Long Forward Probabilities, Recovery and the Term ...Arrow-Debreu prices (via Perron-Frobenius Theorem) Carr and Yu (2012) I 1D di usions on bounded intervals with regular boundaries

Long Forward Probabilities, Recovery and theTerm Structure of Bond Risk Premiums

Vadim Linetsky

Northwestern University

Supported in part by NSF grants CMMI 1536503 and DMS 1514698.

Vadim Linetsky Long Forward Probabilities, Recovery and the Term Structure of Bond Risk Premiums

Based on:

Likuan Qin and V.L., Positive Eigenfunctions of MarkovianPricing Operators: Hansen-Scheinkman Factorization, RossRecovery and Long Term Pricing, Operations Research (PE).

Likuan Qin and V.L., Long Term Risk: a MartingaleApproach, http://ssrn.com/abstract=2523110 (LT).

V.L., Yutian Nie and Likuan Qin, Long Forward Probabilities,Recovery and the Term Structure of Bond Risk Premiums,http://ssrn.com/abstract=2721366.


http://ssrn.com/abstract=2523110

http://ssrn.com/abstract=2721366

Long Term Factorization

Stochastic discount factor (pricing kernel) S assigns prices torisky future payoffs:

Pt,T (Y ) = EP[STY

St

∣∣∣∣Ft

].

Long-term factorization of PK:

St =1

BtMt ,

where Bt is the long bond so that 1/Bt discounts at the rateof return on the long bond and Mt is a martingale.Alvarez and Jermann (Econometrica, 2005) first introducedthis factorization in a discrete-time ergodic setting.Hansen and Scheinkman, Long Term Risk: An OperatorApproach, Econometrica 2009, gave a study incontinuous-time ergodic Markovian environments andidentified the factors in terms of the principal eigenfunctionand eigenvalue of the Markovian pricing operator.


Ross’ Recovery Theorem

Ross’ question: can we uniquely recover physical probabilitiesP from currently observed asset prices?

Ross’ Recovery Theorem (J of Finance, 2015):

I All uncertainty is generated by a finite-state, discrete timeirreducible Markov chain

I Transition-independent pricing kernelI Then there is a unique recovery of transition probabilities from

Arrow-Debreu prices (via Perron-Frobenius Theorem)

Carr and Yu (2012)

I 1D diffusions on bounded intervals with regular boundaries

Walden (2013)

I 1D diffusions on RQ & L (2014, PE)

I Detailed analysis for general Markov processes (Borel rightprocesses)


Connection between Long Term Factorization and Ross’Recovery

Hansen and Scheinkman (2014), Borovicka, Hansen andScheinkman (Misspecified Recovery, 2014, to appear in J ofFinance), Q & L (2014, LT and PE) for generalcontinuous-time Markov models (also closely related resultsfor discrete-time, finite state Markov chains in Martin andRoss (2013)):

I Connect Ross’ recovery to the long-term factorization of thepricing kernel

I Identify Ross’ transition independence assumption with settingthe martingale component in the long-term factorization tounity:

Mt = 1, St =1

Bt.


Q & L (2014) Long Term Risk: A Martingale Approach

General (non-Markovian) semimartingale environment

Give a sufficient condition for convergence in semimartingaletopology of trading strategies investing in zero-coupon bondsand rolling over to the long bond

Show convergence in total variation of T -forward measuresQT to the long forward measure L = Q∞

Obtain long-term factorization of HJM models

Restricting to ergodic Markovian environments, recoverHansen & Scheinkman factorization in terms of the principaleigenvalue and eigenfunction of the Markovian pricingoperator

Show that Ross’ recovery identifies P = L and implies growthoptimality of the long bond


Semimartingale Pricing Kernels

Start with (Ω,F , (Ft)t≥0,P) with the usual hypothesis.

A semimartingale pricing kernel (St)t≥0 is assumed to satisfy:

Strict positivity: S and S− are strictly positive,Normalization: S0 = 1,Integrability: EP[ST

St] <∞ for all T ≥ t ≥ 0.

Price at time t of a payoff Y ∈ FT at time T > t:

Pt,T (Y ) = EP[STY

St

∣∣∣∣Ft

].


Bonds and Forward Measures QT

Zero-coupon bonds:

PTt := EP

[STSt

∣∣∣∣Ft

].

For each T > 0, a roll-over strategy BT invests $1 at timezero in PT

0 , at time T rolls over into P2TT for [T , 2T ], etc.:

BTt =

P(k+1)Tt∏k

i=0 P(i+1)TiT

, t ∈ [kT , (k + 1)T ).

