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Term Structure ModelsWorkshop at AFIR-ERM Colloquium, Panama, 2017

Michael SherrisCEPAR and School of Risk and Actuarial Studies

UNSW Business SchoolUNSW Sydney

[email protected]

UNSW

August 2017

Michael Sherris (UNSW) AFIR − ERM Workshop August 2017 1 / 137

Workshop Aims

Overview of commonly used term structure models, both continuoustime and discrete time and methods use to fit models.

Provide participants with an understanding of models and financialand actuarial applications rather than full mathematical details.

Key topics covered (some more detailed than others) will be:

Yield curves, Market instruments, Static Term StructureConcepts of no-arbitrage and risk neutral versus real world probabilities,Main continuous time single factor short rate models,Affine models, Dynamic Nelson-Siegel model, Arbitrage-free NelsonSiegel model,HJM (Heath, Jarrow, Morton) framework,Market models - LMM (Libor Market Models), BGM (Brace, Gatarek,Musiela), SABR (stochastic alpha, beta, rho),Discrete time models and binomial implementation including forwardinduction.


References:

Cairns, A. J. (2004), Interest Rate Models: An Introduction,Princeton University Press.

Diebold, F. X. and Rudebusch, G. D., (2013), Yield Curve Modelingand Forecasting: the Dynamic Nelson-Seigel Approach, PrincetonUniversity Press.

Filipovi´c, Damir (2009), Term-Structure Models A Graduate Course,Springer.

van Deventer, D. R. and Imai, K. (1997), Financial Risk Analytics: ATerm Structure Approach for Banking, Insurance and InvestmentManagement, Irwin Professional Publishing.

Svoboda, S. (2004), Interest Rate Modelling, Palgrave Macmillan.


My Background

Late 70’s and early 80’s: Industry experience in banking, life insuranceand pensions - bond markets (RBA open market operations),corporate finance (tax based financing), swaps, bill futures arbitrage.

Mid 80’s through late 90’s: Professional actuarial education - financeand investment courses for Actuaries Institute in Australia, co-authorof text for SOA ”Financial Economics”, author of text ”Money andCapital Markets: Pricing, Yields and Analysis”.

Since 1985: Academic teaching and research - quantitative finance,financial economics, insurance, longevity risk.

Sherris, M. (1994), A One Factor Interest Rate Model and theValuation of Loans with Prepayment Provisions, Transactions of TheSociety of Actuaries, Vol 46, 251-320Sherris, M. and A. Ang, (1997), Interest Rate Risk Management:Developments in Interest Rate Term Structure Modeling for RiskManagement and Valuation of Interest-Rate-Dependent Cash Flows,North American Actuarial Journal, Vol 1, No 2, 1-26.


Assumed knowledge

Actuarial students studying actuarial professional courses withfinancial mathematics, probability and statistics background.

Practitioners with foundation background in interest rate modelsseeking a deeper understanding of different model approaches.

Practitioners seeking an overview of interest rate models.

Practitioners seeking a refresher of interest rate models.


Actuarial Applications




Term structure models are used in many actuarial applications.

Valuation of interest dependent cash flows - expected claims atdifferent future dates using a spot rate yield curve.

Interest rate risk management using the models to quantify cash flowduration and convexity using a spot rate yield curve

Valuation of option and guarantee features in insurance products thatdepend on uncertain future yield curves.

Determination of solvency requirements for economic and regulatorycapital purposes.


Yield Curves and Market Instruments

Yield Curves and Market Instruments


Term Structure Models

Yields for differing maturities for: coupon paying government bonds,par coupon bonds, zero coupon bonds, FRA’s, interest rate futures,interest rate swaps.

Default-free interest rates (yields to maturity, spot interest rates,forward interest rates) and bond prices for different maturities.

Market instruments and methods to construct the interest rate termstructure to be modelled.

Model for random changes in (risk-free) interest rates and prices of(default-free) discount bonds.

Historical yield curve time series (P measure) and market pricing (Qmeasure).


Spot Rates

Zero coupon bond prices

P (t,T ) = exp [− (T − t)R (t,T )] =1

1 + (T − t) L (t,T )

where P (t,T ) is the price at time t of $1 payable at time T (zerocoupon bond), R (t,T ) is the continuous compounding yield tomaturity and L (t,T ) simple interest yield to maturity (similar toLIBOR rate).

Spot interest rates (spot rates - continuous compounding)

R (t,T ) = − logP (t,T )

T − t


Forward Rates

Forward rate agreement (FRA) - agreement to fix an interest rate fora future time period (time T to S) at time t

P (t,T ) = P (t, S) exp [(S − T ) F (t,T ,S)]

= P (t, S) [1 + (S − T ) L (t,T , S)]

Forward rate at time t (continuous compounding) for time T to S(S > T )

F (t,T , S) =1

S − Tlog

P (t,T )

P (t,S)

and simple forward rate (similar to LIBOR forward rate)

L (t,T , S) =1

S − T

[P (t,T )

P (t,S)− 1

]


Coupon Paying Bonds

Most bonds traded are coupon paying bonds including governmentissued bonds (regarded often as default free)Price of coupon paying bond paying N coupons (in arrears) with Ci

paid at time Ti and nominal (face value) N paid at maturity TN

PTN(t) =

N

∑i=1

P (t,Ti )Ci + P (t,TN)N

=N

∑i=1

1

[1 + Y (TN)]tiCi +

1

[1 + Y (TN)]TNi

N

where Y (TN) is the yield to maturity for the TN maturity couponbond.Market conventions - most government coupon bonds paysemi-annual coupons and are quoted as semi-annual compoundingyields. Times for price quotes are usually between coupon dates(broken periods) and sometimes quoted with accrued interest andsometimes without (dirty and clean prices).


Par Coupon Bonds

Yields to maturity for coupon paying bonds often used to constructdefault free term structure using ”bootstrapping”.

Quoted market yields are impacted by many factors such as differentsize of coupons, tax, liquidity.

Smoothing is usually used to extract a ”par yield curve” - the yieldcurve for coupons bonds with coupons equal to the yield to maturitypayable on the same coupon dates and all priced on a coupon date,hence priced at par (face value usually taken as N = 100). Par yieldto maturity (equal to par coupon rate) given by

N = cTNN

N

∑i=1

P (t,Ti ) + P (t,TN)N

cTN=

1− P (t,TN)N

∑i=1

P (t,Ti )


Futures and Swaps

Interest rate futures (Eurodollar futures, futures on bank bills andbonds) are similar to forwards and have mark to market and depositrequirements

Futures rates require a convexity adjustment

Forward rate = Futures rate − 1

2σ2 (T − t)2

Interest rate swaps - swap fixed for floating coupons, value of payerswap is:

Πp (t) = N(P (t,T0)−

[δK ∑n

i=1P (t,Ti ) + P (t,Tn)

])= N

(δ ∑n

i=1P (t,Ti ) [F (t,Ti−1,Ti )−K ]

)where N is nominal value, K is fixed rate, δ is time between swappayments.

Par swap rates have Πp (t) = 0, similar to par coupon bond rates.


Bootstrapping Par Yields

Given the par yield to maturity curve for coupon paying bonds thenspot interest rates and forward interest rates can be determined.

All of these are equivalent but spot rates and forward rates can beused more flexibly to value more general cash flows.

Spot rates are determined working forward by maturity usingbootstrapping.

The first spot rate (continuous compounding) is given using the firstpar coupon rate:

1 + cT1 = exp [−R (t,T1)]

R (t,T1) = − log [1 + cT1 ]

Then having solved for R (t,Tj ) for j = 1 to i − 1, solve period byperiod, for R (t,Ti ) using

1 = cTi

i

∑j=1

exp [−R (t,Tj )] + 1 exp [−R (t,Ti )]


Bootstrapping Par Yields

Rearranging gives spot rates (continuous compounding)

R (t,Ti ) = − log

1 + cTi

1− cTi

i−1∑j=1

exp [−R (t,Tj )]

i = 2, . . . ,N

Then forward rates (continuous compounding) are derived from thespot rates

F (t,Ti−1,Ti ) =1

Ti − Ti−1[R (t,Ti )− R (t,Ti−1)]


Yield Curve Smoothing

Methods used to smooth market bond price/yield curve data include:

Weighted Least Squares or Regression, MLE or Bayesian

Fit discount factors, spot rates or forward rates with:

Splines,Nelson-Siegel or Svensson Parametric Curve, and similarexponential-polynomial class (see Cairns (2004) Chapter 12),Smith-Wilson Kernel method


Yield Curve Smoothing

General approachP = Cd + ε

where

P is (column) vector of market prices,C is the cash flow matrix with cash flows for the market instrumentsused to fit term structure,d discount factors for the term structure (determined from smoothedyield curve in terms of yields to maturity, spot rates or forward rates),ε vector of errors to be minimised in the smoothing of market data

Issues:

smoothing spot rates results in saw tooth forward rates so better tosmooth forward rates,market instruments include coupon bonds (short maturities, illiquidity)or futures and swaps (longer maturities, more active trading),need for interpolation and extrapolation.


