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)
What is the risk premium in the Vasicek model
If drt = γ(rbar-rt)dt + σdXt,
λ(rt, t) =[γ(rbar-rt)-γ*(rbar*-rt)]/σ
The risk premium = E(dZ/Z)/dt - r = -B(t,T)σλ(rt, t)
How are current spot rates and instant forward rates related under Ho Lee
FIS 19
Let f(0,t) = instantaneous forward rate, r(0,t) = ctsly compounded spot rate
We have the relationship f(0,t) = r(0,t) + t ∂r(0,t)/∂t
What are some drawbacks of the Ho Lee model
- Since the rate process follows a random walk, rates can go to +/- infinity as T increases
- The model implies a term structure of volatility that is a straight line. However we expect short rates to be more volatile than long term rates
Describe the RN dynamics of the Hull White model
drt = (θt - γ*rt)dt + σdXt with inital value r0
* The model looks like Vasicek with a time varying parameter θt
* This parameter allows the model to reproduce initial term structure like Ho Lee
* This model is tractable
* The short rate is normally distributed
* Efficient numerical procedures can be derived and built from HW
For the normal short rate models, what is the value of Euro Calls/Puts
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)
Vasicek, HoLee, HW only differ in their specification of SZ
How is SZ defined for the 3 normal short rate models?
Vasicek: SZ(T0, Ti) = B(T0, Ti) sqrt[σ2/2γ* (1-e-2γ*T0]
Ho Lee: SZ(T0, Ti) = sqrt[σ2 T0 (TB - T0)2]
Hull White: SZ(T0, Ti) = B(T0, Ti) sqrt[σ2/2γ* (1-e-2γ*T0]
Compare how well Ho Lee and Hull White do in pricing caps
- The calibrated HW model does better than Ho Lee
- Short horizon cap prices are generally hard to replicate, even with Hull White
- The pricing error in HW is smaller than under Ho Lee, especially for short rate caps
- Calibration can sometimes give a negative value of the speed of mean reversion of HW γ* implies that the RN model generates exploding rates
- A smaller value of γ* would make HW similar to Ho Lee
Describe the Black Karasinski model
dyt = (θt - γtyt)dt + σtdXt
* The parameters γ and σ are chosen to fit market option prices
* θ ensures the model can replicate initial term structure of rates
* The model does not yield a closed form solution to standard options
* Because no analytical solution is available, calibrating model paramters to market data is more complex than for normal models
* We need to implement numerical procedures
* A downside is that certain cases allow the chance to make infinite money in a small amount of time
What are 2 lognormal rate models and what are 2 key characteristics
Two models: BDT (Black Derman Toy); Black and Karasinski
* It is not possible to obtain a closed form solution to ZCB and standard interest rate derivatives like caps, bond options, swaptions
* These models guarantee rates can never be negative unlike the normal models
What is the Black formula for the price of a caplet
FIS 20
Caplet(0, Ti+1) = NΔZ(0, Ti+1) [fn(0,Ti, Ti+1)N(d1) - rKN(d2)]
d1 = 1/σfsqrt(Ti) ln(fn(0,Ti, Ti+1)/rK) + 1/2 σf sqrt(Ti)
What is the payoff of a receiver swaption
Suppose we have a swaption maturing at T0 on a swap maturing at TS with payment period Δ
The payoff of the receiver swaption is NΔΣZ(T0, Ti) Max(rK - c(T0, TS),0)
What is the Black formula for price of a European receiver swaption
V(0, T0, TS) = NΔΣZ(T0, Ti) (rKN(-d2) - fns(0, T0, TS)N(-d1))
d1 = 1/σfssqrt(T0) ln(fns(0,T0, TS)/rK) + 1/2 σfs sqrt(T0)
What is the difficulty in valuing interest rate sensitive claims in an RN world?
FIS 21
The discounting term and the payoff term will be correlated to some defreee, and it is difficult to evaluate terms like
Cov(e-∫(t,T) rudu; g(rT, T))
What is the expected payoff under the forward measure?
