Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

BR: The Binomial Representation Theorem – I

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The second half of the second chapter of BR’s book uses the binomial tree model discussed so far to introduce some of the basic probabilistic concepts in the theory of mathematical finance (in particular, the ones they need to build the theory in continuous time)

1. Process: The set of of possible values the underlying can take. The random variable S_i, then, denotes the value of the process at time i.

2. Measure: How probabilities evolve over the tree

3. Filtration: The history of the asset price S_i up until the time \displaystyle i on the tree: F_i. It corresponds to the history of S_i on a particular node as of time i. The binomial structure ensures that there is only history corresponding to any node. Given a node and a point in time filtration fixes the history “so far”. It is a useful construct to talk about where we are at a point in time and how we reached there.

4. Claim: The derivative in question being priced, which gives a payoff at “maturity”: X – it’s value is a function of the filtration F_T.

5. Conditional Expectation of a Claim: Given a claim X, we can talk about it’s expectation given the history “so far”, i.e. a filtration F_i – or in other words, the expectation E^Q[X|F_i]. Note that this makes this conditional expectation also a random variable, as it depends on the filtration – i.e. where we are on the tree.

For each node/point in time, given the probability measure Q, E^Q[X|F_i] denotes the expectation of the claim X if we have observed a filtration F_i. It is clear that the unconditional expectation is the same as \displaystyle E^Q[X|F_0]

6. Previsible process: It’s a process \phi_i whose value at any time is dependent only the history upto one time earlier, i.e. F_{i - 1} (e.g. \phi_i = S_{i - 1})

7. Martingale: A process S is a martingale with respect to a measure Q and a filtration F if \forall j\leq i:

E^Q[S_i | F_j] = S_j

That is, the process S, if it is a martingale does not move up or down systematically – i.e. it has no drift.

8. Tower Property: \forall k\leq j\leq i

E^Q[E^Q[S_i | F_j]|F_k] = E^Q[S_i|F_k]

It’s a useful mathematical property of expectations not only in mathematical finance but also in time series analysis. Another way to interpret the Tower property is to say that the conditional expectation process of a claim is always a martingale (obviously, this is just a consequence of the mathematical property of conditional expectations).

Please note that previsibility is closely related to the notion of predictability. It essentially says that a process whose value is “known” at any time given the “history so far” is previsible. Technically, a process X_tis previsible, if given a filtration F_t over a probability space (\Omega, P, F), X_t is adapted to F_t or, alternatively, is F_t – measurable. On the other hand, the martingale property is a probabilistic concept relating best forecast to conditional expectation of the random variable (in finance with a clear physical interpretation of “no drift”).


Written by Vineet

August 27, 2010 at 4:26 pm

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