P-Martingales and T -forward factorizations:

MTt := StB

Tt , St =

1

BTt

MTt , t ≥ 0.

For each T > 0, the T -forward measure QT (Jarrow 1987,Geman 1989, Jamshidian 1989):

QT |Ft = MTt P|Ft , t ≥ 0.


The Short-term Limit: Risk-Neutral Measure Q = Q0

If S is a special semimartingale, due to our strict positivityassumption there is a multiplicative decomposition:

St = e−DtMt ,

M is a local martingale and D is predictable.

If M is a martingale, then eDt can be interpreted as therisk-free asset (“implied savings account” Doberlein andSchweizer (2001)), and M defines the RN measure:

Q|Ft = MtP|Ft , t ≥ 0.

Under technical conditions, Doberlein and Schweizer (2001)prove that eDt coincides with the limit B0

t of roll-overstrategies BT

t as T ↓ 0 (“classical savings account”).

Further, when S is a supermartingale, D is non-decreasing.


The Long-Term Limit: Long Forward Measure L = Q∞

Theorem (Q& L (2014, LT))

Assume that for each t > 0 there exists a positive random variableM∞t > 0 such that

limT→∞

EP[|MTt −M∞t |] = 0.

(i) Positive P-martingales (MTt )t≥0 converge to a positive

P-martingale (M∞t )t≥0 in Emery’s semimartingale topology.(ii) Positive semimartingales (BT

t )t≥0 converge to a positivesemimartingale (B∞t )t≥0 in semimartingale topology.(iii) Forward measures QT converge in total variation to anequivalent measure Q∞ on each Ft , and

Q∞|Ft = M∞t P|Ft , t ≥ 0.


The Long-Term Limit: Long Forward Measure L = Q∞

We call Q∞ the long forward measure and denote it L.

We call (B∞t )t≥0 the long bond.

Long-term factorization of the semimartingale pricing kernel

St =1

B∞tM∞t

into discounting at the rate of return on the long bond and amartingale component.

Extends Alvarez and Jermann (2005) and Hansen andScheinkman (2009) to general semimartingale environmentswithout Markovian assumption.

Under L, the long bond B∞t is the numeraire asset.

The long term factorization is a general semimartingalephenomenon, not an artifact of Markovian models.


Risk-Return Trade-off under the Long Forward Measure

Theorem (Q & L (2014, LT))

(i) The long bond is growth optimal under L, that is, it has thehighest expected log return under L among all assets priced by thepricing kernel S .(ii) The Sharpe ratio of any asset priced by the PK S takes theform under L:

ELt

[RVt,t+τ

]− R f

t,t+τ

σLt

(RVt,t+τ

) = −corrLt(RVt,t+τ ,

1

R∞t,t+τ

)R ft,t+τσ

Lt

(1

R∞t,t+τ

)where RV

t,t+τ , R ft,t+τ and R∞t,t+τ is the return from holding asset

V , risk-free zero-coupon-bond and long bond from t to t + τ .


Risk-Return Trade-off under the Long Forward Measure -Cont.

Proposition (Q & L (2014, LT))

(i) Under diffusion setting,

limτ↓0

corrLt(R∞t,t+τ , 1/R

∞t,t+τ

)= −1.

(ii) If furthermore the risk free asset exists,

limτ↓0

corrLt(R0t,t+τ , 1/R

∞t,t+τ

)= 0,

where R0t,t+τ = B0

t+τ/B0t is the return on the risk free asset from t

to t + τ .


A Quartet of Measures


Markovian Pricing Kernels

X is a conservative Borel right process (Borel topology on thestate space, strong Markov, right continuous paths). (Ft)t≥0is generated by X .

Pricing kernel S is a positive semimartingale multiplicativefunctional of X :

St+s(ω) = St(ω)Ss(θt(ω)),

where θs is the shift operator, θs : Ω→ Ω,

Xs(θt(ω)) = Xt+s(ω).

Pricing operators (Pt)t≥0:

(Pt f )(x) = EPx [St f (Xt)]

for any Borel payoff f (x) for which expectation is well defined.


Positive Eigenfunctions and Eigen-Measures

Suppose Pt has an eigenfunction 0 < π(x) <∞:

Ptπ(x) = e−λtπ(x)

for each t > 0, x ∈ E , and some λ ∈ R.

The process

Mπt = Ste

λt π(Xt)

π(X0)

is a positive P-martingale, and PK admits aneigen-factorization (Hansen-Scheinkman (2009)):

St = e−λtπ(X0)

π(Xt)Mπ

t .

We can define an eigen-measure Qπ,

Qπ|Ft := Mπt P|Ft , (Pt f )(x) = e−λtπ(x)EQπ

x

[f (Xt)

π(Xt)

].