Term Structure Models - R packages

R packages for Term Structure

termstrc cubic splines approach of McCulloch (1971, 1975), theNelson and Siegel (1987) method with extensions by Svensson(1994),Diebold and Li(2006) and DePooter(2007). Weightedconstrained optimization procedure with analytical gradients and aglobally optimal start parameter search algorithm.YieldCurve fits Nelson-Siegel, Diebold-Li and Svensson.


Instantaneous Forward Rates and the Short Rate

Instantaneous forward rate at time t

f (t,T ) = limS→T

F (t,T , S) = − ∂

∂TlogP (t,T ) = −∂P (t,T ) /∂T

P (t,T )

which gives

P (t,T ) = exp

[−∫ T

tf (t, u) du

]= exp [−R (t,T ) (T − t)]

and

f (τ) = R (τ) + τ∂

∂τR (τ)

where τ = T − t.

Short rate at time t is

r (t) = limT→t

R (t,T ) = R (t, t) = f (t, t)

:


Static Nelson-Siegel

Parametric formula for instantaneous forward yield curve (crosssectional)

f (τ) = β1 + β2e−λτ + β3λτe−λτ

Nelson-Siegel spot rate yield curve (integrate forward yield curve)

y (τ) = β1 + β2

(1− e−λτ

λτ

)+ β3

(1− e−λτ

λτ− e−λτ

)Desirable features:

P (0) = 0 and limτ→∞ P (τ)→ 0instantaneous short rate limτ→0 y (τ) = f (0) = r (equals β1 + β2)limτ→∞ y (τ) = β1 (the long term interest rate)flexible shapes - flat, increasing, decreasing, humped, U-shaped (β3determines size and shape of hump)


Dynamic Nelson-Siegel

Static Nelson Siegel fits constant parameters β1, β2, β3, λ to producea best fit term structure

Dynamic Nelson Siegel treats parameters β1, β2, β3 as variables, orlatent factors, and the coefficients as factor loadings

yt (τ) = β1t + β2t

(1− e−λτ

λτ

)+ β3t

(1− e−λτ

λτ− e−λτ

)Factor loadings

1, the loading on β1t , impacts on all yields, but impacts long termyields relatively more compared to other factors(1−e−λτ

λτ

), the loading on β2t , starts at 1 and decreases quickly to zero

- impacts short term yields most, a short term factor,(1−e−λτ

λτ − e−λτ)

, the loading on β3t , starts at 0, increases and then

decreases to zero - impacts medium term yields most, a medium termfactor


Svensson Yield Curve

Parametric formula for instantaneous forward yield curve (crosssectional)

f (τ) = β1 + (β2 + β3λ1τ) e−λ1τ + β4λ2τe−λ2τ

Svensson spot rate yield curve (integrate forward yield curve)

y (τ) = β1 + β2

(1− e−λ1τ

λ1τ

)+ β3

(1− e−λ1τ

λ1τ− e−λ1τ

)+β4

(1− e−λ2τ

λ2τ− e−λ2τ

)allows for second hump in the yield curve.


Discussion - Static Term Structure

Term structure from a static perspsective (P and Q measures).

Different ways of representing the yield curve (level, slope, curvature).

Different market instruments used to fit the yield curve (bonds,futures, forwards, swaps).

Methods of fitting term structure yield curves (yield curve smoothing).

Parametric form of cross sectional yield curve and links to dynamicmodels (Nelson-Siegerl).


Concepts of no-arbitrage and risk neutral versus real worldprobabilities

Concepts of no-arbitrage and risk neutral versusreal world probabilities


PDE Approach - Short Rate Models

Start from stochastic process for interest rates (short interest rate),

Use Ito formula to derive stochastic process for bond prices (bondsare contingent claims or derivatives on interest rates),

Require no-arbitrage and derive partial differential equation (PDE) for(zero coupon) bond prices,

Solve PDE subject to boundary conditions.


Brownian Motion

Sometimes called a ”Wiener process”A stochastic process {W (t) , t ≥ 0} is said to be a standard Brownianmotion process, or simply Brownian motion, if:W (0) = W0 = 0;W (t) has stationary and independent increments;and for every t > 0,

W (t) ∼ N (0, t) .

A stochastic process {X (t) t ≥ 0} is said to be a Brownian motionprocess with drift coefficient µ and variance parameter σ2 if:X (0) = X0 = 0;{X (t) t ≥ 0} has stationary and independent increments; andfor every t > 0, X (t) ∼ N

(µt, σ2t

).


Standard Brownian motion

Integral from

W (t) = W0 +∫ t

0dW (u)

Properties

W (t) is continuous

W (t) is nowhere differentiable

W (t) is process of unbounded variation

W (t) is process of bounded quadratic variation

Conditional distribution of W (u) |W (t) for u > t is normal withmean W (t) and variance (u − t)

Variance of a forecast of W (u) increases indefinitely as u → ∞


Generalized univariate Wiener process

Drift and volatility depend on X (t) and t

dX (t) = α (X (t) , t) dt + σ (X (t) , t) dW (t)

Special cases:Arithmetic Brownian motionGeometric Brownian motionMean reverting process

Dynamics for function of Brownian motion f (X (t) , t).


Arithmetic Brownian Motion

dX = αdt + σdW

α (X (t) , t) = α

σ (X (t) , t) = σ

The process defined by

W (t) =X (t)− αt

σ,

is a standard Brownian motion.If {W (t) , t ≥ 0} is a standard Brownian motion, then by defining

X (t) = αt + σW (t) ,

the process {X (t) , t ≥ 0} is a Brownian motion with a drift α andvolatility σ.


Arithmetic Brownian Motion

X (t) may be positive or negative

distribution of X (u) given X (t) for u > t is normal with meanX (t) + α (u − t) and standard deviation σ

√(u − t)

variance tends to infinity as u goes to infinity (variance grows linearlywith time)


Geometric Brownian Motion

If X (t) starts at positive value then it remains positive

X (t) has an absorbing barrier at 0

Conditional distribution of X (u) given X (t) for u > t is lognormal.Conditional mean of lnX (u) given bylnX (t) + α (u − t)− 1

2σ2 (u − t) and conditional standard deviation

of lnX (u) is σ√(u − t).

Often used to model values - positive and increases at exponentialrate.


Geometric Brownian Motion

E (X (t) |X (u) , 0 ≤ u ≤ s )

= E(eY (t) |X (u) , 0 ≤ u ≤ s

)= E

(eY (s)+Y (t)−X (s) |X (u) , 0 ≤ u ≤ s

)= eY (s)E

(eY (t)−Y (s) |X (u) , 0 ≤ u ≤ s

)= Y (s) · exp [α (t − s)] .

since (Y (t)− Y (s)) ∼ N((

α− 12σ2)(t − s) , σ2 (t − s)

), then

E(eY (t)−Y (s)

)= e[(α− 1

2 σ2)(t−s)+ 12 σ2(t−s)]

= e [α(t−s)]


Mean Reverting Process

dX = κ (µ− X ) dt + σX γdW

α (X (t) , t) = κ (µ− X ) κ > 0

σ (X (t) , t) = σX γ

κ is speed of adjustment parameter, µ long run mean and σ is volatilityparameter

X (t) is positive as long as X (t) starts positive

As X (t) approaches zero, the drift is positive and volatility vanishes

As u → ∞ the variance of X (u) is finite


Mean Reverting Process

If γ = 12 the conditional distribution of X (u) given X (t) for u > t is

non-central chi-squared with mean

(X (t)− µ) exp [−κ (u − t)] + µ

and variance

X (t)

(σ2

κ

)(exp [−κ (u − t)]−exp [−2κ (u − t)]

)+µ

(σ2

2κ

)(1− exp [−κ (u − t)])2

CIR or square root process


One-Dimensional Ito’s Formula

Consider the SDE

dX = α(X (t), t)dt + σ(X (t), t)dW (t) = αdt + σdW

and suppose that we have another stochastic process f (X (t) , t) = ftwhich depends on X (t) and t. Ito’s formula tells us this process satisfiesthe following:

dft =∂f

∂XdX +

∂f

∂tdt +

1

2

∂2f

∂X 2σ2tdt.

