Number of Paths from (0,0,0) to (n,n,n)

Q: How many paths are there to go from (0,0,0) to (n,n,n)? i.e. walk in a certain direction by one step each time.

A: (3n)C(n)*(2n)C(n)*(n)C(n). In general, for k-dimensional space, the answer would be a product of k terms. Things to ponder:

Why combination, not permutation? -> There are 3n steps to walk in total, and there are 3 dimensions x, y and z. We see the steps, not the dimensions, as the space to pick from. For example, for n = 2, there are 6 steps to make, 1, 2, 3, 4, 5 and 6. We pick two from these to assign to the x-direction, and that's why the order is not important (assigning x-direction to positions 1,5 is the same as assigning x-direction to positions 5,1).

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Merton's Jump Diffusion Model: market completeness etc.

Recall:
Existence of martingale measure <=> no-arbitrage
Uniqueness of martingale measure <=> market completeness

When there are jumps, the market is no longer complete because the jump process 'creates' many more states so that the number of asset becomes too few. Hence we are left with the unknown "market price of jump risk". Merton proposes that this price of risk should be zero because the jump in a stock is non-systematic, i.e. diversifiable.

Note that empirical study suggests that Merton's assumption is quite wrong.

Correlation and Dependence

http://en.wikipedia.org/wiki/Correlation_and_dependence
http://mathforum.org/library/drmath/view/64808.html

- There are many flavors of correlation measures
- Non-zero correlation => Dependence
- Independence => Zero correlation

BUT

- Zero correlation =X=> Independence