do interest rate options contain information about excess returns?

SAFE ( c©2010 Anna Cieslak) 1

New Directions in Term Structure Modeling, SAFE, 2010

Do Interest Rate Options Contain Information about

Excess Returns?

by Caio Almeida, Jeremy Graveline and Scott Joslin

Discussion by Anna Cieslak

University of LuganoInstitute of Finance

June 29, 2010

This paper...

⊲ This paper

Factors

Predictability

Conclusions


... is an important attempt to answer this question by estimating3-factor ATSMs on swaps and interest rate caps jointly!

Thought-provoking results...

� For premia, purely Gaussian models and stochastic vol (SV) mod-els estimated on options deliver similar results, but... SV modelsestimated w/o options lag behind.

� For options, different models give similar, relatively large, pricingerrors.

� For conditional yield volatilities, the best fit comes from a modelestimated w/o options.

How to interpret and reconcile these findings?

Factors in yields and capsSomething in common?

This paper

⊲ Factors

Caps vs swaps

IV vs pc’s

Joint estimation

OLS

Pricing errors

Intuition

JK fit

Predictability

Conclusions


Caps versus swap zeros


� In the last 2 decades, IVs havebeen high in recessions...

1995 1997 1999 2001 2003 2005 2007 20090

10

20

30

40

50

60

a. Caps implied vol

%p.a

.

1y2y3y




� i.e. ... when Fed eases financialconditions to support real econ-omy / fight unemployment.

1995 1997 1999 2001 2003 2005 2007 20090

10

20

30

40

50

60

a. Caps implied vol

%p.a

.

1y2y3y

1995 1997 1999 2001 2003 2005 2007 20090

2

4

6

8b. Swap zero yields

%p.a

.

6m1y2y3y





⇒ Let’s regress cap IVs on yields,one-by-one:

IV nt = β0 + β1y

mt + ut

n = {1, 2, 3, 4, 5, 7, 10y}, m =

{6m, 1y, ..., 10y}.

02

46

810

12

34

57

100.2

0.3

0.4

0.5

0.6

0.7

0.8

yield maturity

R2 from regression of cap IVs on yields

cap maturity

R2





⇒ Let’s regress cap IVs on yields,one-by-one:

IV nt = β0 + β1y

mt + ut

n = {1, 2, 3, 4, 5, 7, 10y}, m =

{6m, 1y, ..., 10y}.

02

46

810

12

34

57

100.2

0.3

0.4

0.5

0.6

0.7

0.8

yield maturity

R2 from regression of cap IVs on yields

cap maturity

R2

Short-term yields explain up to 80% of variation in cap IVs across all cap maturities. The

explanatory power of the yield curve for IVs declines with maturity of yields.

Explaining variation in cap IVs with yield PCs


Initially surprising result... that a 3-factor model can fit both yields and IVs

T-stats and adj. R2 in a regression of cap IVs on yield PCs

IV n

t= β0 + β′

1PCt + ut

Cap maturity, n 1y 2y 3y 4y 5y 7y 10y

A. tstat (NW)

pc1 13.2 14.7 15.0 15.2 15.2 15.9 16.7pc2 12.0 9.1 6.6 5.7 5.0 4.3 3.7pc3 6.4 8.8 8.3 7.4 6.5 5.0 3.6

B. Adj. R2

pc1 61% 66% 68% 69% 69% 71% 71%∆ pc2 18% 13% 11% 10% 9% 7% 6%∆ pc3 7% 10% 10% 9% 8% 6% 4%

Total 87% 89% 89% 87% 86% 84% 81%

...becomes clear from the perspective of this table:

⇒ Factors in yields (esp level) fit a majority of variation in cap IVs!

Do we need the joint estimation?

This paper

Factors

Caps vs swaps

IV vs pc’s

⊲ Joint estimation

OLS

Pricing errors

Intuition

JK fit

Predictability

Conclusions


For simplicity, let’s consider an OLS model for yields without imposing no-arbitrage restrictions:

ymt = am + b

′

mPCt + eyt (1)

The coefficients am and bm give the best linear fit in the sense of minimizingthe mean cross-sectional squared error.

Also, let’s neglect non-linearities in caps, and fit:

IVnt = cn + d

′

nPCt + ect . (2)

Q: How does this simple exercise compare to the performance of the complex

ATSMs estimated using both swap zeros and caps?

Caps IVs explained by pure yield curve factors


95 97 99 01 03 05 07 090

20

40

60

80Implied vol 1y cap

%p.a

.

