Vasicek Model in Interest Rates

Transcription

1 Vasicek Model in Interest Rates In case of Stocks we can model them as a diffusion process where they can end up anywhere in the universe of Prices, but Interest Rates are not so. Interest rates can not be modelled as a simple diffusion process, in Interest Rate Scenarios there can be Government intervention, Change in Econmic Scenarios which will either push the Interest Rates up or down and this cycle continues. We can also say that Interest Rates are a Mean Reverting, where the rates hover around some Long Term Mean. So here we are going to look at such a model which incorporates the Mean reversion behaviour of Interest rates: - The Vasicek Model The discrete time version of the Vasicek Model will look like :-.(i) κ Rate of Mean Reversion, Level of Mean Reversion, ( 1) Bernoulli Random Variables, with Q Probabilities 0.5 If you think back to the CRR Model for Stocks, we could exactly determine the Q probabilities based on the Vol, Ir etc. But in case of IR tress it s not the case, we have more than 1 free parameter and hence we can choose the Q Probabilities as we want as long as its satisfies all the constraints. For simplicity Q Probability 0.5 is chosen for the Vasicek Model. The above eq. is the base Vasicek Model, where κ, & σ are constant. The base model is easiest to calibrate but then it won t be able to match the exact Bond Prices in the market. So it makes sense to make these parameters as time dependant which will enable the Model to match the Market

2 exactly. But the 1 st parameter that needs to be time dependant is always your mean level is evolving with time, so eq (i) can be re-written as :- as this determines how.(ii) Now you can always visualize this as Binomial IR Tree Model. Let s re arrange eq(i):- κ Δt (1 κδt) +.(iii) Let s see the 1 period Binomial Model κ Δt (1 κδt) r n + σ Δt r n κ Δt (1 κδt) r n - σ Δt This is the Binomial Way of looking at it, where in each Time Step Δt, there are 2 States for the IR to achieve, and when you expand this for many time steps you will get your IR tree which will show you your all possible Price Paths. And for each state the Q probability 0.5. Now we have to move from a Discrete Time to a Continuous Time Setting and also we need to Price Bonds under this Model. So we need to find the limiting distribution of the Vasicek Model. If you look at Eq(iii) closely you will realize that this is an AR(1) process. AR(1) - A process considered AR(1) is the first order process, meaning that the current value is based on the immediately preceding value So eq.(iii) can we be re-written as a + b +.(iv) Where, a κ Δt, b (1 κδt), a + b (a + b + ) + a (1 + b) b a (1 + b) + ( b ) + + b a(1 + b + ) b +..(v).

3 .. a + +..(vi) We know th t Δt, as we move to continuous time, (n -> ) nd (Δt 0) Now let s look t how the bove equation evolves with the above Limits. (1 Δt) (1 Δt) Now this Limit is not so straight forward to compute, so let me show you the Maths behind it Let, (1 Δt) z Taking Ln() on both sides, Ln,(1 Δt) - Ln(z) Ln(1 Δt) Ln(z), We can use l Hôpital s rule here, which gives us ( ) Ln(z) -κt Ln(z), z lim lim lim lim ( ) lim (1 (1 Δt) ) ( 1- ) ( 1- ) lim We know that this will behave like a Random Variable from the Normal Distribution, so we need to figure out the Mean and Variance of this Random variable and we will be done. Let this R ndom V ri ble be X, - 0, the expectation of the Bernoulli R.V. s were 0, -, lim -, lim - lim, -, *, - 1, basic Expectations Theory} lim lim lim ( ) lim lim (1- ) Let call this So now we can re-write eq.(vi) as:- ( 1- ) + + Z, where Z N(0,1).. (vii)

4 Few interesting facts around. When T is small, this reduces to,. {Do the Taylor expansion around T} When T is large, this reduces to. Now once we have the limiting distribution of IR in the Vasicek Model we need to see how we can price a Bond. Generally a 0 Coupon $1 bond will be priced as P(0). This relationship holds when IR is constant, but IR are not constant and they are Stochastic in Nature, so we nned to modify our Bond Pricing Equation as follows:- P (0), - { Pricing under Risk Neutral Probabilities Q} Now the challenge is to find a solution for this Integrand and that is what we are going to do here. Let s c ll this Integrand Integrations are nothing but summations as long as there is convergence. lim, we need to solve this. Let s rec ll Eq(i) - + N - + T - +.(viii) Now we know that, a + + T + - a - - T - a Let s collect the 2 Stoch stic terms nd work with those - ( - ),( +. ) (. ) ] (1 ) T - a - + (1 ) { Tκ - a - + } (ix) κ From our previous analysis we know that this Integrand is just some Normal Distribution., ] { T - ( 1- e ) - } * (1 ) + T ( 1 e ) } * + Tκ } κ, ] lim (1 ), - lim (1 2 ) lim (N ) (T - ( 1- ) + (1- ) )

5 So now we have all the pieces together, we have determined to be Normally Distributed with some mean and Variance, and now we can find out the Bond Price. P (0), - { Use the Moment Generating Function to find out the expectation of the exponential of a Normally Distributed Variable} P (0), -., - *This c n further be simplified but let s le ve it t th t+ We can see that this is a Linear Function in the Exponent, this is called an Affine Model. Affine Model Any model which leads to Bond Prices which are exponential of a Linear Function. So Vasicek is an Affine Model.