The Vasicek Interest Rate Process Part I - The Short Rate

Transcription

1 The Vasicek Interest Rate Process Part I - The Short Rate Gary Schurman, MB, CFA February, 2013 The Vasicek interest rate model is a mathematical model that describes the evolution of the short rate of interest over time. The short rate is the annualized interest rate at which an entity can borrow money for an infinitesimally short period of time. Vasicek models the short rate as a Ornstein-Uhlenbeck process. An Ornstein-Uhlenbeck process is a mean-reverting process where the short rate is allowed to incorporate random shocks but is pulled back to it s long-term mean whenever it moves away from it. Interest rates exhibit mean reversion, which is the tendency for a stochastic process to return over time to a long-term mean. Vasicek s stochastic differential equation that describes the evolution of the short rate R t in continuous time is... δr t = θ(λ R t δt + σ δw t (1 When the short rate moves below its long-term mean λ the short rate drift becomes positive and the short rate is pulled upward. When the short rate moves above its long-term mean the short rate drift becomes negative and the short rate is pulled downward. The speed at which the drift is pulled upward of downward is given by the positive valued parameter θ, which measures the speed of mean reversion. The greater the speed the faster the process reverts toward the long-term mean. Random shocks are introduced via the variables σ, which is the annualized short rate volatility, and δw t, which is the change in the driving Brownian motion over the infinitesimally short time interval [t, t + δt. We will define the short rate at time period t > s to be R t, which is the short rate at period s (known plus the sum of the changes in the short rate from period s to period t (unknown. The equation for the short rate R t in continuous time is... t R t = R s + δr u (2 In this white paper we will develop the mathematics of the Vasicek short rate stochastic process and use the mathematics to answer the three questions applicable to the hypothetical problem below... Our Hypothetical Problem We are tasked with pricing a pure-discount bond and for that task we need a short rate curve. Our go-forward interest rate assumptions are as follows... Description Symbol Value Current short rate R Long-Term short rate mean R 0.08 Mean reversion rate θ 0.40 Annualized short rate volatility σ 0.02 s Question 1: Graph the short rate curve from year 0 thru year 10. Question 2: What is the expected short rate at the end of years 1 and 3? (include confidence interval. Question 3: What is the correlation between the short rate at the end of years 1 and 3? 1

2 The Vasicek quation For The Short Rate Of Interest We will define the function f(r t, t to be a function of time t and the short rate of interest at time t. As we did in quation (1 above we will define the variable θ to be the rate of mean reversion, the variable R t to be the short rate of interest at time t, the variable λ to be the long-term short rate mean and the variable σ to be the annualized short rate volatility. Given these definitions we will further state that the equation for f(r t, t is... f(r t, t = e θt (R t λ (3 The derivatives of quation (3 with respect to time t and the short rate at time t are... δf(r t, t δt = θe θt (R t λ...and... δf(r t, t δr t = e θt...and... δ 2 f(r t, t δr 2 t = 0 (4 Per Ito s Lemma the equation for the change in the stochastic short rate R t over the infinitesimally small time interval [t, t + δt is... δr t = a(r t, tδt + b(r t, tδw t (5 Using quations (1 and (5 above, which both define the change in the short rate R t, and given that... δr t = a(r t, tδt + b(r t, tδw t...and... δr t = θ(λ R t δt + σ δw t Then after mapping the two equations we can make the following definitions... a(r t, t = θ(λ R t...and... b(r t, t = σ (6 Per Ito s Lemma quation (3 is once differentiable with respect to time t and twice differentiable with respect to the driving Brownian motion W t. Using a Taylor Series xpansion the equation for the change in f(r t, t is... ( δf(rt, t δf(r t, t = + a(r t, t δf(r t, t + 1 δt δr t 2 b(r t, t 2 δ2 f(r t, t δrt 2 δt + b(r t, t δf(r t, t δw t (7 δr t Using quations (4 and (6 above we can rewrite quation (7 as... δf(r t, t = (θe θt (R t λ + θe θt (λ R t δt + σ e θt δw t Taking integrals of both sides of quation (8 above we get... = σ e θt δw t (8 δf(r u, u = f(r t, t f(r s, s = σ σ After substituting quation (3 into quation (9 we can rewrite quation (9 as... e θt (R t λ e θs (R s λ = σ e θt R t e θt λ e θs R s + e θs λ = σ e θt R t = e θs R s + e θt λ e θs λ + σ (9 R t = e (e θt θs R s + e θt λ e θs λ + σ quation (10 is therefore the equation for the short rate at time t (R t given the short rate at time s (R s, the rate of mean reversion (λ and the annualized short rate volatility (σ. (10 2