This formula is also known as Ito’s formula (stochastic Taylor series).

df =∂f

∂XdX +

∂f

∂tdt

+1

2

[∂2f

∂X 2dX 2 + 2

∂2f

∂X ∂tdXdt +

∂2f

∂t2dt2]+ o (dt)

=

[α

∂f

∂X+

∂f

∂t+

1

2σ2 ∂2f

∂X 2

]dt + σ

∂f

∂XdW

dX = α (X (t) , t) dt + σ (X (t) , t) dW (t)

dX 2 = (αdt)2 + 2ασdtdW + (σdW )2 = σ2dt


Bivariate Ito’s Formula

Consider the SDE’s

dX = α (X ,Y , t) dt + σ (X ,Y , t) dW1

dY = β (X ,Y , t) dt + ν (X ,Y , t) dW2

dW1dW2 = E [dW1dW2] = ρdt

Probabilistically, dW2 = ρdW1 +√(1− ρ2)dZ

df (X ,Y , t) =∂f

∂XdX +

∂f

∂YdY +

∂f

∂tdt

+1

2

[∂2f

∂X 2dX 2 + 2

∂2f

∂X ∂YdXdY +

∂2f

∂Y 2dY 2

]+ o (dt)

=

[α

∂f

∂X+ β

∂f

∂Y+

∂f

∂t+

1

2

[σ2 ∂2f

∂X 2+ 2ρσν

∂2f

∂X ∂Y+ ν2

∂2f

∂Y 2

]]dt

+σ∂f

∂XdW1 + ν

∂f

∂YdW2


Covariation Process and Quadratic Variation

ConsiderdXi (t) = ai (t) dt + bi (t) dWi (t)

then quadratic covariation process for Xi (t) and Xj (t) is

〈Xi ,Xj 〉 (t) =∫ t

0bi (u) bj (u) du

d 〈Xi ,Xj 〉 (t) = bi (t) bj (t) dt sometimes written as dXi (t) dXj (t)

Quadratic variation is 〈X ,X 〉 (t) =∫ t0 b (u)2 du.

Product rule

d (X (t)Y (t)) = X (t) dY (t) + Y (t) dX (t) + d 〈X ,Y 〉 (t)

Integration by parts formula

X (t)Y (t) = X (0)Y (0)+∫ t

0X (s) dY (s)+

∫ t

0Y (s) dX (s)+ 〈X ,Y 〉 (t)


Feynman-Kac formula

Consider the pde[α (X , t)

∂f

∂X+

∂f

∂t+

1

2σ2 (X , t)

∂2f

∂X 2

]− rf + c (X , t) = 0

defined on interaval [0,T ] with terminal condition

f (X ,T ) = Ψ (X )

Then Feynman-Kac formula is solution in terms of conditional expectation

f (X , t) = E

[ ∫ Tt exp

(−∫ st rdτ

)c (X , s) ds

+ exp(−∫ Tt rdτ

)Ψ (X ) |Xt = x

]

where X is a diffusion process

dX = α (X , t) dt + σ (X , t) dW


PDE for Bond Pricing

Model short interest rate with a diffusion process (single factor)

dr (t) = α (r , t) dt + σ (r , t) dZ

A bond has a value P (r , t,T ) = P (r , τ) with dynamics (from Ito’sformula)

dP (r , t,T ) =∂P (r , t,T )

∂rdr +

1

2

∂2P (r , t,T )

∂r2dr2 +

∂P (r , t,T )

∂tdt

=∂P

∂r(αdt + σdZ ) +

1

2

∂2P

∂r2σ2dt +

∂P

∂tdt

=

[α

∂P

∂r+

1

2

∂2P

∂r2σ2 +

∂P

∂t

]dt +

∂P

∂rσdZ

= P [mdt + sdZ ]

where

m =1

P

(α

∂P

∂r+

1

2

∂2P

∂r2σ2 +

∂P

∂t

)and s =

1

P

∂P

∂rσ



Construct a portfolio, V , of two bonds, V1 and V2, with differentmaturities τ1 > τ2

V = V1 (r , τ1) + V2 (r , τ2)

dV = dV1 + dV2 = V1 [m1dt + s1dZ ] + V2 [m2dt + s2dZ ]

= [V1m1 + V2m2] dt + [V1s1 + V2s2] dZ

Now if we choose V1 and V2 so that the volatility of the portfolio is zero

[V1s1 + V2s2] = 0

V1

V2= − s2

s1

and

(V − V2) s1 + V2s2 = 0

V2 =Vs1

s1 − s2and V1 = −

Vs2s1 − s2



We then have, since the portfolio is now riskless,

[V1m1 + V2m2] = rV = r (V1 + V2)

− Vs2s1 − s2

m1 +Vs1

s1 − s2m2 = rV

or, rearranging,(m1 − r)

s1=

(m2 − r)

s2= λ

where λ is the market price of (interest rate) risk. The excess return perunit of risk is the same for all maturity bonds.We vary V1 so the portfolio is self financing

dV = V

(m2s1 −m1s2

s1 − s2

)dt



Substituting expressions for m and s we obtain the PDE for bond prices

1

P

(α

∂P

∂r+

1

2

∂2P

∂r2σ2 +

∂P

∂t

)− r =

1

P

∂P

∂rσλ

∂P

∂t+ (α− σλ)

∂P

∂r+

1

2

∂2P

∂r2σ2 − rP = 0

Then solve subject to boundary conditions.Solving PDEs using known solutions for particular cases, guesses orFeynman-Kac.


PDE for Bond Pricing - Example

Consider random walk for yield (y (0) = r)

P (τ) = e−yτ and dy = σdZ

∂P

∂t= yP

∂P

∂y= −τP

∂2P

∂y2= τ2P

From Ito formula

dP =

(y +

1

2σ2τ2

)Pdt − στPdZ

Hence

m = y +1

2σ2τ2 and s = στ

and risk premium for interest rate risk gives(y + 1

2σ2τ2)− r

στ= λ


PDE for Bond Pricing - Example

In this case

y (τ) = r + λστ − 1

2σ2τ2

= risk free rate + price of risk × risk− convexity adjustment

Note that

yield curve is quadratic in maturity

maximum at λσ , so humped shaped

can have negative yields

instantaneous forward rates are quadratic

f (τ) = − ∂

∂τlogP (τ) = − ∂

∂τ(−y (τ) τ) = y (τ) + τ

∂

∂τ(y (τ))

= r + λστ − 1

2σ2τ2 − τ

(λσ− σ2τ

)= r + 2λστ − 3

2σ2τ2


Martingale Approach for Bond Pricing

Background on Martingales

Change of Measure

Fundamental Theorem of Asset Pricing

Bond Pricing as an Expectation under Risk Neutral Measure


Continuous-Time Martingales

We denote the filtration generated by the process at time t by Ft and isinterpreted as the information (or history) of the process up until time t.We shall assume the process {W (t)} is Ft-adapted, which means thatgiven this history Ft , the value of the process at t, W (t), is known.The stochastic process {W (t) , t ≥ 0} is said to be a martingale if:E [|W (t)|] < ∞, i.e. finite expectation of the absolute value, and for alls < t, we have

E (W (t) |Fs ) = W (s) .

The best forecast of unobserved future values is the most recentlyavailable observed value of the process.



Example (Brownian Motion)Consider a Brownian motion {Xt , t ≥ 0} with drift parameter α andvariance parameter σ2. Then, for s < t, we have

E (Xt |Fs ) = E [Xs + (Xt − Xs) |Fs ]

= E [Xs |Fs ] + E [Xt − Xs |Fs ]

= Xs + α (t − s) .

since(Xt − Xs) ∼ N

(α (t − s) , σ2 (t − s)

).

