Interest Rate Models
A survey of current frameworks for yield-curve modeling and the pricing of fixed-income options
Marco Avellaneda New York University
Summary of this presentation
Term-structure of Interest Rates Fixed-income markets Yield-curve stripping and smoothing Stylized facts about yield curve motions Stochastic interest rate models Ho & Lee Vasicek Hull-White CIR
1 Summary of this presentation
Pricing European-style options Forward Measure – Black 1976 methodology Parametric volatility versus Black-Scholes volatility
Forward Rate Models Heath-Jarrow-Morton Brace-Gatarek-Musiela Correlation and Covariance Implementation Issues & Methodologies
Conclusions
Term Structure of Interest Rates
2 Fixed-Income Markets
Government Debt (U.S., G7) Tax-exempt debt (Municipalities) Mortgage-Backed Securities Corporate Debt – Investment Grade
High-Yield Corporate Debt (Junk bonds) Emerging Market Debt (U.S. / local currencies)
Derivatives Listed Markets Dealer (OTC) Market
US Treasury Debt Benchmarks On the run securities: 3m, 6m, 1y, 2y, 5y, 10y, 30y
Previous Current Bills Mat Date Price/Yield P rice/Yield Yld Chg P rc Chg 3month 7/26/01 3.65 (3.74) 3.70 (3.79) 0.05 +5 6month 10/25/01 3.64 (3.76) 3.65 (3.77) 0.01 +1 1year 2/28/02 3.59 (3.73) 3.60 (3.74) 0.01 +1 Notes/ Previous Current Bonds Coupon Mat Date Price/Yield P rice/Yield Yld Chg P rc Chg 2year 4.250 3/31/03 100-07+ (4.12) 100-06+ (4.14) 0.02 --0-01 5year 5.750 11/15/05 104-11 (4.68) 104-08+ (4.70) 0.02 --0-03 10year 5.000 2/15/11 98-21 (5.18) 98-17 (5.19) 0.02 --0-04 30year 5.375 2/15/31 95-03+ (5.72) 94-30 (5.73) 0.01 --0-06
Inflation Previous Current Indexed Treasury Coupon Mat Date Price/Yield P rice/Yield Yld Chg P rc Chg 5year 3.625 7/15/02 102-02 (1.91) 102-05+ (1.82) -0.09 +0-03+ 10year 3.500 1/15/11 101-15 (3.32) 101-24 (3.29) -0.03 +0-09 30year 3.875 4/15/29 105-17 (3.56) 105-31 (3.54) -0.02 +0-14
Source: Bloomberg
3 (XURGROODU)XWXUHV'DLO\3ULFHV
Contract Las t Change Open High Low Volume Open Int. May '01 (EDK01) 95.7875 0.0525 95.78 95.795 95.765 0 0 Jun '01 (EDM01) 95.88 0.055 95.84 95.89 95.83 0 0 Jul '01 (EDN01) 95.92 0.065 95.925 95.93 95.92 0 0 Aug '01 (EDQ01) 95.93 0.08 0 95.93 95.93 0 0 Sep '01 (EDU01) 95.945 0.08 95.87 95.95 95.87 0 0 Oct '01 (EDV01) 95.795 0.115 95.71 95.795 95.71 0 0 Dec '01 (EDZ01) 95.69 0.095 95.635 95.7 95.625 0 0 Mar '02 (EDH02) 95.495 0.1 95.435 95.505 95.425 0 0 Jun '02 (EDM02) 95.185 0.09 95.14 95.2 95.125 0 0 Sep '02 (EDU02) 94.92 0.08 94.87 94.93 94.87 0 0 Dec '02 (EDZ02) 94.645 0.085 94.595 94.65 94.59 0 0 Mar '03 (EDH03) 94.52 0.085 94.47 94.535 94.465 0 0 Jun '03 (EDM03) 94.365 0.085 94.3 94.37 94.3 0 0 Sep '03 (EDU03) 94.265 0.085 94.2 94.27 94.2 0 0
Forward-rate-Agreements FRA = agreement to enter into a loan in the future
$100
t ∆t
$100*(1+∆ t F)
$100*(1+ ∆ t R)
FRA= 1 period swap t ∆t in arrears
$100*(1+ ∆ t F)
4 Interest Rate Swap
6-M LIBOR Firm 6.52 % 2
Swap Dealer 6-M LIBOR Firm 1 6.42 %
Swap Benchmark Rates
Se curity S wp Rt S wp S pd S pd Chg
US D 2 YR S WAP 4.7 57.25 -0.75
US D 5 YR S WAP 5.474 77.75 0
US D 10 YR S WAP 6.047 84.5 0.25
US D 30 YR S WAP 6.506 75.5 -0.25
Fixed Rate
6-M LIBOR
5 Swaps & Par Bonds
Swaps are ``derivatives’’ in the sense that the holder is not purchasing debt. Cash-flows depend on the difference between financing rates at inception and financing rates in the future.
