Beyond Black's model

To address the concern that the assumptions behind the Black model can be violated in practice and might affect the effectiveness of our hedges, we analyze the robustness of our main findings by re-running our strategies on the Vasicek (1977) model and the SABR model.

6.5.3.A Vasicek model

The assumptions behind the Vasicek model are almost equal to the Black model. The only difference is that the Vasicek model assumes a constant absolute volatility (normal distribution) while the Black model assumes a constant relative volatility (lognormal distribution).74 The consequence is that the gamma of a straddle in the Vasicek model differs from the gamma of the Black model:

ΓVasicek=

wkN

wFN=_Vaq‰sUh_√bXx(Y) + x(−Y)Z (6-14) where E_{Jy is the absolute volatility that is equal to E. Numerically the delta-gamma neutral hedge on basis of the Vasicek model is exactly the same as the delta-gamma neutral hedge on basis of Black's gamma (see Section 6.2.3). The delta-vega neutral hedge ratio is different though. The vega risk parameter on basis of the Vasicek model is equal to:

vVasicek=

wk waq‰s=

Uh√b

V Xx(Y) + x(−Y)Z (6-15)

To obtain a delta-vega neutral hedge we can now derive that the hedge ratio on basis of the Vasicek model should be equal to √b

MbŠ‹Œ‹

Table 6-9 shows the return statistics for the delta-vega neutral strategies. For all markets we find comparable statistics to our main findings reported in Table 6-4. This suggests that our findings are robust for assuming a constant absolute volatility.

74 _{In the Vasicek model the volatility is measured as an absolute value and not relative to the underlying swap}

forward as in the Black model. For example, a volatility of 50% for a swap rate of 2% implies an absolute volatility of 100 bp. The same absolute volatility of 100 bp leads to a relative volatility of 20% for a swap rate of 5%.

Table 6-9: Summary statistics for trading strategies on the Vasicek model Vega neutral

Mean t-stat Stdev Sharpe

USD 3 vs 6 0.86% 2.60 1.53% 0.56 6 vs 9 0.38% 2.10 0.85% 0.45 9 vs 12 0.25% 1.70 0.68% 0.37 3 vs 12 1.28% 2.44 2.41% 0.53 EUR 3 vs 6 0.52% 2.21 0.88% 0.59 6 vs 9 0.11% 0.79 0.52% 0.22 9 vs 12 0.15% 1.40 0.43% 0.35 3 vs 12 0.69% 1.85 1.36% 0.51 JPY 3 vs 6 0.78% 3.32 1.01% 0.78 6 vs 9 0.23% 1.41 0.69% 0.34 9 vs 12 0.27% 2.04 0.48% 0.56 3 vs 12 1.10% 2.84 1.61% 0.68 GBP 3 vs 6 1.09% 5.06 0.90% 1.21 6 vs 9 0.44% 3.57 0.54% 0.82 9 vs 12 0.22% 1.76 0.44% 0.50 3 vs 12 1.52% 4.54 1.32% 1.16 EQW 3 vs 6 0.81% 4.34 0.74% 1.09 6 vs 9 0.29% 2.73 0.41% 0.71 9 vs 12 0.22% 2.53 0.33% 0.67 3 vs 12 1.15% 3.82 1.16% 0.99

Note: This table reports annualized summary statistics of non-overlapping monthly returns on long-short

swaption straddles strategies. Each month a swaption straddle with the lowest maturity is sold with a notional of one, and held for one month. At the same time an additional long straddle position, with a higher maturity, is bought at a notional that neutralizes the vega exposure of the combined long-short position according to the Vasicek (1977) model. The straddles are initiated on the last business day of the month. In addition to the USD, EUR, JPY and GBP markets, we compute the summary statistics of an equally-weighted (EQW) portfolio. Summary statistics include annualized sample mean (Mean), annualized standard deviation (Stdev), skewness (Skew), kurtosis (Kurt), first-order autocorrelation (AC(1)), annualized Sharpe ratio and the percentage of months with a positive return (Hit). t-statistics are adjusted according to Newey and West (1987, 1994) to correct for heteroscedasticity and serial correlation up to four lags. The number of observations in each series is 189 and runs from April 1996 to December 2011.

