as this is clearly rubbish. It's good that you have to do stuff for yourself as it often quotes results without many of the intervening steps that more modern texts would put in - something I clearly fell foul of leading to this thread.
I think that you should probably read b&r first to get comfortable with the subject and then read hull as it is more in depth and the end of chapter Q&A's help entrench the knowledge. I did it the other way round and am now returning to Hul before getting into the acted material again. All in all, I think they complement each other well.

As an entirely uninteresting detour, I was a little miffed when I ordered hull ed. 5 from the institute in their "big sale" (why sell off all the old stock just because the shop is going online?), as I soon realised that you couldn't get ed.5 solutions for love nor money. To their credit however they swapped it for ed. 6 and sourced the relevant solutions for me with no extra postage charge.

And ES, I'm sure I don't need to tell you but I realise you both were equally helpful with the solution (and indeed you got there first! ), I just didn't mean to offend you as I wanted to question Gareth directly.

 

[Gareth]

i think that now we have insurance bonds and mortality bonds, you can argue that expected value methods of valuing liabilities are no longer valid - e.g. a life office selling annuities could hedge their mortality risk through use of mortality bonds (this could also apply to DB pension schemes). a general insurer could hedge their exposure to catastrophes by holding cat bonds.

swiss re seems to be leading the way in opening new investment markets in mortality and insurance bonds, and investors are pouring money into these new areas (since they have zero correlation to the rest of the investment market, it provides good diversification).

so still early days, but it does not seem implusible that in 10 years time, all actuaries valuations will be done on a no arbitrage basis, requiring all the complex techniques currently used by investment banks.

I agree ST6 is wholly inadequate to provide us with enough knowledge of the subject, hopefully IOA/FOA will improve the syllabus and add more derivative related subjects.

I don't think a phd is necessary to work in this field. I did a msc in pure maths, and I feel comfortable with the level of maths used in most of the literature - although I suspect that we would have to rule out wider disciplines from the actuarial profession, as they would not have the mathematical understanding to succeed in this area.

Other good books:

Steven Shreve - Stochastic Calculus and Finance
Oksendal B. - Stochastic differential equations

There are 100's of papers on the subject which you can find with google for free. Also see wilmott.com for links to papers.

I personally prefer Baxter and Rennie, as it takes a more mathematical approach to derivative pricing, and is written in a more appealing way. That said, I am sure Hull is the key to passing - the core reading seems to be a copy and paste job to me (with a few word changes here and there). It surprises me how badly it is copied in places - maybe IOA / Acted have a deal with the publisher to do this or something?

Wilmott's book is not much use for our exam imo. It is rather applied and focused on PDE's - not a bad thing since this is the most important part of the practical work from what i understand, but doesnt help for passing ST6. In practice, from speaking to people who work in the field, no one uses the martingale approach for actual pricing, it's really just a theoretical tool.

anyway, enough rambling...ST6 calls!