Long Forward Measure as an Eigen-Measure


Suppose the Markovian pricing kernel satisfies the sufficientcondition for long term factorization under Px for each initial statex ∈ E . Then, under some regularity condition, the long bond isidentified with a positive multiplicative functional of X in thetransition independent form:

B∞t = eλLtπL(Xt)

πL(x),

where πL(x) is a positive eigenfunction of the pricing operators(Pt)t≥0 with the eigenvalues e−λLt for some λL ∈ R. The longforward measure L is identified with the correspondingeigen-measure QπL .


Uniqueness of Recurrent Eigenfunction πR

Theorem (Q & L (2014, PE))

There is at most one positive eigenfunction πR such that X isrecurrent under the corresponding eigen-measure QπR .

Proof is essentially based on the fact that for a recurrentMarkov process excessive functions are constant.

Q & L (2014, PE) give several sets of sufficient conditions forexistence

and explicitly verify existence in many financial models (incl.affine, quadratic).


Ergodicity Identifies QπR = L


Assume X has a stationary distribution µ under QπR and thereexist c > 0, α > 0 and T0 > 0 s.t. for each T ≥ T0

|EQπR

x [f (XT )]− Eµ[f (X )]| ≤ c

πR(x)e−αT

for each f s.t. |f (x)| ≤ 1πR(x)

. Then (BTt )t≥0 converge in

semimartingale topology to the positive semimartingale

B∞t = eλtπR(Xt)

πR(X0),

MπRt = M∞t , QπR = L, and, assuming

∫E (1/πR)dµ <∞,

PTt = Ce−λ(T−t)πR(Xt) + O(e−(λ+α)(T−t)), C = Eµ[1/πR(X )].


Ross’ Recovery under Recurrence: P = QπR

Ross’ assumption of transition independence of PK:

St = e−λth(Xt)

h(X0)

for some positive h.1h is a positive eigenfunction, and PK has the factorization

St = e−λtπ(X0)

π(Xt)Mπ

t ,

where π = 1h and Mπ

t = 1.

P = Qπ is an eigen-measure.

If we further assume X is recurrent under P, then

P = QπR .


Ross’ Recovery under Ergodicity: P = L

Combining Ross’ transition independence assumption withrecurrence yields a unique recovery

P = QπR .

Strengthening to ergodicity we arrive at the furtheridentification

P = QπR = L.


Ross’ Recovery in Semimartingale Models without theMarkov Property

Ross’ assumption can be extended to non-Markoviansemimartingale environments by directly assuming that

M∞t = 1.

This leads to

St =1

B∞t

and idenfiticationP = L.

Under this assumption, the long bond is the numeraire assetunder P and is growth optimal by Jensen’s inequality.


Empirical Exploration

Data: 1993-2015 US Treasury yield curves from FRED (St. LouisFed web site; same data available from US Treasury).

Figure: Bootstrapped zero-coupon yield curves


Modeling Term Structure at the Zero Lower Bound (ZLB)

The zero interest rate regime Dec 2008-2015 is a challenge toconventional interest rate models.

Affine models cannot deal with ZLB.

Gaussian models admit negative rates, while CIR-type havevanishing volatilities for bonds of all maturities at the ZLB.


Modeling Term Structure at the ZLB Cont.

Shadow rate idea due to F. Black (J. of Finance, 1995)“Interest Rates as Options”: nominal short rate is a positivepart (due to the option to convert to currency) of a shadowrate that can get negative.

Gorovoi and Linetsky (Risk 2003, Mathematical Finance2004): solved with Vasicek shadow rate and calibrated toJapanese government bonds.

Adopted by the Bank of Japan in 2005 (Baba et al. 2005).

Kim and Singleton (J of Econometrics, 2012): extended andestimated 2-factor shadow rate models on JGB data.

Shadow models in use by central banks post-crisis.

We are working on a more general class based on SDEs withsticky boundaries that nest shadow rate models.


B-QG2 Shadow Rate Model

Estimate the same specification for 2-factor shadow ratequadratic Gaussian model as Kim and Singleton.

The state variable X is a 2D Gaussian diffusion under P.

The market price of Brownian risk is affine in X , so that Xremains 2D Gaussian diffusion under Q.

The short rate is a positive part of the shifted quadraticfunction in X .

Estimation: extended Kalman filter, with the bond pricingPDE solved via ADI finite difference scheme, and KNITROnon-linear optimizer.


Estimation Results: P and Q Dynamics

Short rater(Xt) = (−0.0046 + 0.27X 2

1,t + 0.18X1,tX2,t + 0.05X 22,t)

+.