Let gT be lognormal under the T-forward measure
ln(gT) ~ Normal(μT, σT2)
Then Ef*[max(gT - K; 0)] = F(0, T)N(d1) - KN(d2)
d1 = 1/σT ln(F(0, T)/K) + 1/2 σT
What is the value of Euro vanilla options unter the forward measure
The value of European vanilla options maturing at T written on an instrument with maturity value T with strike K is
Call: Z(0, T) Ef*[max(gT - k; 0)) = Z(0, T)[F(0, T)N(d1) - KN(d2)]
Put: Z(0, T)[KN(-d2) - F(0, T)N(-d1)]
What is the distribution of the spot rate under the T-forward measure
The LIBOR rate rn(τ, T) is lognormal
ln(rn(τ, T)) ~ Normal[ln(fn(0, τ, T) - σf2τ/2 ; σf2τ]
Describe the lognormal distribution of rn(τ, T) under the T-forward measure
rn(τ, T) ~ LogN(fn(0, τ, T); ∫(0, τ)σf(t) σf(t)dt
What is the RN drift under HJM and what requirements does this need
Let the RN SDE for instant forward rates be
d(f(t, T)) = m(t, T)dt + σf(t, T)dXt
The risk neutral drift m(t, T) = σf ∫(t,T) σf(t, τ)dτ
This requires that as T goes to infinity, the volatility converges to 0
How are forward and futures rates related?
f(0, τ, T) = ffut(0, τ, T) - ∫(0,τ) (σZ(t, T+τ)2 - σZ(t,T)2)/2τ dt
What are some weaknesses of one factor models
FIS 22
- At every time instant, rates for all maturities in the yield curve are perfectly correlated since they all derive from the short rate dynamics
- This contradicts our findings in the real world (FIS4) where 3 factors explain yield curve movements better: level, slope, and curvature
- Points on the YC are known to exhibit some de-correlation so a more satisfactory model of YC evolution needs to be defined
- Such a model would require more than one factor
Compare option prices under 1-factor and 2-factor Vasicek
Both have the standard BS structure and d1 is defined the same way in terms of SZ. But the models differ in SZ
1 Factor: SZ(T0) = B(T0, TB) sqrt[σ2/2γ* (1-e-2γ*T0]
2 Factor: SZ(T0)2 = (B1(T0, TB) sqrt[σ12/2γ1* (1-e-2γ1*T0])2 + (B2(T0, TB) sqrt[σ22/2γ2* (1-e-2γ2*T0])2
What is the formula for an option price on a ZCB in 2 factor Vasicek
What is the volatility of long term rate under 2 factor Vasicek
The long term rate rt(τ) = -A(τ)/τ + B1(τ)/τ rt + (B2(τ) - B1(τ))/τ Φ2,t
We use Ito’s Lemma to derive the long term rate volatility as
σt(τ) = sqrt[σ12(B1(τ)/τ)2 + σ22(B2(τ)/τ)2]
What are three measures of swaption implied volatility
QFIQ 116 17
- Black Volatility is the implied volatility from conventional Black model
- Absolute/normal implied volatility which is Black volatility times the prevailing swap rate
- A displaced lognormal version of Black’s formula
Describe Black Volatility pre- and post-2008
Pre: Black implied volatilities typically ranged between 10-30% with a clear inverse relation with rates. They performed reasonably well
Post: Since 2009, BV has failed to perform well, so swaption market participants have chosen to use absolute/normal volatilities. Why?
* Absolute/normal volatilities are more stable than Black
* Absolute volatilities are less sensitive to rate levels than Black
* The new view is that interest rate distributions are closer to normal than Lognormal (as implied by Black)
What are 3 properties of the displaced lognormal Black model
A simple variant that can be adopted with little extra effort
* There is an extra degree of freedom in specifying the displacement paramet which can be used to vary the underlying rate distribution
* The model is closely related to other displaced stochastic rate models that are commonly used for simulations
* The model is convenient just as the Black model is
What are 4 criteria in selecting a good interest rate model
QFIQ 129 21
- Ability to reproduce intitial ZC curve
- Ability to reproduce volatility surfaces and smiles
- Ability to accurately price “out of calibration sample” implied volatilities
- Good balance between adequate number of parameters and too many (flexibility vs fit)
Describe the paradigm of lognormal swap rates
QFIQ 129 21
- We assume swap rates follow a lognormal distribution
- This raises an issue when swap rates are very low because lognormal volatility scales inversely with rates
- The lognormal implied volatility can explode for very low rates
- Lognormal volatility can even be undefined for negative rates
Describe the paradigm of normal swap rates
- Assume swap rates follow a normal distribution
- This paradigm is more stable
- It enables full implied volatility matrices to be generated at any time, whatever the level of the swap rate
- Normal implied volatilities are more independent of the swap rate level and over time than lognormal volatility
What are the two approaches for imposing a floor level, and give examples of rate models for each approach
A Posteriori: The Gaussian models HW and G2++
A Priori: CCIR++, CIR2+, DD-LMM, and LMM+ models
Define the a posteriori approach for floors and any issues
- We impose a lower absolute level for interest rates
- When rates fall below this level in a simulation, the rate is set to the floow level
- The issue is that it truncates part of the simulation set
- Therefore, it affects the properties of the economic scenario as far as martingality and market consistency are concerned
What are 4 reasons for imposing a negative interest rate?
