Fixed Income Part 2

Question 1

Ornstein Uhlenbeck process

Answer 1

short rate rt is mean reverting and evolves according to
drt = a(b-rt)dt + σdzt*
for constant a >0, b, and σ(<0) and a winer process z*

Question 2

What is the Vasicek model?

Answer 2

d r_t=a(b^*-r_t ) dt+σ dz_t

an AR(1) process in discrete time, normal distribution

Vasiciek model incorporates mean reversion, where the short rate is pulled to a level b at a rate a. This means on average, r_t increases when r_t is smaller than b
and decreases when r_t is larger than b.

On average rt reverts towards b. a defines the speed of the mean reversion (most time low)

Interest rates can become negative in Vasiciek.

Question 3

Short rate dynmaics in Vasiciek

Answer 3

Can use the short rate dynamics from historical data: Yields a, b, σ, i.e., physical parameters.

Can use market prices: Yields a, b∗, σ, i.e., risk-neutral parameters.
Back and forth: λ ≈ −1.2 (AHMAD AND WILMOTT (2007))

λ = interest rate risk premium; b* = b-λσ/a

Question 4

Bond prices in Vasicek

Answer 4

When a =/ 0 bond prices in the Vasicek model satisfy:

P(t, T) = A(t, T) e^(-B (t, T) rt)

Both A(t,T) and B(t,T) can be found in closed form.

B(t,T) depends on mean recersion coefficient
A(t,T) depends on mean recersion coefficient, long run level and volatility

Question 5

Hull-White Model

Answer 5

Hull White is a one factor model and an extension of Vasiciek. It provides an exact fit to the initial term structure.

The short rate evolves according to dr_t=(θ(t)-ar_t ) dt+σ dz_t

This can be characterised as the Vasicek model with a time-dependent reversion level, as at time t, the short rate reverts to θ(t)/a at a rate a. Let F(0,t) being the instantaneous forward rate at time 0 for the maturity t.

If we choose
θ(t)=∂F(0,t)/∂t+aF(0,t)+σ^2/2a (1-e^(-2at) )
then the time-0 term structure is exactly matched. Ignoring the final term (as it is small), it shows that, on average, r follows the slope of the initial instantaneous forward rate curve. When it deviates from that curve, it reverts back to it at rate a.

Question 6

Jamshidians Trick

Answer 6

An option on a coupon bond is the same as a portfolio of options on ZCBs. So, to price an option on a coupon-bearing bond:
Find the threshold r^*, which is the critical value of r for which the price of the coupon-bearing bond equals the strike price of the option on the bond at the option maturity T.
Price the options on the ZCBs comprising the coupon bearing bond using Black’s formula.
Sum the prices to get the price of the option on the coupon bond.

Question 7

What can we price exactly in the Vasiciek and Hull-white model?

Answer 7

Remember earlier:
Caps and floors are effectively ZCB options.
Swaptions are effectively options on coupon-bearing bonds.
In the Vasicek and Hull-White models, we can price exactly ZCB options (Black’s formula) ⇒ caps, floors
Coupon bond options (Jamshidian’s trick) ⇒ swaptions

Question 8

Vasiciek vs. Hull White

Answer 8

Yield curve is an output (conditional on a, b, and r0) when you use Vasicek, an input when you use Hull-White.

Mathematically, Vasicek is a particular case of Hull-White
Pick θ(t) ≡ ab and Hull-White becomes Vasicek.
Bond options in Vasicek? Calculate the yield curve from the model, and use it as an input for Hull-White prices

Question 9

Cox, Ingersoll and Ross

Answer 9

Cox, Ingersoll, and Ross (CIR) have proposed an alternative model with the same mean-reverting drift as Vasicek, but where the standard deviation to the change in the short rate in a short period of time is proportional to √r. This means that as the short-term interest rate increases, the standard deviation increases. Let a, b, and σ are non-negative constants, then
dr_t=a(b-r_t ) dt+σ√(r_t ) dz_t^*

Question 10

Comparison Vasicek vs. CIR

Answer 10

Vasiciek (simpler)
- Mean reversion
- constant vol
- interest rates become negative
- simple closed-form pricing formulas for bonds, futures, eur options, caps, floors, swaptions

CIR (more complicated)
- mean reversion
- volatility increasing with interest rate
- interest rate cannot become negative
- still analytically tractable, although slightly more complicated

Question 11

Further interest rate models:

Answer 11

Black-Karasinki - keeps interest model positive
Multi Factor models
Heath-Jarrow-Morton (HJM), Libor Market Models (LMM): More flexibility for the volatility structure.
But less analytical tractability: Need Monte Carlo simulations.

Question 12

Interest rate tree

Answer 12

(1) we construct a tree matching the dynmaics of a centered ornstein uhlenbeck process
(2) We shift the tree to match the term structure of zero coupon rates. Hull does this by introducing state prices, but we focus on the intuition via risk-neutral pricing