Matematica e statistica applicata - Notes

Lecture 20-3-2017

The indicator function is useful when you want to price options. Vasicek process: μ is the long-run mean, α is the speed of the process, and σ is the scale parameter. This process is a mean reverting process. It is useful to model interest rates. μ is a positive parameter if you decide to use the Vasicek model for interest rates; it is fixed by the central bank. α depends on the market: if the market is reactive to the information, α is high. σ is the fluctuation of the path. If we can determine the condition probability, we can simulate trajectories for the Vasicek model.

Computing Itô's Lemma

If we want to compute Itô’s Lemma:

! = ! !! "!= ∙ !! "!= ! ! =0! 1"! "! #( ) = ∙ ∙ + ∙ + ∙ 0 ∙! ! !2"! "$ "!( ) = + − + ! !

This process is also Markovian. To compute the conditional distribution of X, we observe that the first part of equation 5 is deterministic. The increments of Brownian motion are independent and normally distributed. If I have a summation of independent normal random variables, the solution is normal too. We are able to establish that the distribution of the integral is normal, but we need to know the mean and the variance. I can also verify that the stochastic integral satisfies the integrability condition because the integral is finite.

To compute the variance, isometry property is used. The isometry property states:

#% % #[ )]) 678 ( : ; = 8 ( ! ! !& &

The conditional distribution of the Vasicek process is normal. CondMean is a function of (X, T, S, μ, α, σ). sCondVar requires (σ, α, T, S).

It is possible to obtain an analytical price for the zero-coupon bond using an analytical formula of the Vasicek process. The interest rate we consider in the model is not the same as the Black-Scholes model because in BSM it is deterministic. In the Vasicek model, the short-rate is an observable process. The price of a ZCB is the discount factor we have from the zero-rate term structure.

Drawback of Vasicek

Short-term interest rate is the internal return of an investment when you give money to an institution, but it should be positive. The condition distribution of Vasicek is normal and it is possible to obtain a negative value. To overcome this problem, μ should be positive and higher than 0. An alternative model is the Cox-Ingersoll-Ross model.

Brownian motion with drift and volatility is useful if we need prices based on historical values.

= + ! ! = ' '( ( (8 = 8 + 8 ! !' ' ' − = + ( − )( ' ( ' > = 0' ' ' ) = + + (( ' (Chan-Karolyi-Longstaff-Sanders is a generalization of the geometric Brownian motion and of the Coz-Ingersoll-Ross model and of the Vasicek model and of the Brownian motion with drift and volatility too.

Lecture 21-3-2017

The density is lognormal. In Vasicek, the probability is normal. The Markovian property means that all the information until s is the value contained in the value of the process at s. If we consider the Brownian motion at t, we have a normal distribution where the mean is the value of the process at time s.

We can use the cumulative density function for the construction of a random number generator:

• () = ( ≤ )) * () for continuous random variable
• = 8 )+,~[0,1]
• Simulate using runif (n=1; 0,1)-+. ( )
• = - -

For some processes, the transition density is unknown. ≅ − ! ! !+. ∆ = ∙ 1 = 0, … , - Euler scheme approximates the transition density with a normal distribution. The exact transition density is lognormal. If the distribution is normal you have a non-zero probability that x can be negative. The transition