Standard swaps (with LIBOR resets) can be priced as par bonds.
Floating rate bond is worth par
= 100
100 + = LIBOR
100 + F 100
= 100 + F
Fixed rate payer =Long $100, short bond with face value $100
6 Summary of US Dollar market: a dealer’s perspective
T-Bond and T-Bill futures (CBOT)
Interest rate futures (ED)
U.S. Treasury bonds
Swaps
Options: bond options, swaptions
No-arbitrage: existence of discount factors
One interest rate per maturity => Existence of a discount curve
P(T ) = present val. of cash flow of $1 in T
P(T)
T
7 Spot Curve of Discount Factors Nov 1997
1.2
1
0.8 1 $ f o 0.6
V P 0.4
0.2
0 1 9 7 5 3 1 9 7 5 3 1 9 7 5 3 1 9 7 5 3 1 0 1 3 5 7 9 2 4 6 8 9 1 3 5 7 8 0 2 4 6 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 month
Forward-rate curve and Discount curve f(T)
T d f (T ) = − log P(T ) dT
T P(T ) = exp− ∫ f (t)dt 0
8 Spot Forward Rate Curve f(0;T) Nov 1997
6.8
6.6
6.4 t n 6.2 e c r e
P 6
5.8
5.6
5.4 5 4 3 2 1 0 9 8 7 6 5 4 3 0 9 8 7 6 1 1 3 5 7 9 1 2 4 6 8 0 2 4 2 3 5 7 9 1 1 1 1 1 2 2 2 2 2 3 3 3
Month
Pricing Future Cash-Flows
f(T)
c T 1 c3 c4
Portfolio represented t by future cash-flows c2 c5
9 Pricing Cash-Flows f(T)
T
Ti − ∫ f (t )dt = 0 Fair Value of Portfolio ∑cie i
Yield curve stripping Find a piecewise-constant FR curve that prices a universe of benchmark bonds correctly
Tij Ni = − ( ) = Bi ∑cij exp ∫ f t dt i 1,..., M j=1 0
f f2 3 f τ = T 1 i iNi
τ τ τ Bond 1 2 3 maturities
10 Bootstrapping: solve for forward rates recursively
− = f1T1 j B1 ∑c1 je ≤τ T1 j 1
− − τ − ( −τ ) = f1T2 j + f1 1 f2 T2 j 1 B2 ∑c2 je ∑c2 je ≤τ τ < ≤τ T2 j 1 1 T2 j 2 ......
M −1 − ∑ f (τ −τ − )− f (T −τ − ) − f T i i i 1 M Mj M 1 = 1 Mj + i=1 BM ∑cMje ∑cMje ≤τ τ < ≤τ TMj M −1 M −1 TMj M
At each step, we have a decreasing function of the unknown rate: unique solution
Pricing Date: 17-Apr-01 Last Next Effective Payment Payment Approximate Annual Bond Coupon Date Date Date Price Yield Invoice Price Yield
3m 0 7/18/01 NA NA 4.660 4.660 NA 4.714% 6m 0 10/18/01 NA NA 4.560 4.560 NA 4.612% 2y 5.00 1/12/03 1/12/01 7/12/01 100.790 4.504 102.102 4.555% 3y 4.50 7/12/03 1/12/01 7/12/01 99.980 4.501 101.161 4.552% 4y 3.50 7/12/04 1/12/01 7/12/01 96.880 4.555 9 7.799 4.607% 5y 5.00 1/12/06 1/12/01 7/12/01 101.390 4.701 102.702 4.756%
5.100%
5.000% 4.900% 4.800% Example: 4.700% 4.600% Stripping a 4.500% dataset with 6 4.400% bonds 4.300%
4.200% 4/17/01 - 7/18/01 - 10/18/01 - 1/12/03 - 7/12/03 - 7/12/04 - 7/18/01 10/18/01 1/12/03 7/12/03 7/12/04 1/12/06
11 Yield curve smoothing
Find f(t) which is smooth (no jumps) and consistent with the market.
