Black, LMM, Interest rate model

[Crumple]

Hi,

-How can the stochastic nature of interest rates be overlooked when pricing interest rate derivatives using the black model

-How is the Radon-Nikodym derivative a random variable. Is it not a set of known constant ratios of probabilities in discrete terms for every path of a random variable?

-LMM in acted notes first starts of with a P(t,T) as a numeraire then there is a conversion to a rolling CD as the numeraire. Can someone expand on this please. or an online paper?

-The notes say that the statistical properties of the process exp(integral_of(-r(u)du)) need to be determined to price an interest rate derivatives using the expectation V(t,T) = E_q[exp(integral_of(-r(u)du))X(T)|r(t)]. page 11 chapter 13.
Is there not a second uncertainty with regards to the X(T) which is the payoff and may depend on for example the deposit rate. So how will this be dealt.

Thanks.

 

[David Hopkins]

The Black model assumes that the underlying variable has a particular lognormal distribution at time T and a payoff directly linked to that value. If you have a derivative where the payoff depends on other variables, eg previous interest rates, you wouldn't be able to use this model.

Similarly, the formula in your last question can only be used if the final payoff XT can be expressed purely in terms of the values of the short rate r over the period from time t to T.

I think you've answered your own question about the RN derivative. In a binomial tree model the future stock price is a random variable because it has different values on different branches of the tree. So does the RN derivative!

I'm afraid I'm not an expert on the LMM model. I can't tell you any more than is in the reading.