95 97 99 01 03 05 07 090

20

40

60


%p.a

.

95 97 99 01 03 05 07 090

10

20

30

40


%p.a

.

95 97 99 01 03 05 07 090

10

20

30

40


%p.a

.

Note: Observed, fitted ATM cap IVs, and 95% confidence bands. The IVs are fitted with the 3 first PC obtained from swap zero coupon curve with maturities6m to 10y. All data is from Datastream.

Caps IVs explained by pure yield curve factors


95 97 99 01 03 05 07 090

20

40

60


%p.a

.

95 97 99 01 03 05 07 090

20

40

60


%p.a

.

95 97 99 01 03 05 07 090

10

20

30

40


%p.a

.

95 97 99 01 03 05 07 090

10

20

30

40


%p.a

.

observedfitted95% CI

Note: Observed, fitted ATM cap IVs, and 95% confidence bands. The IVs are fitted with the 3 first PC obtained from swap zero coupon curve with maturities6m to 10y. All data is from Datastream.

Model performance... not so different


Pricing errors in 3 factor models: Regressions vs ATSM

τ 6m 1y 2y 3y 4y 5y 7y 10y

A. Yields (RMSE in bps)

OLS 2.4 3.9 2.0 1.5 2.1 2.0 0.9 2.5Ao

2(3) AGJ 6.8 9.3 — 4.5 6.2 6.6 5.3 —

B. Caps (RMSRE in %)

OLS — 22.7% 17.6% 15.2% 13.6% 12.4% 10.7% 9.2%Ao

2(3) AGJ — 35.5% 14.4% 10.9% 9.6% 9.0% 8.3% 8.9%

Note: I compare results obtained from OLS regressions of yields and cap IVs on three yield PCs with those of theAo

2(3) model reported by Almeida, Graveline and Joslin (AGJ). AGJ estimation uses both yields and caps. Sample

used for the regressions is weekly, 1995:01–2008:06. AGJ results are based on weekly data, 1995:01–2006:02.

My intuition...

� OLS gives a similar fit to cap IVs as Ao2(3) although it does not exploit any information

in derivatives. Neither does great on this front...

� Ao2(3) trades off a bit of the fit of yields to improve on IVs.

� This is how much a 3-factor model can do.

More intuition

This paper

Factors

Caps vs swaps

IV vs pc’s

Joint estimation

OLS

Pricing errors

⊲ Intuition

JK fit

Predictability

Conclusions


Let’s consider A1(3) model with state Xt = (X1t, X2t, Vt) where Vt is the volstate, and ym

t = Am +B′

mXt.

Conditional vol of a m-maturity yield becomes:

vart(ymt+h) = (B′

m ⊗B′

m)× vec [vart(Xt+h)] (3)

= b0 + b1Vt (4)

But... In estimation w/o caps, Vt often becomes the level factor [misspeci-fication, Feller condition].

Q: Does inclusion of caps change this rotation, i.e. Vt is allocated correctly?

Level in yields versus volatility, A1(3)


Volatility fit of the essentially affine A1(3) model, Treasury curve 1970–2003

1970 1975 1980 1985 1990 1995 2000

5

10

15

x 10-3 x 10-3

x 10-3x 10-3

x 10-3x 10-3

3-m

onth

6-m

onth

1970 1975 1980 1985 1990 1995 2000

5

10

15

1970 1975 1980 1985 1990 1995 2000

5

10

15

1970 1975 1980 1985 1990 1995 2000

4

6

8

10

12

1970 1975 1980 1985 1990 1995 2000

3

4

5

6

7

1970 1975 1980 1985 1990 1995 2000

2

3

4

5

6

1-y

ear

2-y

ear

5-y

ear

10-y

ear

Source: Jacobs & Karoui, JFE, p.300. Note: Solid line shows the EGARCH(1,1) volatility estimates with AR(1)

mean equation, dotted line depicts the conditional volatility implied by the essentially affine model A1(3).

Level in yields versus volatility, A1(3)


Volatility fit of the essentially affine A1(3) model, swap curve 1991–2005

x 10-3 x 10-3

x 10-3 x 10-3

x 10-3 x 10-3

6-m

onth

3-m

onth

1991 1996 2001 2005

0

2

4

6

1991 1996 2001 2005

0

1

2

3

4

5

1991 1996 2001 2005

0

1

2

3

1-y

ear

2-y

ear

5-y

ear

10-y

ear

1991 1996 2001 2005

1

1.5

2

2.5

1991 1996 2001 2005

1

1.5

2

2.5

1991 1996 2001 2005

1

1.5

2

2.5

Source: Jacobs & Karoui, JFE, p.300. Note: Solid line shows the EGARCH(1,1) volatility estimates with AR(1)

mean equation, dotted line depicts the conditional volatility implied by the essentially affine model A1(3).