Continuous model Minimize over f:
2 Tij − ∫ f (s)ds 2 Tmax − 0 + ε ( ( )) ∑ Bi ∑cije ∫ f ' t dt 0 i j
Sum of squared errors ``Penalty’’ term
Smoothing Algorithm: curve is piecewise constant with small intervals between jumps
( ) = − ∆ < ≤ ∆ = f t fk (k 1) t t k t k 1,2,..., N max
(f ,..., f ) Minimize over 1 Nmax
2 Nij − ∆ N 2 ∑ fk t max − k =1 + ε ( − ) ∆ ∑ Bi ∑cije ∑ fk fk−1 t = i j k 1
12 Minimization can be done with standard tools: e.g. Solver
Delta T = 3 months; Epsilon=0.333
5.00% 4.90% 4.80% 4.70% 4.60% 4.50% 4.40% 4.30% 4.20%
1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 l- t- - r- l- t- - r- l- t- - r- l- t- - r- l- t- - u c n p u c n p u c n p u c n p u c n J O Ja A J O Ja A J O Ja A J O Ja A J O Ja 3 month period ending
Error is less than 0.001 on all bonds & zeros
Regularization and splines
Splines=piecewise-polynomial curves
Penalty/ regularization methods correspond to polynomial splines
The optimal ``splining knots’’ are located at the coupon dates of the input bond/swap data.
Splining methods with too few knots can give rise to unstable curves
13 Stochastic Interest Rate Models & Option Pricing
Evolution of Benchmark Swap Rates
9.00 8.50 8.00 7.50 1 YR 7.00 5YR 6.50 10YR 6.00 30YR 5.50 5.00 4.50 4.00 5 5 6 6 7 7 8 8 9 9 0 0 1 9 9 9 9 9 9 9 9 9 9 0 0 0 / / / / / / / / / / / / / 3 3 3 3 3 3 3 3 3 3 3 3 3 / / / / / / / / / / / / / 1 7 1 7 1 7 1 7 1 7 1 7 1
14 US Treasury Yield Curve: Mar 1986 -June 1988
9-11 7-9 5-7
1 3-5 1
9 ) % (
7
e t a r 6 -8 7 r 5 a -8 n 8 -M u -8 3 J p 9 - e -8 2 S c 1 - e -9 2 D r 2 - a -9 1 M n 3 - u -9 3 1 J 5 - p -9 2 e 6 -S n a 9 y 2 J r- 7 - a 9 3 2 - m M 8 - n 9 3 1 u - -J n 2 u -J 3
Caps and Floors
Series of options on 3-month LIBOR rates
Series of options on 3-month discount bonds
Swaptions
Option on a swap
Option on a series of 3-month discount bonds
15 Caps
X 1 X2 X3 X4
where xi = max ( Ri –F, 0)
Receiver 5x10 Swaption
exercise t =15 6-M LIBOR
0 t =5
no exercise 0 t =15
Swaption expiration date
16 Markovian one-factor models
Evolution of sort-term interest rates is modeled using a diffusion process
dX (t) = σ (X (t),t)dZ(t)+ µ(X (t),t)dt
r(t) = R(X (t)) = short - term interest rate
Markovian one-factor models
Evolution of all forward rates is obtained from the fact that discount factors are functions of the state variable (hence of r(t))
P(t;T ) = present value of a future cash - flow of $1 paid at time T, evaluated at time t
T −∫ r(s )ds P(t;T ) = Ee t X (t) = Z(t, X (t);T )
17 Ho-Lee, 1986 Ho-Lee Model J. Finance
Short rate follows Brownian motion (continuous random walk) with drift
Drift is adjusted to fit the current term structure of interest rates