Thus, we see that E (Xt |Fs ) = Xs only if α = 0.A B.M. with a zero drift is a martingale.



Example (Geometric Brownian Motion)Consider a geometric Brownian motion {Yt , t ≥ 0} defined as

Yt = exp (Xt) ,

where Xt is a B.M. with drift parameter µ and variance parameter σ2.Then, for s < t, we have

E (Yt |Fs ) = E [exp (Xt) |Fs ]

= E [exp (Xs + (Xt − Xs)) |Fs ]

= E [exp (Xs) |Fs ]× E [exp (Xt − Xs) |Fs ]

= Ys × E [exp (Xt − Xs) |Fs ]

= Ys × exp

[(µ +

1

2σ2

)(t − s)

]



Recalling the moment generating function of a normal distribution, wehave

E (Yt |Fs ) = Ys × exp

[α (t − s) +

1

2σ2 (t − s)

]= Ys × exp

[(α +

1

2σ2

)(t − s)

].

Thus, we see that the geometric Brownian motion is not a martingaleunless of course if α = − 1

2σ2.



We can transform the geometric B.M. to form a martingale by defining theprocess

Zt = exp

(W (t)− 1

2σ2t

).

By following the same argument as above, it is left as an exercise to provethat the process {Zt , t ≥ 0} is a martingale if the Brownian motion haszero drift.


Stochastic Differential Equations

Definition: A stochastic differential equation, or simply SDE, has thefollowing form:

dXt = µ (Xt , t) dt + σ (Xt , t) dBt ,

where Bt is the standard Brownian motion process.The drift µ (Xt , t) and σ (Xt , t) can be functions of the process Xt andtime t.A stochastic process Xt that satisfy the above SDE is sometimes called adiffusion process.


Stochastic Integration

{W (t) , t ≥ 0} is a Brownian motion with zero drift and varianceparameter σ2

Let f be a function with continuous derivative on the interval [a, b].∫ b

af (t) dW (t) = lim

n→∞max(tk−tk−1)→0

n

∑k=1

f (tk−1) (W (k)−W (k − 1))

where a = t0 < t1 < · · · < tn = b is a partition of [a, b].This definition of a stochastic integral is sometimes called the ”Itointegral”.



To simplify notations, W (k) is the value of the stochastic process at timetk .The integral is itself another stochastic process which we can label as

Yt =∫ b

af (t) dW (t) .

The integration by parts formula applied to sums is given by

n

∑k=1

f (tk−1) (W (k)−W (k − 1)) = [f (b)W (b)− f (a)W (a)]

−n

∑k=1

[f (tk)− f (tk−1)]W (k) .



Now, we take appropriate limits on both sides, we see that∫ b

af (t) dW (t) = [f (b)W (b)− f (a)W (a)]−

∫ b

aW (t) df (t)

= [f (b)W (b)− f (a)W (a)]−∫ b

aW (t) f

′(t) dt.

Take the expectation of both sides,

E (Yt) = [f (b)E (W (b))− f (a)E (W (a))]︸︷︷︸0

−∫ b

aE (W (t)) df (t)︸︷︷︸

0

= 0.

No matter what the form of the function is, the mean of the ”Ito integral”is always zero.



Its variance can also be computed as follows.

Var

[n

∑k=1

f (tk−1) (W (k)−W (k − 1))

]

=n

∑k=1

f 2 (tk−1)Var (W (k)−W (k − 1))

= σ2n

∑k=1

f 2 (tk−1) (tk − tk−1) ,

Var (Yt) = limn→∞

max(tk−tk−1)→0

σ2n

∑k=1

[f 2 (tk−1)(tk − tk−1)

]

= σ2∫ b

af 2 (t) dt.


Stochastic Integration - Example

Let the process {Yt , t ≥ 0} be defined by

Yt =∫ t

0eα(t−u)dW (u) , for t ≥ 0,

where α is some constant parameter. Find the mean and variance of thisprocess.Solution: The process has zero mean (Ito integral).Variance is

Var [Yt ] = σ2 ×∫ t

0

[eα(t−u)

]2du

= σ2 ×∫ t

0e2α(t−u)du

=σ2

2α

(e2αt − 1

).


Cameron-Martin-Girsanov Theorem - Change of Measure

If W (t) is a Brownian motion under measure P and λ (t) is a (previsible)

process satisfying EP exp(12

∫ T0 λ (t)2 dt < ∞

), the Novikov condition,

then there exists a measure Q such that

Q is equivalent to P

dQdP = exp

(−∫ T0 λ (t) dW (t)− 1

2

∫ T0 λ (t)2 dt

)W (t) = W (t) +

∫ t0 λ (t) dt is a Q Brownian motion

This can be applied to change measure as follows

EQ [X ] = EP

[dQ

dPX

]= EP

[exp

(−∫ T

0λ (t) dW (t)− 1

2

∫ T

0λ (t)2 dt

)X

]


Fundamental Theorem of Asset Pricing

Define the money market account (or cash account) as

B (t) = B (0) exp

(∫ t

0r (s) ds

)The Fundamental Theorem of Asset Pricing as applied to Bond Pricesstates that

Bond prices are arbitrage free if and only if there exists a measure Q,equivalent to P, under which, for each T , the discounted bond priceprocess P (t,T ) /B (t) is a martingale for all t : 0 < t < T .

The market is complete if and only if Q is the unique measure forwhich P (t,T ) /B (t) are martingales.

Q is referred to as the risk-neutral measure with the cash account asnumeraire since the expected return on a bond under this measure is therisk-free rate.



Assume SDEs for the risk free short rate and the bond price are (notechange in notation from earlier)

dr (t) = a (t) dt + b (t) dW (t)

dP (t,T ) = P (t,T ) [m (t,T ) dt + s (t,T ) dW (t)]

Money market account

dB (t) = r (t)B (t) dt

Market price of risk

λ (t) =m (t,T )− r (t)

s (t,T )

Discounted bond price process

Z (t,T ) =P (t,T )

B (t)= P (t,T ) exp

(−∫ t

0r (s) ds

)Michael Sherris (UNSW) AFIR − ERM Workshop August 2017 60 / 137


We have (Ito product rule)

dZ (t,T ) =1

B (t)dP (t,T ) + P (t,T ) d

(1

B (t)

)+ d

⟨1

B,P

⟩(t)

and using Ito formula

d

(1

B (t)

)= − 1

B (t)2dB (t) +

1

2

2

B (t)3d 〈B〉 (t) = − r (t) dt

B (t)

Hence

dZ (t,T ) =P (t,T ) (m (t,T ) dt + s (t,T ) dW (t))

B (t)− r (t)P (t,T ) dt

B (t)

= Z (t,T ) [(m (t,T )− r (t)) dt + s (t,T ) dW (t)]

= Z (t,T )

[(m (t,T )− r (t)− λ (t) s (t,T )) dt

+s (t,T ) (dW (t) + λ (t))

]= Z (t,T ) s (t,T ) dW (t)

and Z (t,T ) is a martingale under measure Q.Michael Sherris (UNSW) AFIR − ERM Workshop August 2017 61 / 137


Discounted bond priceP(t,T )B(t)

is a martingale under risk neutral measure

P (t,T )

B (t)= EQ

[P (T ,T )

B (T )|Fs

]P (t,T ) = EQ

[B (t)

B (T )|Fs

]and for 0 < S < T

P (t, S) = EQ

[exp

(−∫ S

tr (u) du

)|Fs

]


Discussion - Bond Pricing

Use of PDE approach versus Martingale Approach.

Difference between P and Q measure for interest rate dynamics versusbond prices.

Importance of market price of risk.

Single factor versus multiple factor models.


