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2 The Vasicek model for interest rates In the Black-Scholes model, the assumption was made that the interest rate was fixed and risk-free. In practice this is not the case and several models exist that attempt to describe interest rates mathematically. The Vasicek model defines the price, Q, at time 0 of a bond paying one unit at time f as Q. = exp (- [ Reds). where { R, : s > 0), the spot-rate process, is an Ornstein-Uhlenbeck process. More precisely, R. =e Ro+ (1 -e ")u+X. where Ro is the initial spot rate, # is the long-term mean level, 0 > 0 is the speed of reversion and X, is an Ornstein-Uhlenbeck process with volatility a > 0 and reversion parameter 0 > 0. That is E[X,] = 0 and Cov (X,. X,) = me (att) (20 mins!) - 1). Simulate (R. : 0