dr(t) = σdW (t)+ µ(t)dt
t r(t) = r(0)+σW (t)+ ∫ µ(s)ds 0 = σW (t)+ b(t)
Ho-Lee Model
T T −∫ f (s)ds −∫ r(s)ds P(0;T ) = e 0 = Ee 0 Known from yield curve data
18 Ho-Lee Model
T T −∫ f (s)ds −∫ r(s)ds P(0;T ) = e 0 = Ee 0 Unknown T T −∫σW (s)ds −∫ b(s)ds = Ee 0 e 0
T σ 2T 3 −∫ b(s )ds =e 6 e 0 Solving for b(s)
T σ 2T 3 T σ 2t 2 ∫ f (s)ds = + ∫b(s)ds b(t) = f (t)+ 0 6 0 2
Ho-Lee Model
Drift of short rate process is a function of the derivative of forward rate curve
σ 2t 2 r(t) = σW (t)+ f (t)+ 2
dr(t) = σdW (t)+ µ(t)dt, µ(t) = f '(t)+σ 2t
19 Discount factors & Forward Rates
Zero-coupon bond T − f (s)ds σ 2 price is obtained in ∫ −σ (T −t )W (t )+ (T −t )Tt ( ) = t × 2 explicit form P t;T e e
σ 2t 2 Simple forward rate f (t;T ) = f (T )+σW (t)− +σ 2tT dynamics 2
Simple structure for ( ) dP t;T = −σ ( − ) ( )+ the volatility of discount ( ) T t dW t r(t)dt factors P t;T
Ho-Lee Model: standard positions of f(1,T), W(1)=+1 the forward rate curve f(1,T) W(1)= -1 f(0,T) 7.5
7 t
n 6.5 e c r e p 6
5.5
5 1 23 45 67 89 111 133 155 177 199 221 243 265 287 309 331 353 month (T*12)
20 O. Vasicek, Vasicek’s Model 1977, J.F.E.
``Equilibrium model’’ as opposed to an ``arbitrage model’’: seeks to explain the shape of the yield curve
dr(t) = σdW (t)+ κ (r∞ − r(t))dt
t ( ) = −κt ( )+ ( − −κt ) + σ −κ (t−s) ( ) r t e r 0 1 e r∞ ∫ e dW s 0
σ = volatility, κ = speed of mean - reversion,
r∞ = long - term average rate
Mean-reversion
Short rate has an asymptotic distribution with finite variance
r∞
µ = r r(0) ∞ ∞ σ 2 2 σ ∞ = 2κ
t
21 Parameter Estimation
Chen & Scott: Estimation of parameters with Treasury data 1988-1993
κ = 0.2456
r∞ = 6.48% σ = 2.9%
Discount Factors & Forward Rates
Explicit formulas
Simplistic forward rate structure
− ( ) ( )− ( ) P(t;T ) = e A t;T r t B t;T
−κ ( − ) 1− e T t A(t;T ) = ; κ
σ 2 σ 2 ( )2 ( ) = − ( − − ( ))+ A t;T B t;T r∞ T t A t;T 2κ 2 4κ
22 Volatility Structure in the Vasicek model
( ) df (t;T ) = σe−κ T −t) dW (t)+ drift term
Forward rate volatility is ``damped’’ as maturity increases
( ) dP(t;T ) 1− e−κ T −t = σ dW (t)+ r(t)dt P(t;T ) κ
Bond volatility is bounded as T increases
Vasicek Model: initial forward rate curve
σ 2 −κt −κt −κt 2 f (t) = r(0)e + r∞ (1− e )− (1− e ) 2κ 2
Exponential spline Convexity correction connecting current spot rate to asymptotic rate (expected spot rate
23 Vasicek forward rate curves r_0=4.5% r_5yr=4.79%, sig=3% 4.85
4.8
4.75
4.7
4.65
4.6 kappa=0.25 4.55 kappa=0.01
4.5
4.45
0 .3 .6 .9 .2 .5 .8 .1 .4 .7 3 .3 .6 .9 .2 .5 .8 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4
Hull & White 1992 Hull-White / Modified Vasicek Model
Retains the desirable mean-reverting property of the Vasicek model