6.5.3.B SABR model

This section considers a stochastic volatility model as alternative to the deterministic volatility models that we have used so far. We select the SABR model, introduced in Hagan et al. (2002), because it is widely used in the market (Rebonato et al., 2009).

The SABR model allows for a relation between the implied volatility (E) and the underlying swap forward rate F. For this reason the delta in the SABR model (∆_khŽj) differs from the delta in the Black model. We use the (Bartlett, 2006) modified SABR delta.

∆khŽj= ∆ +vwa wF+ wa w ‘ν F’“ (6-16)

where ∆ and v on the right hand side are the original Black delta (7) and Black vega (9) respectively.

According to the SABR model, the at-the-money straddle position is not necessarily delta neutral. Therefore, we analyze a SABR delta neutral strategy using a long-short combination of two straddles, instead of a delta-gamma neutral strategy. Likewise, we also analyze a SABR vega neutral strategy using the adjusted vega formula from Bartlett (2006) with the caveat that the vega hedge on basis of the SABR model is not necessarily delta neutral under the SABR model.

v_khŽj=vwa

w+wawF‘F

’

ν “ (6-17) Similarly to our main analysis, we take a short position of one straddle contract in the shorter-term maturity and a long position in the longer-term maturity with a hedge ratio that either neutralizes the estimated SABR delta or SABR vega. For the empirical analysis we have gathered swaption smile data for USD swaptions from JP Morgan because we need smile data to estimate the SABR model. We estimate the parameters of the SABR model with a fixed beta parameter of 0.5, which seems to be market practice in the U.S. according to Rebonato et al. (2009).

Table 6-10: Summary statistics for trading strategies on SABR model

Delta neutral Vega neutral

Mean t-stat Stdev Sharpe Mean t-stat Stdev Sharpe

USD 3 vs 6 0.96% 2.36 1.75% 0.55 0.89% 2.42 1.48% 0.60

6 vs 9 1.27% 1.79 3.42% 0.37 0.31% 1.77 0.91% 0.34

9 vs 12 0.46% 1.19 1.19% 0.39 0.19% 1.01 0.63% 0.30

3 vs 12 1.77% 2.14 2.84% 0.62 1.21% 2.10 2.40% 0.50

Note: This table reports annualized summary statistics of non-overlapping monthly returns on long-short

swaption straddles strategies using the SABR model for hedging. The combined long-short straddles position either neutralizes the vega or the delta exposure according to the SABR greeks. The straddles are initiated on the last business day of the month. Summary statistics include annualized sample mean (Mean), annualized standard deviation (Stdev), skewness (Skew), kurtosis (Kurt), first-order autocorrelation (AC(1)), annualized Sharpe ratio and the percentage of months with a positive return (Hit). t-statistics are adjusted according to Newey and West (1987, 1994) to correct for heteroscedasticity and serial correlation up to four lags. The number of observations in each series is 189 and runs from April 1996 to December 2011.

Table 6-10 tabulates the summary statistics for the delta neutral and vega neutral long-short strategies using the SABR model for hedging. The average returns are positive across all maturities for both delta and vega neutral strategies. Both strategies have statistically significant means for 3 vs 6 and 6 vs 9-month maturity combinations at a 90% confidence level. In general, the vega neutral returns and Sharpe ratios under the SABR

model seem comparable to the returns and Sharpe ratios under the Black model. For example the 3 vs 6-month strategy has a return (Sharpe ratio) of 0.89% (0.60) under the SABR model and 0.85% (0.54) under the Black model. Based on these results we conclude that our main findings are robust.