Interest Rate Models & Hedging Flashcards
What is the payoff of an FRA
These instruments are convenient for hedging interest risk. With 0 inception value, the buyer of an FRA receives(pays) N(Ft - Lti) δ if it is positive (negative)
What are some complications with pricing interest rate derivatives
The price of a bond depends on the stochastic behavior of the current and future spot rates. We need to assume that bond prices are a function of current and future rates and that the spot rate cannot be assumed constant
Other complications
* Many interest rate derivatives are American and can be exercised early
* The payouts of underlying securities may be different for interest rate derivatives
* We need to replace the drift in drt dynamics with something other than the risk free rate, because r is what we’re trying to model. Applying Girsanov is not straightforward
What are 3 fundamental equations in fixed income derivatives
Neftci 18
- Ctsly compounded rate: R(t,T) = -ln(z(t,T))/(T-t)
- Stochastic rate bond prices: Z(t,T) = EtQ[e-∫(t,T) r(s)ds]
- Ctsly compounded forward rate: F(t,T,U) = [ln(Z(t,T) - ln(z(t,U))]/(U-T)
What are 5 common shapes of the Yield curve
- Normal: Increasing, concave down (r shape)
- Decreasing, concave up (L shape)
- Flat
- Humped (spike early on)
- Inverted Humped (valley early on)
How are ZCB prices and forward rates connected
Neftci 19
The arbitrage free relation uses the instantaneous forward rate to find future ZCB prices
Z(t,T) = e-∫(t,T) F(t,s)ds
F(t,s) = f(t,t,s) = instant forward curve as of time t
Describe the HJM model for F(t,T) (ZCB prices are GBM)
Let dBt = rtBtdt + σ(t,T,Bt)BtdWt. Then the SDE for F is
dF(t,T) = σ(t,T,B(t,T))[∂σ/∂T]dt + [∂σ/∂T]dWt
Compare the classical vs traditional approach
Classical: Takes expectations in RN world. Future values of r fluctuate with time, but bond prices are martingales in RN world. The Yield Curve at t contains all info needed for bond prices based on no arbitrage. Given a proper model for r, we can obtain the observed prices
HJM: No expectations, all values of F(t,s) are known at t. F is the instant fwd rate and the forward curve at t contains all info needed for bond prices based on no arbitrage. Given a proper model fro F(t,T) we can obtain observed prices
What is the market price of interest rate risk
Assume GBM process for ZCB
dBi/Bi = μi(Bi, t)dt + σi(Bi, t)dWt
The market price of interest rate risk is λ(rt, t) = λt = (μi - rt)/σi
What is the fundamental PDE for interest rate derivatives
Let drt = a(rt, t)dt + b(rt, t)dWt. Then any fixed income instrument B must satisfy
Br (a(rt, t) - λtb(rt, t)) + Bt + 1/2 Brrb(rt, t)2 - rtB = 0
The solution to the PDE for interest rate derivatives
We use Feynman Kac to find that B(t,T) = EtQ[e-∫(t,T) rsdsB(T,T)
Define the key equation for a constant discount model
Neftci 21
Consider a function F(xt), x stochastic such that for β>0
F(xt) = EtP[∫(t,inf) e-βsg(xs)ds]. This function solves the following PDE
μFx + 1/2 Fxxσ2 - βF + g = 0
What is the fundamental PDE of a bond price?