Matches the current term structure of interest rates
dX (t) = −κX (t)dt +σdW (t), X (0) = 0
Ornstein-Uhlenbeck ( ) = ( )+ ( ) process with drift r t X t b t correction
dr(t) = σdW (t)+ κ (r∞ (t)− r(t))dt
24 Spot & forward rate dynamics: Modified Vasicek
t −κ ( − ) X (t) = σ ∫ e t s dW (s) 0 2 σ 2 r(t) = X (t)+ f (0;t)+ (1− e−κt ) 2κ 2
−κ ( − ) f (t;T ) = e T t X (t)+ f (0;t)+ C(t;T ) 2 σ −κ 2 −κ ( − ) 2 C(t;T ) = [(1− e T ) − (1− e T t ) ] 2κ 2
Application of Short-rate Models ``Convexity adjustment’’ in futures vs. forward rates
Rates implied by ED futures contracts are higher than forward rates Φ(T,∆T ) = E[R(T ,∆T )] Futures-implied rate (settles on term rate)
1 P(T ) Forward rate F(T,∆T ) = −1 ∆T P(T + ∆T )
T − ( ) 1 ∫ r s ds = E e 0 R(T,∆T ) P(T ) t
25 Approximate term rates with instantaneous rates and use explicit formula
R(T,∆T ) ≅ r(T ) Φ(T,∆T ) ≅ E(r(T )) F(T,∆T ) ≅ f (0;T ) σ 2 −κt 2 From r(t) = X (t)+ f (0;t)+ (1− e ) Vasicek model 2κ 2
2 σ −κ 2 Φ(T,∆T ) = E X (T )+ f (0;T )+ (1− e T ) 2κ 2 2 σ 2 ≅ F(T,∆T )+ (1− e−κT ) 2κ 2
November 1997 data CONVEXITY ADJUSTMENT: EURODOLLAR FUTURES
Maturity (months)Futures price Convexity Adjustment (%) 4 94.19 0.0012 10 94.14 0.0023 13 94.08 0.003 16 93.98 0.0044 19 93.98 0.0092 22 93.94 0.0131 25 93.91 0.0176 28 93.85 0.0234 31 93.87 0.0232 34 93.85 0.0371 37 93.83 0.0447 40 93.74 0.0522 43 93.79 0.0637 46 93.77 0.073 49 93.75 0.083
26 Future-Implied minus Forward ED 3-month rates
9 8 7
s 6 t Theoretical n i
o 5 p 2 2 s
i 4 σ
s t a Φ − F ≅ B 3 2 2 1 0 4 10 13 16 19 22 25 28 31 34 37 40 43 46 49 Forward Maturity (in months)
Sqrt(Futures-Forwards)/Time Implied sigma
0.1 0.09 y = 0.0001x + 0.0538 0.08 R2 = 0.0039 2 2 − σ 49 0.07 8.3×10 4 = 0.06 2 12 0.05 0.04 2 −4 0.03 2 × (12) ×8.3×10 σ = ≅ 1% 0.02 (49)2 0.01 0 4 10 13 16 19 22 25 28 31 34 37 40 43 46 49 Forward Maturity (in months)
f(1;T); X(1)=+sig; Hull-White 1-factor model: 1-std displacements of f(0;T) the forward rate curve in one year f(1;T); X(1)= -sig
6.8 6.6 6.4 6.2 )
% 6 (
e t 5.8 a r 5.6 5.4 5.2 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 months (T*12)
27 Cox, Ingersoll & Ross, 1985
CIR/Modified CIR
Short rate follows square-root diffusion
dr(t) = σ r(t)dW (t)+ κ (r∞ − r(t))dt
Square root diffusion prevents the rate from becoming negative
Distribution of rate is non-central chi squared
Non-central Chi-squared vs. Gaussian
Non-central chi-squared is positively skewed; it can be thought as the distribution of a sum of n squares of independent normals
n = number of degrees of freedom κ = 4 r∞ σ 2
× × ≅ 4 0.24 0.0648 = Essentially From Chen-Scott: n 2 79.34 (0.028) Gaussian!