Main Continuous Time Single Factor Short Rate Models

Main Continuous Time Single Factor Short RateModels


Main continuous time single factor short rate models

One factor models with constant interest rate volatility (affine models)

Vasicek (1977) dr (t) = α (µ− r (t)) dt + σdW (t)

Ho and Lee (1986) dr (t) = θ (t) dt + σdW (t)Extended Vasicek or Hull and White Model (1990, 1993)

dr (t) = α (µ (t)− r (t)) dt + σdW (t)

One factor models with rate-dependent interest rate volatility;

Cox, Ingersoll and Ross (1985) -

dr (t) = α (µ− r (t)) dt + σ√

r (t)dW (t)Black, Derman and Toy (1990) - log-normal short rate -

d log r (t) =

(θ (t)− σ

′(t)

σ(t)log r (t)

)dt + σ (t) dW (t) or with

constant volatility d log r (t) = θ (t) dt + σdW (t)Black and Karasinski (1991) - log-normal short rate -

d log r (t) = α (t) (log µ (t)− log r (t)) dt + σ (t) dW (t)


Vasicek model

Risk neutral dynamics

dr (t) = α (µ− r (t)) dt + σdW (t)

mean reverting process

µ long term mean risk free rate under risk neutral measure, α rater (t) reverts to long term mean

σ local volatility of short term rate

r (t + s) given r (t) is normally distributed under Q with mean

µ + (r (t)− µ) e−αs

and varianceσ2[1− e−2αs

]2α


Vasicek model

Bond prices determined by:

using PDE for bond price of zero coupon bonds, guess form ofsolution and derive form of bond price by solving PDE, or

using martingale approach and the dynamics for r (t) to evaluatedscounted expected value under risk neutral measure (often solvedusing Laplace transforms).

Model does not give a perfect fit to an initial yield curve.


Vasicek model

Bond price takes the form

P (t,T ) = exp [A (t,T )− B (t,T ) r (t)]

where

B (t,T ) =1− e−α(T−t)

α

A (t,T ) = (B (t,T )− (T − t))

(µ− σ2

2α2

)− σ2

4αB (t,T )2

Use this to construct the zero coupon bond yield curve and derive spotrates and forward rates.


Ho-Lee model

Gives a perfect fit to an initial yield curve.Risk neutral dynamics

dr (t) = θ (t) dt + σdW (t)

time varying drift allowing to fit initial yield curve

short rate normally distributed.

Bond price for zero coupon bonds

P (t,T ) = exp [A (t,T )− B (t,T ) r (t)]

whereB (t,T ) = (T − t)

A (t,T ) =σ2

6(T − t)3 −

∫ T

tθ (s) (T − s) ds


Ho-Lee model

Setting

θ (t) =∂

∂Tf (0,T ) + σ2T

where

f (0,T ) = − ∂

∂TlogP (0,T )

We have

r (t) = r (0) +∫ t

0θ (s) ds + σW (t)

= f (0, t) +σ2t2

2+ σW (t)

which allows a perfect fit to the initial yield curve.

We then have

f (t,T ) = − ∂

∂TlogP (t,T ) = r (t)+ f (0,T )− f (0, t)+σ2t (T − t)


Extended Vasicek or Hull and White Model

Risk neutral dynamics

dr (t) = α (µ (t)− r (t)) dt + σdW (t)

Bond price for zero coupon bonds, matching initial term structure,

µ (t) =1

α

∂

∂tf (0, t) + f (0, t) +

σ2

2α2

(1− e−2αt

)P (t,T ) = exp [A (t,T )− B (t,T ) r (t)]

where

B (t,T ) =1− e−α(T−t)

α

A (t,T ) = logP (0,T )

P (0, t)+ B (t,T ) f (0, t)

− σ2

4α3

(1− e−α(T−t)

)2 (1− e−2αt

)Michael Sherris (UNSW) AFIR − ERM Workshop August 2017 71 / 137

Extended Vasicek or Hull and White Model

We also have

r (t) = e−αtr (0) + α∫ t

0e−α(t−s)µ (s) ds + σ

∫ t

0e−α(t−s)dW (s)

= f (0, t) +σ2

2α2

(1− e−αt

)2+ σ

∫ t

0e−α(t−s)dW (s)

Special cases

µ (t) = µ, a constant, gives Vasicek

if α→ 0 and αµ (t)→ θ (t) as α→ 0 then gives Ho-Lee.


Cox-Ingersoll-Ross Model

Instantaneous interest rate r(t) dynamics under the risk neutral Q

measure given by:

dr(t) = α(µ− r(t))dt + σ√

r(t)dW (s)

where α > 0 is the speed of mean reversion of r(t), µ > 0 is the long-runmean of r(t), σ

√r(t) ≥ 0 is the volatility of the short rate process.

Require 2αµ ≥ σ2 to ensure the process is positive.


Cox-Ingersoll-Ross Model

Zero coupon bond price is given by:

P(t,T ) = EQt [e−∫ Tt r (u)du ] = exp [A (t,T )− B (t,T ) r (t)]

where A(t,T ) and B(t,T ) are given by:

A(t,T ) =2αµ

σ2log

(2γe(γ+α)(T−t)/2

(γ + α)(eγ(T−t) − 1) + 2γ

)

B(t,T ) =2(eγ(T−t) − 1)

(γ + α)(eγ(T−t) − 1) + 2γ

withγ =

√α2 + 2σ2

with boundary conditions A(T ,T ) = 0 and B(T ,T ) = 0.


Other Short Rate Models

Many other models with differing assumptions, for example:

Brennan and Schwartz (1979) - two-factor model with short rate andlong term rate on a consol bond.

Longstaff and Schwartz (1992) - two-factor version of CIR with shortrate and instantaneous volatility of short rate as factors


Affine Term Structure Models - Short Rate Models

Affine short rate term structure models have bond prices of affine form inthe short rate

P (t,T ) = exp [A (t,T )− B (t,T ) r (t)]

and for bond prices to have this form the risk neutral drift and volatility ofthe short rate take the form

m (t, r (t)) = a (t) + b (t) r (t)

s (t, r (t)) =√

δ (t) + γ (t) r (t)

where a (t) , b (t) , δ (t) , and γ (t) are deterministic functions.



Consider the risk neutral dynamics for the short rate

dr (t) = m (t, r (t)) dt + s (t, r (t)) dW (t)

Then an application of Ito’s formula gives

dP (t,T ) = P (t,T )

[(∂A

∂t− ∂A

∂tr (t)− Bm+

1

2(Bs)2

)dt − BsdW (t)

]Under the risk neutral dynamics all bonds have instantaneous expectedreturns equal to the short rate so that

dP (t,T ) = P (t,T )[r (t) dt + S (t,T , r (t)) dW (t)

]where S (t,T , r (t)) is the volatility of the bond price P (t,T ).



Define

g (t, r) =

(∂A

∂t− ∂A

∂tr (t)− Bm (t, r (t)) +

1

2[Bs (t, r (t))]2

)− r

then this must be zero for all t and r for the bond prices to bearbitrage-free.Differentiating twice with respect to r gives

∂2g (t, r)

∂r2= −B ∂2m (t, r)

∂r2+

1

2B2 ∂2s (t, r (t))2

∂r2= 0

and for this to hold

∂2m (t, r)

∂r2= 0 and

∂2s (t, r (t))2

∂r2= 0

which gives the affine form for the short rate dynamics.



Special cases include a number of the spot rate models already covered.Vasicek

a = αµ, b = −α, δ = σ2, γ = 0

dr (t) = α (µ− r (t)) dt + σdW (t)

Cox, Ingersoll and Ross

a = αµ, b = −α, δ = 0, γ = σ2

dr (t) = α (µ− r (t)) dt + σ√

r (t)dW (t)


Discussion - Short Rate Models

Some models were developed as equilibrium models (Vasicek andCIR) and others as arbitrage-free or relative valuation models.

Models with constant parameters, drift and volatility, do not providean exact fit to the current yield curve, so models with time varyingdrift and volatility were developed to do this.

Model fitting requires both the current yield curve and a volatilitystructure for time varying volatility models.

Gaussian models have geneally small chance of negative interest rates.

Gaussian models have closed form expressions for bond prices (andoptions on bonds) because of log-normal distrubution of bond prices.


Discussion - Short Rate Models

Log-normal short rate models require numerical implementation tocompute bond prices usually with a lattice.

Many models developed as discrete time lattice models havecontinuous time limits (Ho-Lee, Black-Derman-Toy).

Many are single factor models, with perfect correlation ofinstantaneous bond returns across maturities, but can be extended tomultiple factors.

Matching volatility and correlations between different maturity bondreturns and option prices requires time varying volatility and multiplefactor models.