Assume the time t price of a ZCB
B(t,T) = EtQ[e-∫(t,T) rsds]. This function solves the following PDE:
-rtB + Bt + Br[a(rt,t) - λtb(rt, t)] + 1/2Brrb(rt, t)2 = 0
With the normal boundary B(T,T) = 1
Describe the Ho Lee model
FIS 14
drt = θtdt + σdXt with inital value r0
* The dynamics have some attractive characteristics
* Drift is given by θtdt
* Diffusion comes from σdXt
* Future rates rt are normally distributed
* However, this means rates can be negative
* As t increases, the volatility of future rates rt also increases
Describe the Vasicek model
drt = γ(rbar-rt)dt + σdXt with initial value r0
* This equation is linear and can be solved directly
* It is a Mean reverting process
* rt is normal
* γ regulates the speed of mean reversion; if γ = 0 then the process has no reversion and is completely unpredictable
Solution to the SDE in Vasicek
rt = r0e-γt + rbar(1-e-γt) + σ∫(0,t) e-γ(t-u)dX(u)
Ito Lemma for one factor interest rate model
Assume drt = m(rt, t)dt + s(rt, t)dXt. Then
dPt = [∂P/∂t + (∂P/∂r)m(rt, t) + 1/2 (∂2F/∂r2)s(rt, t)2]dt + (∂F/∂r)s(rt, t)dXt
Pricing PDE for Vasicek model
FIS 15
Let drt = γ(rbar-rt)dt + σdXt and the price of any interest security be Z(r,t,T) with payoff g(rT, T). The price Z must satisfy the PDE
∂Z/∂t + m*(r,t)∂Z/∂r + 1/2 (∂2Z/∂r2)σ2 = rZ
With boundary condition Z(rT,T,T) = g(rT,T)
General one factor model Pricing PDE for interest securities
Let drt = m(rt, t)dt + s(rt, t)dXt and the price of any interest security be Z(r,t,T) with payoff g(rT, T). The price Z must satisfy the PDE
∂Z/∂t + m*(r,t)∂Z/∂r + 1/2 (∂2Z/∂r2)s(rt, t)2 = rZ
With boundary condition Z(rT,T,T) = g(rT,T)
European Option prices under Vasicek
Call on ZCB = Z(r0,0,TB)N(d1)-KZ(r0,0,T0)N(d2)
Put on ZCB = KZ(r0,0,T0)N(-d2)-Z(r0,0,TB)N(-d1)
d1 = 1/SZ(T0)ln(Z(r0,0,TB)/KZ(r0,0,T0)) + SZ(T0)/2 ; d2 = d1 - SZ(T0)
SZ(T0) = B(T0,TB)sqrt[σ2/(2γ*)(1-e-2γ*T0)]
Describe the CIR model
drt = γ(rbar - rt)dt + sqrt(art)dXt
* This is another one factor equilibrium model, reverting to a constant level rbar
* This model stops negative rates by decreasing volatility for low rates
* rt follows a non-central chi-square distribution
* Yielf curves generated by this model can take the following shapes: upward sloping, downward sloping, and slightly humped (rare)
* This model is tractable, but involves integrals if non-central chi-square distributions
How do we replicate a ZCB?
FIS 16
The replicating portfolio for a bond Z1(r,0,T1) must have a long position of Δ units of bond Z2(r,0,T2) where
Δ = (∂Z1(r,t)/∂r) / (∂Z2(r,t)/∂r)
By Feynman Kac, what is the general solution to the interest rate PDE
FIS 17
Let drt = m(rt, t)dt + s(rt, t)dXt. The price of any interest security Z(r, t, T) must satisfy the PDE :
∂Z/∂t + m*(r, t)(∂Z/∂r) + 1/2 (∂2Z/∂r2)s(rt, t)2 = rZ with boundary condition Z(rT, T, T) = g(rT, T) = payoff. If V(r, t) satisfies the PDE and boundary condition, then
V(rt, t) = EQ[e-∫(t,T) rudug(rT, T)|rt]
Applying Feynman Kac: What is the no arbitrage forward price
At t = 0, the value of a forward on a bond expiring at T* (solution to the PDE) is EQ[e-∫(0,T) rudu(Z(rT,T,T*) - K)|r0]
The No arbitrage forward price is K = Z(r0,0,T*)/Z(r0,0,T)
Applying Feynman Kac: What is the no arbitrage forward swap rate
fns(0, T, T*) = [Z(r0,0,T) - Z(r0,0,T*)]/ΔΣZ(r0,0,Ti)
What is the market price of interest risk in a general short rate model?
FIS 18
Let drt = m(rt, t)dt + s(rt, t)dXt
The market price of risk = λ(rt, t) = [m(rt, t) - m*(rt, t)]/s(rt, t)
What is the market price of risk in the Vasicek model?
drt = γ(rbar-rt)dt + σdXt
Then λ(rt, t) = MPR =[γ(rbar-rt)-γ*(rbar*-rt)]/σ
What is the risk premium in the general short rate model
If drt = m(rt, t)dt + s(rt, t)dXt, λ(rt, t) = [m(rt, t) - m*(rt, t)]/s(rt, t)
The risk premium = E(dZ/Z)/dt - r = (1/Z ∂Z/∂r s(r, t)) * λ(rt, t)