28 The asymptotic density of rates for different parameter values
1 2 Y0 κr∞ ≥ σ 2
1 2 0 <κr∞ < σ Y0 2
τ 0
Y0 κr∞ ≤ 0
Density for r(t) : square-root model . r0=5.5%, sigma=3%, kappa=0.25, r_inf=6.8%
90 80 70 ) t=0.5 Y
, 60 t t=2 , 50 5
5 40 t=5 0 .
0 30 t=10 ( p 20 t=20 10 0 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 1 1 1 1 2 2 3 3 4 4 5 5 6 . . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 ...... 0 0 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Y(t)
29 Densities for r(t): r0=9.5%, sigma=3%, kappa=0.25,r_inf=5.5%
70 60 t=0.5 ) 50 Y
, t=2 t , 40 5 t=5 9
0 30 . t=10 0 ( 20 p t=20 10 0 6 5 4 3 2 1 9 1 8 7 6 5 4 3 2 1 . 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 ...... 0 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Y
CIR: forward rates & Discount factors
P(t;T ) = e− A(t;T )r(t)−B(t;T )
( ) 2(1− e−α T −t ) A(t;T ) = ( ) α + κ + (α −κ )e−α T −t α = κ 2 + 2σ 2
Exponential-affine structure for zero-coupon bonds
30 Common Features of One-factor Models
Short rates follow 1-D diffusions: easy to implement by Monte Carlo or finite-differences (lattices)
Forward rates shocks are perfectly correlated with short rate shocks (albeit with different amplitudes according to mean-reversion/damping of vol)
Can be calibrated to the current term structure of interest rates easily
Main interest: heuristic value, pricing American option & other PDE problems
Main differences
Ho-Lee and its discrete analogue, B-D-T, fit the term structure of interest rates but do not have mean-reversion of short rates.
Vasicek and CIR are ``equilibrium models’’ that do not fit the current term structure of interest rates. However, they can be modified by adding a time-dependent drift to fit the term structure (Hull-White)
Modified Vasicek has Gaussian short rates and lognormal zero-coupon bonds. Very tractable.
CIR ``enforces’’ positive rates but is analytically cumbersome. High number of degrees of freedom in practice.
31 Pricing Future Cashflows
T −∫ r(s)ds V (r,t) = Ee t F(r(T ))r(t) = r
∂V σ 2 (r,t) ∂2V ∂ V + + µ(r,t) − rV = 0 ∂t 2 ∂r2 ∂r
V (r,T ) = F(r)
Fokker-Planck equation
Markov 1 factor models: numerical algorithm
Stochastic differential equation dr(t) = σ (r(t))dW (t)+ µ(r(t),t)dt
= +σ ( ) ∆ + µ( )∆ Discrete approximation rn+1 rn rn Nn+1 t rn ,t t
Implementation: Monte Carlo of finite-differences (PDE)
32 Finite-difference (PDE) implementation of 1-factor models
3-pt stencil probabilities or weights
+σ ∆ 1 σ 2 (r) µ(r,t) r 0 t p = − ∆t u σ 2 σ 2 0 0 σ 2 (r) p = 1− r r m σ 2 0 1 σ 2 (r) µ(r,t) p = + ∆t d σ 2 σ −σ ∆ 2 0 0 r 0 t
Explicit/ Implicit Schemes, CFL