Affine Term Structure Models - Multi-Factor Models

More generally affine term structure (ATS) models are arbitrage-freemultifactor model of interest rates in which the yield on any risk-freezero-coupon bond is an affine function of a set of unobserved latentfactors.Affine term structure (ATS) models have bond prices of affine form

P (t,T ) = exp

[A (t,T ) +

n

∑j=1

Bj (t,T )Xj (t)

]= exp

[A (t,T ) + B (t,T )

′X (t)

]where X (t) = (X1 (t) ,X2 (t) , . . . ,Xn (t))

′is a vector of state variables

and B (t,T ) = (B1 (t,T ) ,B2 (t,T ) , . . . ,Bn (t,T ))′.

We will only consider time homogeneous models where A (t,T ) andB (t,T ) are functions of (T − t) only and the state variables X (t) aretime homogeneous.


Affine Term Structure Models - Multi-Factor Models

If bond prices have the form

P (t, t + τ) = exp[A (τ) + B (τ)

′X (t)

]then X (t) must have SDE’s dX (t) = (α + β)X (t) + SD (X (t)) dW (t)

where W (t) is an n-dimensional Brownian motion under Q,

α = (α1, α2, . . . , αn)′

is a constant vector, β =(

βij

)ni .j=1

is a constant

matrix, S = (σij )ni .j=1 is a constant matrix and D (X (t)) a diagonal matrix

√γ′1X (t) + δ1 0 · · · · · · 0

0√

γ′2X (t) + δ2 0 · · · 0

... 0. . . · · ·

...

...... 0

. . ....

0 · · · · · · 0√

γ′nX (t) + δn

where δ1, δ2, . . . , δn are constants and γ = (γ1, γ2, . . . , γn) is a constantvector.Michael Sherris (UNSW) AFIR − ERM Workshop August 2017 83 / 137

Affine Term Structure Models - Gaussian Model

As an example, a 3-factor Gaussian ATS model has

r (t) = µ + 1′X (t)

where 1 = (1, 1, 1)′

and X (t) = (X1 (t) ,X2 (t) ,X3 (t)) follows azero-mean Ornstein-Uhlenbeck process under the real world probabilitymeasure

dX (t) = −KX (t) dt + ΣdW (t)

with K is lower triangular and Σ is diagonal, both 3× 3 matrices.Market prices of risk for the latent factors are given by

λ (t) = λ (0) + ΛX (t)

where λ (t) = (λ1 (t) , λ2 (t) , λ3 (t))′

and Λ a 3× 3 matrix.


Affine Term Structure Models - Gaussian Model

Bond prices for a τ−maturity bond given by

P (t, τ) = EQ

[exp

(−∫ t+τ

tr (u) du

)|Fs

]and the risk neutral dynamics are gven by

dX (t) = (−K − λ (t)Σ)X (t) dt + ΣdW (t)

= − ((K + ΣΛ)X (t) + Σλ (0)) dt + ΣdW (t)

Zero coupon bond prices can be written as

P (t, τ) = exp [A (τ) + B (τ)X (t)]

where expressions for A (τ) and B (τ) are factor loadings that are functionsof the parameters of the model solved from a set of differential equations.


Dynamic Nelson-Siegel Model

The Dynamic Nelson Siegel (DNS) model is effectively a 3-factor Gaussianmodel, although not arbitrage-free. Recognising the 3 factors as level,slope and curvature of the yield curve, we can write it as

yt (τ) = lt + st

(1− e−λτ

λτ

)+ ct

(1− e−λτ

λτ− e−λτ

)and can be written in state-space form as a measurement equation

yt = Λft + εt

where

yt =

yt (τ1)yt (τ2)

...yt (τN)

, ft =

ltstct

, εt =

εt (τ1)εt (τ2)

...εt (τN)



The parameter matrix is

Λ =

1 1−e−λτ1

λτ1

1−e−λτ1

λτ1− e−λτ1

1 1−e−λτ2

λτ2

1−e−λτ2

λτ2− e−λτ2

......

...

1 1−e−λτN

λτN

1−e−λτN

λτN− e−λτN

where we have observable yields at times t = 1, . . . ,T , and N maturitiesfor zero coupon bond yields at each time.The lt , st , and ct are common factors with dynamics given by thetransition equation.



The transition equation assumes a first-order vector autoregression

(ft − µ) = A (ft−1 − µ) + ηt

where

µ =

µl

µs

µc

, A =

a11 a12 a13a21 a22 a23a31 a32 a33

, ηt =

ηlt

ηst

ηct



Finally the state space formulation requires assumptions for the covariancestructure of the measurement and transition errors. Here we assume whitenoise and orthogonal errors so that(

εtηt

)∼ WN

(00

Q 00 H

)This formulation allows the use of Kalman filter for optimal filtering andsmoothing as well as prediction of yield factors and observed yields.Estimation is then by maximum likelihood with the state spacerepresentation and the Kalman filter.



Kalman filter provides one-step-ahead prediction errors for which theGaussian pseudo likelihood can be evaluated for any set of parameters.The parameter configuration that maximises the likleihood is foundnumerically using some form of either gradient based methods or analyticscore functions.Other model estimation procedures are possible including an EM basedoptimization or a Bayesian approach using Markov-chain Monte Carlo.There are a large number of parameters so that traditonal gradient basednumerical optimization methods may be intractable.If there are N maturities at each time point to fit then the measurementequation requires one-parameter λ, unless a fixed value is used, thetransition equation has 12 parameters (3 means and 9 dynamicparameters), the measurement error covariance matrix has

(N2 +N

)/2

parameters and the transition disturbance covariance marix has 6parameters.So if there were 15 yield maturities to be fitted then there would be 139parameters.


Arbitrage-Free Nelson-Siegel Model

The Dynamic Nelson-Seigel Model is flexible.It can fit cross section and time series of yields well.However it does not impose restrictions to make the model arbitrage-free.The DNS model can be made arbitrage-free and formulated as an affineterm structure model.The arbitrage free Nelson Seigel (AFNS) maintains the DNS factor-loadingstructure and is an affine arbitrage-free term structure model.



Consider a three-factor affine model with Xt =(X 1t ,X 2

t ,X 3t

)then we

require a yield function of the form

y (t,T ) = −A (t,T )

T − t+ X 1

t +

(1− e−λ(T−t)

λ (T − t)

)X 2t

+

(1− e−λ(T−t)

λ (T − t)− e−λ(T−t)

)X 3t

so the factor loadings in the affine model need to be

B1 (t,T ) = − (T − t) , B2 (t,T ) = −1− e−λ(T−t)

λ,

and B3 (t,T ) = −1− e−λ(T−t)

λ+ (T − t) e−λ(T−t)



The required model hasr (t) = X 1

t + X 2t

and risk neutral dynamics for the state variables dX 1t

dX 2t

dX 3t

=

0 0 00 λ −λ0 0 λ

θQ1θQ2θQ3

−

X 1t

X 2t

X 3t

dt

+

σ11 σ12 σ13

σ21 σ22 σ23

σ31 σ32 σ33

dW 1

t

dW 2t

dW 3t



Zero coupon bond prices are

P (t,T ) = exp(C (t,T ) + B1 (t,T )X 1

t + B2 (t,T )X 2t + B3 (t,T )X 3

t

)and the ODEs for the factor loadings are

dB1(t,T )dt

dB2(t,T )dt

dB3(t,T )dt

=

110

+

0 0 00 λ 00 −λ λ

B1 (t,T )B2 (t,T )B3 (t,T )

and

dC (t,T )

dt= −B (t,T )

′θQ − 1

2

3

∑j=1

(Σ′B (t,T )B (t,T )” Σ

)j .,j

with boundary conditionsC (T ,T ) = B1 (T ,T ) = B2 (T ,T ) = B3 (T ,T ) = 0.



The relationship between the real-world dynamics and the risk-neutraldynamics is

dWt = dWt + Γtdt

and use the essentially affine risk premium specification in which Γt isaffine in the state variables so that

Γt =

γ01

γ02

γ03

+

γ111 γ1

12 γ113

γ121 γ1

22 γ123

γ131 γ1

32 γ133

X 1t

X 2t

X 3t

The P measure dynamics are

dXt = KP[θP − Xt

]dt + ΣdWt

and we are free to choose the mean vector θP and the mean reversionmatrix KP under the P measure and preserve the Q dynamics.



