April 2011, Q1iv

[echo20]

This question is asking for reasons why the cost of guarantees calculated with a stochastic model will differ from a Black-Scholes calculation. I’m a bit surprised neither the examiners report nor ASET refer to the fact that Black-Scholes assumes a lognormal distribution, rather than something with fatter tails. Isn’t this a significant cause for under-estimating the cost of guarantees? Would the stochastic asset models that are actually applied use a fatter tailed distribution or would they generally also assume lognormal returns?

 

[scarlets]

I'm surprised that's an SA2 question at all and not something that would be more naturally and fairly asked in ST2 or CT8 etc.

 

[Mark Willder]

This question is asking for reasons why the cost of guarantees calculated with a stochastic model will differ from a Black-Scholes calculation. I’m a bit surprised neither the examiners report nor ASET refer to the fact that Black-Scholes assumes a lognormal distribution, rather than something with fatter tails. Isn’t this a significant cause for under-estimating the cost of guarantees? Would the stochastic asset models that are actually applied use a fatter tailed distribution or would they generally also assume lognormal returns?

I think this point is going beyond the level of understanding that the examiners would expect for SA2. Whenever you see something involving derivatives or stochastic modelling in the exam, the examiners are generally looking for the main practical points rather than technical details.

However, you are quite right. There are a number of holes in Black-Scholes, indeed Fischer Black wrote a paper on the topic over 20 years ago. In practice, people make all kinds of adjustments to Black-Scholes (or use much more advanced techniques). But we shouldn't be too hard on Black-Scholes - it's an enormous leap forwards compared to the old fashioned deterministic techniques.

Best wishes

Mark

 

[echo20]

Thanks Mark. I agree that Black-Scholes is a pretty amazing bit of maths! I was mainly wondering whether stochastic models used in practice would generally be based on normal/lognormal distributions. In this case I can see how you can choose the volatilities to be the same as B-S (which was one of the examiners' points) but if you have a different distribution, it'd be a bit less clear as to whether it was a like-for-like comparison, even if you chose the variance to be the same.

Scarlets - I think it was fair question in that as Mark says, it was trying to test understanding of how stochastic models work in practice which seems decent SA2 fodder. Just wish I'd done better at it!

 

[Mark Willder]

I agree that Black-Scholes is a pretty amazing bit of maths! I was mainly wondering whether stochastic models used in practice would generally be based on normal/lognormal distributions. In this case I can see how you can choose the volatilities to be the same as B-S (which was one of the examiners' points) but if you have a different distribution, it'd be a bit less clear as to whether it was a like-for-like comparison, even if you chose the variance to be the same.

(Note, the following discussion goes beyond what I think you'd be expected to know for SA2.)

In practice some people are still using normal and lognormal distributions because the mathematics is so much simpler. If you are valuing a large number of policies using stochastic simulations (perhaps for Solvency II or MCEV) then any reasonable approximations you can make to save computing time/cost is worthwhile.

However, increasingly more sophisticated techniques are being used. A while back I was involved in some research using Levy jump processes (another criticism of B-S is that it assumes that prices move continusously by tiny amounts, when in reality they make sudden jumps at discrete times). There's a wealth of actuarial literature on the topic suggesting all kinds of models, eg hyper models (where the parameters for each simulation are also random) and regime switching log-normal models (where the volatility assumption moves between low and high risk states).

All this research comes to the same conclusion as you did. The risk of adverse experience is much higher than predicted by B-S and so the corresponding cost of any options and guarantees is undervalued by B-S.

If anyone can comment on the models they use in practice (without giving away company secrets) then it would be interesting to hear from you.

Best wishes

Mark

 

[echo20]

Thanks a lot Mark, sorry for straying off syllabus - but really helpful for me to get some idea of what's done in practice, having no experience in this area.