Solving the backward Fokker- Planck equation
Payoff At Maturity Value Today Finite-difference Scheme C C observed PU
S0 PM
PL C(S , 0) 0 t S t K S S0 K
S0 = today’s price
K = strike price Ct + LC = 0
33 Solving the forward Fokker- Planck equation Probability Density Dirac Mass Finite-difference Scheme Function p p PU
S0 PM
PL
t x Y t x Y
* Pt - L P = 0
Monte Carlo Simulation
= +σ ( ) ∆ × + µ( )∆ rn+1 rn rn ,tn t snorm( ) rn ,tn t
snorm( ) = pseudo - random number generator for standard normals
34 Closed-form solutions for bond options and volatility modeling
Call option on a zero-coupon bond
C = value of the call option K = exercise price T = time to expiration T + ∆T = maturity date of the bond (paying $1)
Pricing formula: using the ``risk-neutral’’ measure
T −∫ r(s)ds C = Ee 0 max(P(T;T + ∆T )− K,0)
35 Pricing formula: using the ``risk-neutral’’ measure
T −∫ r(s)ds C = Ee 0 max(P(T;T + ∆T )− K,0)
Discounting Payoff term (interest rate sensitive) (interest rate sensitive)
Dynamics of Forward Prices
= Vt value of security that pays no divs. until time T = FVt forward value of this security for delivery at time T V FV = t t P(t;T )
dFVt = σ * F dWt under a new probability P FVt
2 2 2 σ F = σ − 2σ σ ρ +σ V V P P (Think about the vol. of the ratio of V and P)
36 Calculation of Forward Volatilities
Since bond movements are perfectly correlated in 1F models
σ = σ σ = σ V P(.;T +∆T ) P P(.;T ) ∴ σ = σ −σ F P(.;T +∆T ) P(.;T )
Ho-Lee
σ = σ ( + ∆ − )−σ ( − ) = σ∆ F T T t T t T
Hull-White-Vasicek
− −κ (T +∆T −t ) − −κ (T −t ) σ ( ) = σ 1 e −σ 1 e F t κ κ − −κ∆T −κ (T −t ) 1 e = σe κ
Term volatility
T − −κ∆T − −2κT 1 2 1 e 1 e σ = σ T(t)dt=σ T ∫ κ κ T 0 2 T
37 = + ∆ FPt T - forward value of the bond with maturity T T = P(t;T + ∆T )/ P(t;T ) = FCt T - forward value of the option with maturity T = C(t)/ P(t;T )
C(0) C(T ) = E* = E* (max(P(T;T + ∆T )− K,0)) P(0;T ) P(T;T )
P(0;T + ∆T ) 1 1 P(T;T + ∆T ) = expσ N T − σ 2T = FP(0)expσ N T − σ 2T P(0;T ) T 2 T T 2 T
( ) = ( )[ × ( )− × ( )]= ( σ ) C 0 P 0;T FP N d1 K N d2 BSCall P, K,T,r,0, T
Volatility Calibration
Forward measure formalism gives rise to Black 76 type pricing for bond options and caplets
(σ κ ) Simple formulas relate the model parameters , to marker prices by inverting the Black Scholes formula
Gaussian models treat bond prices as lognormal. This is somewhat different from the market quoting convention, (lognormal rates) but price-to-price calibration is easy.
(σ ( ) κ ( )) Time-dependent t , t are often used to calibrate to several option prices simultaneously.
38 Forward rate models & multifactor analysis
Why models based on movements of forward rates?