3 Proof We will now prove that short rate quation (10 is the solution to Vasicek s stochastic differential quation (1 above. We will begin by making the following simplifying definitions... X = e θt...and... Y = e θs R s + e θt λ e θs λ + σ (11 Given the definitions in quation (11 above we can rewrite quation (10, which is the equation for the short rate at time t, as... R t = XY (12 The derivatives of X are... The derivatives of Y are... δx δt = θe θt δy δt = θλeθt...and......and... δx δw t = 0...and... δy δw t = σe θt...and... δ 2 X Using a Taylor Series xpansion the equation for the change in the short rate R t is... ( δx δr t = δt Y + δy ( δx δt X δt + Y + δy X δw t + 1 ( δ 2 X δw t δw t 2 Using derivative quations (13 and (14 above we can rewrite quation (15 as ( ( δr t = θe θt Y + θλe θt X δt + (0Y + σe θt X δw t δ 2 Y = 0 (13 = 0 (14 Y + δ2 Y δwt 2 X ( (0Y + (0X (15 = ( θr t + θλ δt + σ δw t = θ(λ R t δt + σ δw t (16 quation (16 is equivalent to Vasicek s stochastic differential quation (1, which concludes the proof. Stochastic Short Mean And Variance We can rewrite quation (10, which is the equation for the stochastic short rate at time t given the short rate at time s, as... R t = R s e θ(t s + λ (1 e θ(t s + σ e θt (17 Given that... [δw u = 0 (18 Using quations (17 and (18 above the equation for the first moment of the distribution of R t is... [ [R t = R s e θ(t s + λ (1 e θ(t s + σ e θt = R s e θ(t s + λ (1 e θ(t s + σ e θt = R s e θ(t s + λ (1 e θ(t s Using quation (19 above we will make the following definition... µ t = [R t = R s e θ(t s + λ (1 e θ(t s e θu [δw u (19 (20 3

4 Using the definition in quation (20 we can rewrite the stochastic short rate quation (17 as... Given that... When u v then [δw u δw v R t = µ t + σ e θt (21 [ = 0...and... When u = v then [δw u δw v = δwu 2 = δu (22 Using the revised short rate quation (21 and the expectations in quations (18 and (22 the equation for the second moment of the distribution of R t, which is the expected value of the square of the short rate R t given the short rate R s, is... [ [( Rt 2 = µ t + σe θt [ = µ 2 t + 2 µ t σ e θt = µ 2 t + 2µ t σe θt = µ 2 t + 2µ t σe θt = µ 2 t + σ 2 e t 2 + σ 2 e t e θu [δw u + σ 2 e t v=t v=s v=t v=s e θu [δw u + σ 2 e t e u δu v=t v=s e θu e θv δw u δw v [ e θu e θv δw u δw v u v + σ 2 e t v=t v=s [ e θu e θv δw u δw v u v + σ 2 e t [ e θu e θv δw u δw v u = v [ e u After solving the integral in quation (23 above the equation for the second moment of the distribution of R t becomes... [ Rt 2 = µ 2 t + σ 2 e t e u δu δw 2 u (23 = µ 2 t + σ 2 e t 1 [ t eu s (e t e s = µ 2 t + σ 2 e t 1 = µ 2 t + e (t s Using quation (19 above the mean of the short rate at time t given the short rate at time s is... mean = [R t = R s e θ(t s + λ (1 e θ(t s (24 (25 Using quations (19, (20 and (24 above the variance of the short rate at time t given the short rate at time s is... [ ( 2 variance = Rt 2 [R t = µ 2 t + e (t s µ 2 t = e (t s (26 4