The independent-factor AFNS model has independent state variablesunder the P measure dX 1

t

dX 2t

dX 3t

=

κP11 0 00 κP

22 00 0 κP

33

θP1θP2θP3

−

X 1t

X 2t

X 3t

dt

+

σ11 0 00 σ22 00 0 σ33

dW 1t

dW 2t

dW 3t



The correlated-factor AFNS model has independent state variables underthe P measure dX 1

t

dX 2t

dX 3t

=

κP11 κP

12 κP13

κP21 κP

22 κP23

κP31 κP

32 κP33

θP1θP2θP3

−

X 1t

X 2t

X 3t

dt

+

σ11 0 0σ21 σ22 0σ31 σ32 σ33

dW 1t

dW 2t

dW 3t



The measurement equation isyt (τ1)yt (τ2)

...yt (τN)

=

1 1−e−λτ1

λτ1

1−e−λτ1

λτ1− e−λτ1

1 1−e−λτ2

λτ2

1−e−λτ2

λτ2− e−λτ2

......

...

1 1−e−λτN

λτN

1−e−λτN

λτN− e−λτN

X 1

t

X 2t

X 3t

−

C (τ1)

τ1C (τ2)

τ2...

C (τN )τN

+

εt (τ1)εt (τ2)

...εt (τN)



Estimation of the AFNS uses a Kalman filter maximum likelihoodapproach.The model needs to be converted from continuous to discrete time.The conditional mean vector is

EP [XT |Fs ] =(I − exp

(−KP (T − t)

))θP + exp

(−KP (T − t)

)Xt

and the conditional variance is

V P [XT |Fs ] =∫ (T−t)

0e−K

P sΣΣ′−(KP)

′sds

The state transition equation is

Xt =(I − exp

(−KP∆t

))θP + exp

(−KP∆t

)Xt−1 + ηt

where ∆t is the time between observations.The variance of ηt is

Q =∫ ∆t

0e−K

P sΣΣ′−(KP)

′sds



The AFNS measurement equation is

yt = BXt + C + εt

and the stochastic error term is(εtηt

)∼ N

(00

Q 00 H

)For the Kalman filter we start with unconditional mean and variance ofstate variables under the P measure

X0 = θP , and Σ0 =∫ ∞

0e−K

P sΣΣ′−(KP)

′sds



Denote model parameters by ψ and the information at time t byYt = (y1, y2, . . . , yT ) , then at time t − 1 assume we have the stateupdate Xt−1 and the mean square error matrix Σt−1.The prediction step is

Xt|t−1 = EP [Xt |Yt−1] = ΦX ,0t (ψ) + ΦX ,1

t (ψ)Xt−1

Σt|t−1 = ΦX ,1t (ψ)Σt−1ΦX ,1

t (ψ)′+Qt (ψ)

where

ΦX ,0t (ψ) =

(I − exp

(−KP∆t

))θP

ΦX ,1t (ψ) = exp

(−KP∆t

)Qt (ψ) =

∫ ∆t

0e−K

P sΣΣ′−(KP)

′sds



The time t update step improves Xt|t−1 using the additional informationcontained in Yt .

Xt = E [Xt |Yt ] = Xt|t−1 + Σt|t−1B (ψ)′F−1t νt

Σt = Σt|t−1 − Σt|t−1B (ψ)′F−1t B (ψ)Σt|t−1

where

νt = yt − E [yt |Yt−1] = yt − B (ψ)Xt|t−1 − C (ψ)

Ft = cov (νt) = B (ψ)Σt|t−1B (ψ)′+H (ψ)

H (ψ) = diag(σ2

ε (τ1) , . . . , σ2ε (τN)

)



A single pass of the Kalman filter allows us to evaluate the Gaussian loglikelihood

log l (y1, y2, . . . , yT |ψ) =N

∑t=1

(−1

2N log (2π)− 1

2log |Ft | −

1

2ν′tF−1t νt

)where N is the number of observed yields.This is then maximised with respect to ψ.


Discussion - Affine Arbitrage-Free and Nelson-Siegel Model

Gaussian versus other affine models.

Fitting current yield curve.

Number of factors.

Benefits of closed form solutions.


HJM Framework

HJM (Heath, Jarrow, Morton)


HJM Framework

General framework for arbitrage-free models.

Starts with initial forward-rate curve as input - guaranteed to fitinitial term structure.

Imposes no-arbitrage conditions to derive relationship between driftand volatility of forward curve.


HJM Framework

Consider a one factor model. We will not cover technical conditionsin details.

Forward curve f (t,T ) 0 ≤ t < T with initial forward curve f (0,T )

For fixed maturity T dynamics of f (t,T ) given by

df (t,T ) = α (t,T ) dt + σ (t,T ) dW (t)

f (t,T ) = f (0,T ) +∫ t

0α (s,T ) ds +

∫ t

0σ (s,T ) dW (s)

For the risk free interest rate

r (T ) = limt→T−

f (t,T ) = f (0,T ) +∫ T

0α (s,T ) ds

+∫ T

0σ (s,T ) dW (s)


HJM Framework

Cash account

dB (t) = r (t)B (t) dt

B (t) = B (0) exp

[∫ t

0r (u) du

]= B (0) exp

[ ∫ t0 f (0, u) du +

∫ t0

∫ ts α (s, u) duds

+∫ t0

(∫ ts σ (s, u) du

)dW (s)

]

Uses technical condition to change order of integration in third term∫ t

0

(∫ u

0σ (s, u) dW (s)

)du =

∫ t

0

(∫ t

sσ (s, u) du

)dW (s)


HJM Framework

Zero coupon bond market prices

P (t,T ) = exp

[−∫ T

tf (t, u) du

]= exp

[−∫ t0 f (0, u) du −

∫ t0

∫ Tt α (s, u) duds

−∫ t0

(∫ Tt σ (s, u) du

)dW (s)

]


HJM, Framework

Discounted bond price

Z (t,T ) =P (t,T )

B (t)

= exp

[−∫ T0 f (0, u) du −

∫ t0

∫ Ts α (s, u) duds

+∫ t0 S (s,T ) dW (s)

]

where

S (s,T ) = −(∫ T

sσ (s, u) du

)and S (t,T ) is interpreted as the volatility of P (t,T ) - zero couponbond price volatility for maturity T .


HJM Framework

Using Ito formula we have

dZ (t,T ) = Z (t,T )

[ (12S

2 (t,T )−∫ Tt α (t, u) du

)dt

+S (t,T ) dW (t)

]

Change of measure to make the discounted bond price a martingale

γ (t) =1

2S (t,T )− 1

S (t,T )

∫ T

tα (t, u) du


HJM Framework

Rearranging gives the no arbitrage condition∫ T

tα (t, u) du =

1

2

2

S (t,T )− γ (t) S (t,T )

Differentiate with respect to T

α (t,T ) = σ (t,T ) (γ (t)− S (t,T ))

The original model was

df (t,T ) = α (t,T ) dt + σ (t,T ) dW (t)

= α (t,T ) dt + σ (t,T )(dW (t)− γ (t) dt

)= −σ (t,T ) S (t,T ) dt + σ (t,T ) dW (t)

Forward rate process is completely specified by the volatility functionsσ (s, t) .


HJM Framework

We have for HJM (single factor)r (t) = f (0, t)−

∫ t0 σ (s, t) S (s, t) ds +

∫ t0 σ (s, t) dW (s) .

In general r (t) is not Markov but special cases for volatilityspecifications correspond to cases considered previously.

Setσ (s, t) = σ

for all s and t then we get the Ho-Lee model

r (t) = f (0, t)− 1

2σ2t2 + σW (t)


HJM Framework

Setσ (s, t) = σe−α(t−s)

Then

S (s, t) = −σ

α

(1− e−α(t−s)

)−∫ t

0σ (s, t) S (s, t) ds =

σ2

2α2

(1− e−αt

)2and

r (t) = f (0, t) +σ2

2α2

(1− e−αt

)2+ σ

∫ t

0e−α(t−s)W (t)

which is a form of Hull-White model.


HJM Framework

For log-normal forward rates set

σ (t,T ) = σf (t,T )

Under this volatility structure we have

df (t,T ) = σ2f (t,T )∫ T

tf (t, s) dsdt + σf (t,T ) dW (t)

Here the drift grows as the square of the forward rate and the forwardrate explodes in finite time. This is a characteristic of all log-normalmodels of instantaneous forward rates (but not discrete froward rates).


Market Models

LMM (Libor Market Models),BGM (Brace, Gatarek, Musiela),SABR (stochastic alpha, beta, rho)


Market Models

Models aim to fit market pricing formulae such as Black’s formulaefor caps and swaptions.

