Select ten standard, normal random numbers using Mathematica, Maple or Matlab. Suppose interest rates follow the SDE:

dr_t = .02r_tdt + .06r_tdW_t.

Assume that the current spot rate is 6%.

(a) Discretize the SDE given above.

(b) Calculate an estimate for the following expectation using a time interval ? = .04,

and the random numbers you selected. Assume that the expectation is taken with respect to the true probability.

(c) Calculate the sample average for

and then multiply this by the sample average for

E[max[rt ? .06, 0]].

Do we obtain the same result?

(d) Which approach is correct?

(e) Can you use this result in calculating bond prices?

(f) In particular, how do we know that the interest rate dynamics displayed in the above SDE are arbitrage-free?

(g) What would happen to the above interest rate dynamics if we switched to risk-neutral measure

(h) Suppose you are given a series of arbitrage free bond prices. How can you exploit this within the above framework in obtaining the arbitrage-free dynamics tot r_t?

Transcribed Image Text:

Ee-d max[7, – .06, 0], ,ds