Predictability of excess returnsLooking for benchmarks

This paper

Factors

⊲ Predictability

Predictability

Conclusions


Predictability of excess bond returns


rxnt+12m = α0 + α

′Factorst + εnt+12m

Factor rx2 rx3 rx4 rx5 rx6 rx7 rx8 rx9 rx10

A. Predictive R2

0. CP, fwd 10 43% 47% 49% 50% 51% 51% 50% 49% 47%1. Treasury premia, X1, X2 41% 48% 54% 59% 62% 64% 66% 67% 68%

2. Treasury vol, V11 17% 16% 14% 12% 11% 9% 8% 7% 5%3. IV cap 1y, 7y 32% 32% 31% 30% 28% 26% 24% 22% 20%

4. X, V, Cap (1.+2.+3.) 57% 62% 66% 68% 70% 71% 71% 71% 72%Improvement 4.-1. 16% 14% 12% 9% 8% 6% 5% 4% 3%

B. t-stat for regression 4

X1 -0.5 -0.4 -0.1 0.4 0.8 1.1 1.5 1.8 2.1X2 5.4 6.4 7.4 8.2 8.7 9.1 9.3 9.5 9.6V11 2.1 1.9 1.6 1.2 0.7 0.3 0.0 -0.3 -0.5IV cap 1y 3.2 4.1 4.6 4.7 4.5 4.2 3.9 3.6 3.2IV cap 7y -4.6 -5.0 -5.0 -4.8 -4.5 -4.2 -4.0 -3.7 -3.4

Note: We report predictive regressions of 1-year holding period bond returns on swap zeros, sample 1995:01–2008:06.Panel A. provides adj. R2 obtained from various regressions: 0. Cochrane-Piazzesi factor, 1. Term premia factorsfrom Treasuries (Cieslak&Povala, 2010), 2. Treasury long-maturity volatility (Cieslak&Povala, 2009), 3. Cap IVswith maturities 1y and 7y. Panel B. reports t-stats when all regressors are used jointly, in regression 4.

Predictability of excess bond returns


rxnt+12m = α0 + α

′Factorst + εnt+12m

Factor rx2 rx3 rx4 rx5 rx6 rx7 rx8 rx9 rx10

A. Predictive R2

0. CP, fwd 10 43% 47% 49% 50% 51% 51% 50% 49% 47%1. Treasury premia, X1, X2 41% 48% 54% 59% 62% 64% 66% 67% 68%

2. Treasury vol, V11 17% 16% 14% 12% 11% 9% 8% 7% 5%3. IV cap 1y, 7y 32% 32% 31% 30% 28% 26% 24% 22% 20%

4. X, V, Cap (1.+2.+3.) 57% 62% 66% 68% 70% 71% 71% 71% 72%Improvement 4. over 1. 16% 14% 12% 9% 8% 6% 5% 4% 3%

� Benchmark... Vast predictability (> 60%) comes from factors within the yield curve, i.e.spanned, no derivatives are needed [Cieslak & Povala, 2010].

� Extra... Vol factors (unspanned) and cap IVs add most predictability at short bondmaturities.

⇒ But... your estimated models seem to do the opposite, i.e. improve on the long end[Table 6 and 7].

⇒ ... and imply a much lower degree of predictability than reported here [after accountingfor the holding period horizon].

Conclusions

This paper

Factors

Predictability

⊲ Conclusions

Sum up


Putting things together

This paper

Factors

Predictability

Conclusions

⊲ Sum up


My simple diagnostics could suggest that:

� The estimated models have difficulties in identifying volatility states cor-rectly, but options help to find the right direction.

� There is little statistically significant difference for explaining term premiawithin a 3-factor model with and w/o options data.

� Providing orthogonal information on vols (e.g. from straddles) wouldhelp identify the volatility states and the marginal contribution of deriva-tives for premia.

I would like to see more on:

� ... the model implied states, esp those driving vols in A1(3), A2(3) esti-mation with and w/o options.

� ... evidence that derivatives are indeed key to improvement of these modelson the premia front.

Overall... An interesting project that can enhance our understanding of both

ATSMs, and vols and premia in bonds.

do interest rate options contain information about excess returns?

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