5 Stochastic Short Rate Covariance Imagine that we are currently sitting at time period 0 and that there are three future time periods x, y and z such that 0 < x < y < z. Given that R x is the short rate at time x the equation for the short rates at time y (R y and at time z (R z can be written as... R y = R x + δr u...and... R z = R x + δr v (27 Because y < z the increments in the driving Brownian motion during the time interval [x, y will be the same for short rates R y and R z and therefore the two short rates are positively correlated. The equation for the covariance between the short rate at time y and the short rate at time z is... Covariance [R y R z = [R y R z [R y [R z (28 Using quation (21 above we can write the equations for the short rate at time y and at time z as... R y = µ y + σe θy...and... R z = µ z + σe θz e θv δw v (29 Per covariance quation (28 we need the expectation of the product of the two short rates R y and R z. expectation in equation form is... [( [R y R z = µ y + σe θy [ = µ y µ z + µ y σe θz = µ y µ z + µ y σe θz ( µ z + σe θz e θv δw v + µ z σe θy e θv [δw v + µ z σe θy e θv δw v + σ 2 e θ(y+z Given the expectation in quation (18 above quation (30 becomes... e θu [δw u + σ 2 e θ(y+z e θu e θv δw u δw v This e θu e θv [δw u δw v (30 [R y R z = µ y µ z + σ 2 e θ(y+z e θu e θv [δw u δw v (31 We can rewrite the double integral in quation (31 as... e θu e θv [δw u δw v = [ e θu e θv δw u δw v u v + Given the expectations in quation (22 above quation (31 becomes... [ e θu e θv δw u δw v u = v (32 z [R y R z = µ y µ z + σ 2 e θ(y+z e u δu = µ y µ z + σ 2 e θ(y+z 1 eu [ z = µ y µ z + σ 2 e θ(y+z 1 (e (y z e x (33 5

6 Using quations (28 and (33 above we can write the covariance of the short rate at time y and the short rate at time z given the short rate at time x as... Covariance (R y R z = [R y R z [R y [R z = µ y µ z + σ 2 e θ(y+z 1 (e (y z e x µ y µ z = (e σ2 e θ(y+z (y z e x (34 Simulating The Stochastic Short Rate Given that R t is the stochastic short rate at time t (unknown given the short rate at time s < t (known, m is the short rate mean per quation (25 and v is the short rate variance per quation (26, we can define the normalized random variable Z via the following equation... R t m v = Z...such that... Z N [ 0, 1 By rearranging quation (35 above the simulated value of the short rate at some future time t given the short rate at time s < t is... R t = m + [ v Z...where... Z N 0, 1 (36 (35 Answers To Our Hypothetical Problem Question 1: Graph the spot rate curve from year 0 thru year 10 - Uses quation (25 above Spot Rate Curve Rate Year Question 2: What is the expected short rate at the end of years 1 and 3? (include confidence interval - Uses quations (25 and (26 above. The expected spot rate at the end of year 1 is... mean = R 0 e θ(1 0 + λ (1 e θ(1 0 = 0.04 exp( (1 exp( = (37 6

7 The expected spot rate at the end of year 3 is... mean = R 0 e θ(3 0 + λ (1 e θ(3 0 = 0.04 exp( (1 exp( The spot rate variance at the end of year 1 is... variance = e (1 0 = (38 = (1 exp( (2(0.40 The spot rate variance at the end of year 3 is... variance = e (3 0 The confidence interval (at 2 std dev is... = (39 = (1 exp( (2(0.40 = (40 Spot rate end of year 1 = ± = ± (41 Spot rate end of year 3 = ± = ± (42 Question 3: What is the correlation between the short rate at the end of year 1 and year 3? - Uses quations (26 and (34 above. The covariance between the spot rate at the end of year 1 and year 3 is... Covariance (R y R z = σ 2 e θ(1+3 1 (e (1 3 e (0 ( = e ( e = (43 Using the variance of the spot rate at the end of years 1 and 3 as calculated in Question 2 above the correlation between the spot rate at the end of year 1 and year 3 is... Correl (R 1, R 3 = Covariance (R 1R = = 0.35 (44 Sdev R 1 Sdev R