- Financial Repression
- Weakening the currency
- Fighting against deflation
- Increase incentive to borrow and invest
Can normal models permit negative rates and reproduce the initial ZC curve?
Both the standard 1 factor Hull White model and the Guassian 2 factors (G2++) models can achieve this
What LIBOR models can fit ZC curves with negative rates?
Some models adapted from the forward diffusion LMM are robust, allowing negative rates while fitting the ZC curve as well as volatility surfaces/smiles
The displaced diffusion LMM (DD-LMM) and the LMM+ which adapts the DD-LMM to include stochastic volatility
What are the CIR model extensions to fit negative rates?
Some models adapted from the CIR model are based on a shifted chi-square diffusion for the instant rate and can reproduce volatility smiles
The shifted CIR (CIR++) and the 2-factor CIR++ (CIR2++)
What are the dynamics of the G2++ model?
Under G2++, the short rate has dynamics
r(t) = x(t) + y(t) + φ(t)
dx(t) = -ax(t)dt + σdW1(t) ; x(0) = 0
dy(t) = -by(t)dt + ηdW2(t) ; y(0) = 0
The two Brownian Motions have correlation ρ
Describe some features of G2++ model
- G2++ is useful in practice even though it theoretically permits negative rates (as it is normal)
- The model is anlytically tractable, we can price plain vanilla instruments and derive spot/forward rates/curves at any point in time
- De Factor: an ‘almost’ negative perfect correlation of factors x(t) and y(t) is introduced, resulting in a more precise calibration to correlation-based products like caps and swaptions
- This is definitely useful for pricing OOM exotic instruments after calibrating to ATM vanilla ones
- G2++ is related to the Hull White two factors model but with a less complicated than the HW and is easier to implement in practice
What are 3 key attributes of a new reference rate
QFIQ 131 21
- Shorter tenor, essentially moving to overnight markets where volumes are larger than e.g. 3 month tenor
- Moving beyond interbank markets to add bank borrowing from a range of non bank wholesale counterparts like cash pools, money market funds, other investment funds, insurance companies, etc
- In some jurisdictions, drawing on secured instead of unsecured transactions
What 3 objectives would a new reference rate seek to achieve
- Provide a robust and accurate representation of interest rates in core money markets that is not susceptible to manipulation. Benchmarks from active liquie markets are the best candidate for this goal
- Offer a reference rate for financial contracts that extend beyond the money market. Such rates shall be used for discounting and pricing cash instruments and interest rate derivatives.
- Serve as a benchmark for term lending and funding
What are 2 approaches for constructing a term benchmark rate
- Is the term rate known at the beginning of the horizon period? Does it reflect expecations about the future or merely past realization of O/N rates (is it backward or forward looking)
- Is the term rate based on pricing of instruments ised to raise term funding or on derivatives used to hedge fluctuations in O/N rates?
Compare forward vs backward looking term rates
Backward term rates are easier to construct, we can build them even without underlying transactions. They do not reflect expectations of future rates. A 3-month backwards SOFR can be used as a reference rate by FRN issuers for floating leg payments. Such a rate is a geometric average daily rate and is less prone to qtr-/yr-end volatility, but they lag movements of O/N rate.
Forward term rates are known at the beginning of the period they apply and are not based on compounding of O/N rates. They are outcomes of a market based price formation process and embed expectations about future rates and conditions.
