RIDING THE SWAPTION CURVE

Johan Duyvesteyn Gerben de Zwart

TopQuants Autumn event Amsterdam – 18 November 2015 MOTIVATION

Volatility risk is priced in swaption markets

– Negative premium: realized volatility is lower than

– Investors are willing to pay a compensation to volatility sellers

– Supported by various research methods and many authors

Term structure across the swaption maturity spectrum

– Higher volatility risk premium for low maturities

– Very little attention in the literature

2 LITERATURE OVERVIEW

Volatility risk premium Term structure VRP

Goodman & Ho (1997) Hedging Based tests THIS STUDY Duarte, Longstaff & Yu (2007)

Almeida & Vicente (2009) Fornari (2010) Model free tests Mueller, Vedolin & Yen (2011) Fornari (2010) Merener(2012) Trolle & Schwartz (2012)

3 HEDGING-BASED TEST

Straddle refresher • Combination of call and put • Very sensitive to changes in volatility • Risk exposures measured by : delta, gamma, vega

Hedging-based test • Isolate movement in volatility and neutralize interference from directional movement in underlying instrument • The return on buying a and dynamically delta-hedging is related to the volatility risk premium (Low & Zhang, 2005)

4 THIS STUDY

We develop a novel hedging-based test to empirically examine the presence of a term structure of the volatility risk premium in the swaption market

- Long-short combinations of two at-the-money swaption with different maturities - Analyse the profit/loss to examine the difference in the VRP between maturities

Short: short term Long: longer term straddle straddle 3M 6M 9M 12M premium Volatility Risk Volatility Swaption maturity

5 DATA

Sample: Four largest and most liquid markets

– U.S., EMU, Japan, U.K.

– Daily pricing data (swap yields, ATM implied volatilities)

– April 1996 – 2011

Data source: Bloomberg

– Cross-validation with anonymous broker

– Swaption maturities: 3, 6, 9, 12 months

– 10 year swap tenor

Black pricing model

– Either Gamma or Vega neutral straddle combinations

6 EMPIRICAL RESULTS – MAIN RESULTS

1.20% Vega neutral Gamma neutral

1.00%

0.80%

0.60%

0.40% Annualized return Annualized

0.20%

0.00% 3 vs 6 6 vs 9 9 vs 12 3 vs 6 6 vs 9 9 vs 12 3 vs 6 6 vs 9 9 vs 12 3 vs 6 6 vs 9 9 vs 12 US UK EMU Japan

7 SUMMARY EMPIRICAL RESULTS

Observations

• Positive returns for all markets and maturity combinations

• Majority of returns significant

• Term structure visible

The empirical results indicate

• Existence of term structure in volatility risk premium across all 4 markets

• Incrementally decreasing term structure

8 ROBUSTNESS

Additional analysis to improve the intuition

• Macro economic announcements (Fornari, 2010; Andersen et al ,2007)

• Jump risk (Branger & Schlag, 2008)

• Volatility risk (Cremers et al. 2015)

• Discrete trading (Branger & Schlag, 2008)

9 DAY-OF-THE-MONTH EFFECT

0.9%

0.8%

0.7%

0.6%

0.5%

0.4%

0.3%

0.2%

Annualized equally weighted return 0.1% 3M vs 6M 6M vs 9M 9M vs 12M 0.0% 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Working day of the month (1 = first day of the month)

10 JUMP RISK

1.20% Vega neutral Vega neutral ex Jumps Gamma neutral Gamma neutral ex jumps 1.00%

0.80%

0.60%

0.40% Annualized return Annualized

0.20%

0.00% 3 vs 6 6 vs 9 9 vs 12 3 vs 6 6 vs 9 9 vs 12 3 vs 6 6 vs 9 9 vs 12 3 vs 6 6 vs 9 9 vs 12 US UK EMU Japan

11 VOLATILITY RISK

Vega neutral Vega neutral excluding volatility Gamma neutral Gamma neutral excluding volatility 1.60%

1.40%

1.20%

1.00%

0.80%

0.60% Annualized return Annualized 0.40%

0.20%

0.00% 3 vs 6 6 vs 9 9 vs 12 3 vs 6 6 vs 9 9 vs 12 3 vs 6 6 vs 9 9 vs 12 3 vs 6 6 vs 9 9 vs 12 US UK EMU Japan

12 DISCRETE TRADING – DELTA EXPOSURE

0.80 Straddle – 3 months US swaption

0.70

0.60

0.50

0.40 Delta 0.30

0.20

0.10

0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Day after opening the straddle

* 3 months US swaption

13 DISCRETE TRADING – DELTA EXPOSURE

Delta - Gamma neutral straddle combination* 0.80 0.70

0.60 0.50 0.40 Delta 0.30

0.20 0.10 0.00 -0.10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Day after opening the straddle

* 3 and 6 months US swaptions

14 DELTA RISK EXPOSURE - VEGA NEUTRAL

Delta - Vega neutral straddle combination* 0.80 0.70 0.60 0.50 0.40

Delta 0.30 0.20 0.10 0.00 -0.10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Day after opening the straddle

* 3 and 6 months US swaptions

15 STOCHASTIC MODELS

1.40% Vasicek Black SABR

1.20%

1.00%

0.80%

0.60%

0.40%

0.20%

0.00% 3 vs 6 6 vs 9 9 vs 12 3 vs 12 16 ECONOMIC IMPORTANCE

• Transaction costs are typically not included in related literature

• Our framework enables analyzing the economic importance

Break-even bid-ask spreads (3 vs 12)

0.70 Bid-offer in volatility points gamma neutral vega neutral 0.60 mix 50/50

0.50

0.40

0.30

0.20

0.10

0.00 17 USD EUR JPY GBP EQW ECONOMIC IMPORTANCE

• Strong diversification vega neutral and gamma neutral strategies 140 Vega neutral 135 Gamma neutral 130Break-even50/50 bid-ask combination spreads (3 vs 12) 125

120

115

110

105

100

95

90

18 SUMMARY AND CONCLUSION

• First paper to provide an hedging-based framework for testing the existence of a term structure in the volatility risk premium

• Our empirical results for the swaption market

– indicate a term structure in the volatility risk premium in all 4 major swaption markets

– suggest that the term structure is affected by both jump risk and volatility risk corroborating the equity market findings of Cremers et al. (2015)

– are robust for Jump Risk, Macro Economic Announcements and discrete trading biases

• Journal of Banking and Finance, Volume 59, October 2015, Pages 57–75 http://www.sciencedirect.com/science/article/pii/S0378426615001545 19