Early 1990’s brings period of low interest rates and emergence of structured products & exotics
Options on the ``slope’’ of the yield curve
Mortgage-backed securities
Long-dated American-style interest rate options (Bermudans)
39 US Treasury benchmark yields
10 9 3m 8 6m 1y ) 7 %
( 2y
d 6 l 3y e i y 5 5y 10y 4 30y 3 2 6 7 8 9 0 1 2 3 4 5 6 7 8 8 8 8 8 9 9 9 9 9 9 9 9 9 / / / / / / / / / / / / / 2 2 2 2 2 2 2 2 2 2 2 2 2 / / / / / / / / / / / / / 1 1 1 1 1 1 1 1 1 1 1 1 1
Three US Treasury Yield Curves in the 1990's
9 8 7
d 6 l e i
y 5
4 31-Dec-90 3 30-Jun-93 27-Jul-98 2 3m 6m 1y 2y 3y 5y 10y 30y maturity
40 Models need to reflect the correlations of forward rates
ν ( ) = α( ) + β ( ) ( )+ df t;T t;T dt ∑ t;T dWk t .... k=1
Changes in the shape of the forward rate curve are driven by several independent factors (having different amplitudes)
Main deformation modes can be estimated from historical data (Principal Component Analysis)
US Treasury yield curve: Normalized Principal Components for quarterly shocks
0.8 0.6
t 0.4 n e 0.2 88.70% m e c 0 8.90% a l p -0.2 1.41% s i d -0.4 -0.6 -0.8 3m 6m 1y 2y 3y 5y 10y 30y maturity
41 Heath-Jarrow-Morton approach
Puts emphasis on the volatility and correlation of forward rates (analogy with Black-Scholes, Black ’76, etc)
For any given volatility/correlation structure, HJM gives the correct drift on the forward rate curve needed to have a risk-neutral measure
A systematic approach to fit the initial term-structure of interest rates for any vol/corr structure
HJM Equation
Assume that the forward rate curve has the following dynamics:
ν ( ) = β ( ) ( )+α( ) df t;T ∑ k t;T dWk t t;T dt k=1
This model is a risk-neutral pricing measure if and only if
ν T α( ) = β ( ) β ( ) t;T ∑ k t;T ∫ k t;s ds k =1 t
42 ``Lognormal’’ HJM
df (t;T ) ν = σ (t;T ) ∑ψ (T − t)dW (t) + µ(t;T )dt ( ) k k f t;T k=1
ν ψ 2 ( ) = ∑ k x 1 k=1
Three correlation factors
1.2 Parallel shift 1
0.8 Psi1(x) 0.6 Psi2(x) Psi3(x) Bend 0.4
Tilt 0.2
0
.1 .4 .7 1 .3 .6 .9 .2 .5 .8 .1 .4 .7 4 .3 .6 .9 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4
43 SWAPTION MODEL
maturity tail strike 2-factor Ho-Lee 1 0.25 4.824 7.6 7.9 2 0.25 5.096 11.8 12.1 5 0.25 5.669 16.9 17.2 1 5 5.273 206.3 209.8 2 5 5.357 256 258.9 5 5 5.684 290.6 294.8 1 10 5.481 506.3 514.2 2 10 5.676 516.6 524.4 5 10 5.998 588.6 599.1
Brace-Gatarek-Musiela model
F F 1 2 Fn
T T T T 1 2 T3 4 N
N fixed consecutive dates (say, 3 months apart)
Consider the evolution of forward rates for loans over periods ( ) = − Tn ,Tn+1 , n 1,..., N 1
1 P(t,T ) F (t) = n −1 n ∆ ( ) T P t,Tn+1
44 BGM Equations
Dynamics for dF (t) n = σ (t)dW (t)+ µ (t)dt ( ) n n n forward rates Fn t
n ∆tF (t) µ (t) = σ (t)∑ m σ (t) ``Risk-neutral’’ drift n n + ∆ ( ) m m=2 1 tFm t
( ) ( ) W1 t ,....,WN t are correlated Brownian motions
BGM / HJM Implementation
Method of choice: Monte Carlo simulation
Advantage: volatilities of forward rates are easy to calibrate from Cap/Floor/Swaption quotes
Disadvantage: correlations between rates are difficult to measure precisely. Dimensional reduction needed.
Disadvantage: Evaluating American options requires sophisticated methods and approximation of the exercise region
45 BGM and HJM are not very different in practice… Discrete lognormal HJM
n ( + ∆ ) = ( )+ ( )σ ( ) ( ) ∆ + ( )σ ( ) ( )σ ( )∆ ∆ fn t t fn t fn t n t N t t fn t n t ∑ fm t m t t t 2
Discrete BGM
n f (t)σ (t)∆t f (t + ∆t) = f (t)+ f (t)σ (t)N (t) ∆t + f (t)σ (t)∑ m m ∆t n n n n n n + ( )σ ( )∆ 2 1 f m t m t t
Difference in drift is higher-order correction term!
``Cherchez la correlation’’
Difficulties in MF models are not in choice of model per se: they reside in the computational complexity and the difficulty of calibration of volatility/correlation
Recent developments such as Least Squares Monte Carlo, allow for pricing American-Style options in MF models (Longstaff & Schwartz, 1998)
Most serious impediment for using MF models: choice of correlation matrix between rates
46