They generally use discrete time interest rates such as LIBOR (marketobserved rates).

Black model is used for pricing interest rate options with anassumption of log-normality of interest rates or log-normality of bondor swap prices.

Pricing models often involve approximations and require changes ofmeasure.

We don’t cover details here where our focus is on term structures ofinterest rates but pricing caps, swaptions is important in practice.


BGM (Brace, Gatarek, Musiela)

BGM derive processes followed by market quoted rates in HJMframework.

Uses forward LIBOR rates assumed to evolve as lognormal.

Consistent with Black’s model for pricing interst rate caps.

Each forward rate has time dependent volatility ands correlations withother forward rate.

Standard LMM model does not produce market observed capletvolatility smile/skew.

Local volatility or stchastic volatility models are used to do this.



Select tenor τ which will be required to be log-normal (only one tenorat a time can be log-normal)

We have

f (t, x) the instantaneous forward rate at time t for time t + xL (t, x) the market quoted forward rate at time t for time t + x oftenor τ

We have

1 + τL (t, x) = exp

(∫ x+τ

xf (t, u) du

)Require L (t, x) to have a log-normal process

dL (t, x) = µ (t, x) dt + γ (t, x) L (t, x) dW (t)



BGM show that µ (t, x) must have the form

∂

∂xL (t, x) + L (t, x) γ (t, x) σ (t, x) +

τL2 (t, x)

1 + τL (t, x)|γ (t, x)|2

and σ (t, x) related to γ (t, x) by

σ (t, x) =0, 0 ≤ x ≤ τ

∑b xτ ck=1

τL(t,x−kτ)1+τL(t,x−kτ)

γ (t, x − kτ) τ ≤ x


SABR model

This is a stochastic volatility model which attempts to capture thevolatility smile.

Stands for ”stochastic alpha, beta, rho”.

Details are relevant for interest rate derivative trading, maybe lessrelevant for most actuarial applications.

We only introduce the concept here.


SABR model

Model for forward rates

df (t,T ) = α (t) f (t,T )β dW1 (t)

dα (t) = υα (t) dW2 (t)

E [dW1 (t) dW2 (t)] = ρdt

Different models β = 0 produces a stochastic Gaussian model, β = 1produces a stochastic log-normal model and β = 1

2 producesstochastic CIR model.


Discussion - Market Models

Fitting models used for pricing caps and swaptions - market models(Black model).

Emphasis on volatility and correlations.

Ability to calibrate models to interest rate options data.

Stochastic volatility and fitting market volatility (smiles etc).


Discrete time models and binomial implementation

Discrete time models



For log-normal models there are no closed form bond price expressionsfor the term structure of bond prices.

Many term structure models were developed as discrete time models,especially log-normal models.

We would like to calibrate the model to both current term structureof interest rates and volatility term structure.

We do this on a binomial lattice for a single factor model.



Work in discrete time steps with time steps of ∆t and nodes on abinomial lattice.

Denote by P (i) the price of a zero coupon bond maturing at timei∆t and σR (i) the volatility of the yield of this bond.

u (i) is the median interest rate at time i∆t.

σ (i) is the volatility of the short term interest rate at time i∆t.

r (i , j) is the short term interest rate at time i∆t. at node j applicablefor period [i∆t, (i + 1)∆t] .

d (i , j) the time i∆t, state j value of a discount bond maturing attime (i + 1)∆t so that d (i , j) = 1/ (1 + r (i , j)∆t) .



Use Arrow-Debreu securities in calibrating the binomial model ofinterest rates to market data.

Q (i , j) is the time 0 value of a security pating 1 if node (i , j) isreached and 0 otherwise with Q (0, 0) = 1.

The time 0 price of a discount bond maturing at time (i + 1)∆t is

P (i + 1) = ∑j

Q (i , j) d (i , j)

Note thatP (i)− exp (−R (i) i∆t)

Calibration uses forward induction and moves UP and DOWN on thelattice to calibrate to volatility.



Arrow-Debreu prices at time i∆t are updated using the time(i − 1)∆t prices using

Q (i , i) =1

2Q (i − 1, i − 1) d (i − 1, i − 1)

Q (i , j) =1

2Q (i − 1, j − 1) d (i − 1, j − 1)

+1

2Q (i − 1, j + 1) d (i − 1, j + 1)

Q (i ,−i) =1

2Q (i − 1,−i + 1) d (i − 1,−i + 1)

The short term interest rate at node (i , j) is

r (i , j) = u (i) exp(

σ (i) j√

∆t)



The lattice is constructed starting at time i = 0 with single statej = 0.

At each subsequent time i there are i + 1 states indexed asj = −i ,−i + 2, . . . , i − 2, i representing net moves to the node.

From the initial node (0, 0) we can go to (1, 1) - an UP move, or to(1,−1) - a DOWN move (binomial tree).

At these nodes PU (i) and PD (i)are the prices of a discount bondwith maturity i∆t and RU (i) and RD (i) are the discount bond yieldsat these nodes.

These values are calibrated to be consistent with observed prices P (i)and volatilities σR (i) .



Require

1

1 + r (0, 0)∆t

(1

2PU (i) +

1

2PD (i)

)= P (i) i = 2, . . . ,N

σR (i)√

∆t =1

2ln

lnPU (i)

lnPD (i)i = 2, . . . ,N

These are solved numericallyusing Newton-Raphson by rewriting theequations as

PD (i)exp(2σR (i)√

∆t) + PD (i) = 2P (i) 1 + r (0, 0)∆t

PU (i) = PD (i)exp(2σR (i)√

∆t)



Need state prices at the UP and DOWN nodes so define these asQU (i , j) and QD (i , j) .

By definition QU (1, 1) = 1 and QD (1,−1) = 1.

Then

PU (i + 1) = ∑j

QU (i , j) d (i , j) i = 1, . . . ,N − 1

PD (i + 1) = ∑j

QD (i , j) d (i , j) j ∈ {−i ,−i − 2, . . . , i − 2, i}



State prices are then updated

QU (i , i) =1

2QU (i − 1, i − 1) d (i − 1, i − 1)

QU (i , j) =1

2QU (i − 1, j − 1) d (i − 1, j − 1)

+1

2QU (i − 1, j + 1) d (i − 1, j + 1)

QU (i ,−i) =1

2QU (i − 1,−i + 1) d (i − 1,−i + 1)

QD (i ,−i) =1

2QD (i − 1,−i + 1) d (i − 1,−i + 1)

QD (i , j) =1

2QD (i − 1, j − 1) d (i − 1, j − 1)

+1

2QD (i − 1, j + 1) d (i − 1, j + 1)

QD (i , i − 2) =1

2QD (i − 1, i − 3) d (i − 1, i − 3)



Algorithm for calibration is then as follows.

Set initial values

r (0, 0) = u (0) =e(R(1)∆t) − 1

∆tQU (1, 1) = 1 and QD (1,−1) = 1

σ (0) = σR (1)

d (0, 0) =1

1 + r (0, 0)∆t



Then for i = 1, . . . ,N − 1

solve for PD (i + 1) using Newton-Raphson (or other numericalmethod) then solve for PU (i + 1)use these to solve for u (i) and σ (i) substituting

d (i , j) =1

1 + u (i) exp(

σ (i) j√

∆t)

to get

PU (i + 1) = ∑j

QU (i , j)

1 + u (i) exp(

σ (i) j√

∆t)

PD (i + 1) = ∑j

QD (i , j)

1 + u (i) exp(

σ (i) j√

∆t)

for i = 1, . . . ,N − 1, j ∈ {−i ,−i − 2, . . . , i − 2, i} .



The one period short term interest rates are determined for each nodeusing the calibrated values for u (i) and σ (i) .

State prices can then be updated.

Any set of cash flows on the lattice can now be valued using the stateprices including options on interest rates.

Many other ways of contructing lattices to include mean reversionand other interest rate models.


Summary and Wrap Up

Summary and Wrap Up


Summary and Wrap Up

Aim has been to give a broad overview of Term Structure Modelsfocussing on useful models for determining discount rate curvesconsistent with market data.

Wide range of models and modelling techniques reviewed andsummarised.

Application to guarantees and options in liabilities not covered indetails.

Participants have a foundation for further study - encourage readingof more advanced texts and research articles.


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