State Vector SDEs:

dXt =

[0.65 00.22 0.04

]([−0.050.77

]− Xt

)dt +

[0.1 00 0.1

]dBP

t .

dXt =

[0.32 0.040.64 0.08

]([0.93−5.92

]− Xt

)dt +

[0.1 00 0.1

]dBQ

t .

Market price of risk

λP(Xt) =

[−0.89−0.96

]+

[−3.33 0.424.21 0.40

]Xt .


Estimation Results: The Filtered Path of Shadow Rate

Year1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016

Yie

ld (

%)

-1

0

1

2

3

4

5

6

7

3 Month RateShadow Rate

Year1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

x1x2


Principal Eigenfunction and Eigenvalue

As time to maturity T increases, the zero-coupon bond pricebehaves asymptotically as (the long bond asymptotics):

P(T , x) ∼ Ce−λTπ(x).

Using estimated Q measure parameters, the principaleigenvalue and eigenfunction are determined numerically byfinite differences: λ ≈ 2.82%

Figure: Shape of principal eigenfunctionVadim Linetsky Long Forward Probabilities, Recovery and the Term Structure of Bond Risk Premiums

Recovering L Measure Dynamics

In diffusion models, the market price of risk under L (Ross’Recovery) is recovered in terms of the principal eigenfunction(σ is the volatility matrix in the SDE):

λLi (x) =∑j

σji (x)∂j log π(x).

Numerical result shows that λL is well approximated by alinear function in the range [−0.3 0.2]× [−0.1 1.2], whichcontains the range of filtered state variables. In particular,

λL(Xt) ≈[

0.16−0.10

]+

[−0.38 0.170.17 −0.12

]Xt .


Comparing P, L and Q

By inspection of market prices of risk, we observe that the Lmeasure dynamics is generally closer to Q (from which it isrecovered), than to the estimated P.

Recall that the transition independence assumption results inthe identification P = L. In contrast, in our results we seesignificant differences between estimated P and recovered L.

This is not surprising. Recall that P = L implies that the longbond is growth optimal.

In contrast, Frazzini and Pedersen (2014, J of FinancialEconomics) document that their “Betting Against theBeta” (BAB) factor levering up shorter maturity bonds torisk parity with longer maturity bonds and shorting longermaturity bonds yields Sharpe ratio of 0.81 in the US Treasurybond market during 1952-2012.


How far apart are P vs. L forecasts? Test of P = L

Using estimated L and P dynamics, we can write down theinstantaneous volatility of the martingale component:

v(x) ≈[−1.055−0.863

]+

[−2.946 0.2464.045 0.525

] [x1x2

].

When P = L, the martingale component is trivial, i.e.v(x) = 0. Thus we can test the hypothesis P = L by testingeach of the component of v(x) being equal to 0.

v1 v2 v11 v21 v220.00% 0.00% 0.08% 0.04% 0.00%

Table: p-values for vi = 0 and vij = 0.


How far apart are P vs. L forecasts? The Timing of FedLift-off

We estimate the implied distribution of the first passage timeof the short rate above 25 bps as the proxy for the timing ofthe Fed zero interest rate policy lift-off as of August 19, 2015.

Median Mean

P 0.33 1.07Q 0.17 0.34L 0.16 0.32

Year0 0.5 1 1.5 2 2.5 3 3.5

Pro

babi

lity

0

0.1

0.2

0.3

0.4

0.5

0.6

P forecastQ forecastL forecast


How far apart are P vs. L forecasts? The Timing of FedLift-off

We estimate the implied distribution of the first passage timeof the short rate above 25 bps as the proxy for the timing ofthe Fed zero interest rate policy lift-off as of Dec. 30, 2011.

Median Mean

P 2.13 2.83Q 1.34 1.47L 1.32 1.46

Year0 1 2 3 4 5 6 7

Pro

babi

lity

0

0.02

0.04

0.06

0.08

0.1

0.12

P forecastQ forecastL forecast


Bond risk premiums

Maturity0 5 10 15 20 25 30

Shar

pe ra

tio

-0.1

0

0.1

0.2

0.3

0.4

0.5

RealizedP forecastL forecast

Figure: Realized, P forecast and L forecast Sharpe ratio over 3 monthholding period.


Further work

Refining interest rate modeling at the zero lower bound:general sticky boundary models, stochastic volatility, linkswith macroeconomic variables.

Explorations of market-implied forecasts.

Looking at other markets.


Documents

Long Forward Probabilities, Recovery and the Term ...Arrow-Debreu prices (via Perron-Frobenius Theorem) Carr and Yu (2012) I 1D di usions on bounded intervals with regular boundaries