Baxter and Rennie - Previsible Process

[joeguirg]

Baxter & Rennie's Financial Calculus was recommended to me as a good introductory book to understanding pricing. I have a math and finance background and was able to get through the first 30 or so pages without major pause. However, on page 32, a definition is provided for a previsible process which caused me some consternation.

In general, I'm wondering whether there's any intuitive guidance to understand this definition?

More specifically, I'm wondering why

1. random bond price process Bi is previsible (bottom of page 32)?
2. previsible process plays the part of trading strategies where one cannot tell in advance where prices will go (top of page 33)?

Any guidance is much appreciated.

 

[bigbadwolf]

In general, I'm wondering whether there's any intuitive guidance to understand this definition?

There is roughly 3/4 of a page's worth of explanation on pp.96-97 of Williams' "Probability with Martingales." "Previsible processes" are also known as "predictable processes" -- under which name you might find more references in the literature. There's a bit of discussion on pp.226-227 of Capinski and Kopp's "Measure, Integral and Probability." Also a bit of discussion on p.169 of Shafer and Vovk's "Probability and Finance." The single best reference I've found is Ch. 25 of Kallenberg's "Foundations of Modern Probability" (2nd edition).

 

[bigbadwolf]

Also a bit of discussion on pp. 77-78 of the 2nd edition of Musiela and Rutkowski's "Martingale Methods in Financial Modelling."

 

[quotes]

Previsible process means a random process whose state is determined before another process shows up the result

i.e. We can buy the stock before it goes up or down, but we cannot go back to past to buy a stock that goes up today. The event of buying a stock relative to the price change is a previsible process. Its occurrence is independent of what goes on in another process but determined right before it.

 

[brotchie]

The best explanation I've heard is in the context of structural vs. reduced form credit modelling.

Consider a structural debt pricing model (a la Black-Cox 1976) where a firm is placed into default if its value hits some exogenous barrier. Firm value X_t follows a GBM and T is the stopping time for when the default barrier is hit. GBM is continuous such that knowing X_t (or the filtration it's adapted to) for all time t < T ensures that X_t -> X_T as t -> T. As we "approach" the default barrier we can see it coming; the closer we get to the stopping time T, the more certain we are of default!

In a reduced form model however (for example Duffie-Singleton 1999) firm default is triggered by a Poisson process. The Poisson "jump to default" arrives randomly, with no predictability; just like a radioactive particle decaying. Given the model assumptions, even if we know the exact state of the universe 1 nanosecond before default (disregarding quantum phenomenon), we cannot "predict" the default. We know the rate of which default will arrive, but the absence of default over some prior period does not make a default more likely over the next period.

Not mathematically rigorous, but that's what I "imagine" in my head when I think about previsibility and predictability.