DELTA HEDGING VEGA RISK? Stéphane CRÉPEY, Évry University stephane.crepey@univ-evry.fr QMF Conference, Sydney, December 17 2004 Figure 1: The Volatility Smile (E. Derman)
Contents 1 Basics of the smile 1 2 Local volatility 4 3 Implied delta versus delta 6 4 Theoretical analysis 9 5 Empirical comparison 15 6 Markets & Models 25 S. Crépey Page 2
1 Basics of the smile One risky asset S (equity, currency or index) with value S 0 at the current date t 0 = 0 Call option with maturity T > t 0 and strike K on S, payoff (S T K) + at T Market price π T,K of the call at t 0 Arbitrage Pricing Theory Existence of a risk-neutral probability P such that (assuming nil interest rates) π T,K = E t 0,S 0 P (S T K) + Black Scholes model with constant volatility σ of S E t 0,S 0 P ds t = σs t dw t under P (S T K) + = Π BS T,K (t 0, S 0 ; σ) S. Crépey Page 1
Implied volatility of the option Constant Σ T,K such that π T,K = Π BS T,K(t 0, S 0 ; Σ T,K ) Market expectations of future realized volatility between (t 0, S 0 ) and (T, K), + spread Hedge slippage and liquidity holes, jumps and stochastic volatility risk premia Empirical smile of volatility Σ T,K Black 1990, I sometimes wonder why people still use the Black Scholes formula Fitting the smile Calibration of the RN dynamics of the underlyer Local volatility Dupire 94 σ σ(t, S); CR-IVF Hull Suo 2002 Spot stochastic volatility Hull White 87, Heston 93 Stochastic volatility + Jumps Bates 96 General Affine Jump-Diffusion models Duffie Pan Singleton 2000 S. Crépey Page 2
0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.2 0.19 0.18 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.2 0.19 0.18 0.13 340 360 380 Strike 400 420 440 0.1 0.110.12 0.07 0.080.09 Maturity 460 0.04 0.050.06 Figure 2: Golden smile, October 6 2003 S. Crépey Page 3
2 Local volatility Generalized Black Scholes model ds t = σ(t, S t )S t dw t E t,s P (S T K) + = Π T,K (t, S; σ) Special case σ σ(t) Black Scholes formula Π T,K (t, S; σ) = Π BS T,K(t, S; Σ) = SN (d + ) KN (d ) where N is the cumulative normal distribution function, d+/ ln( S K )+/ Σ 2 2 (T t) Σ T t and the volatility (squared) Σ 2 1 T t T t σ 2. S. Crépey Page 4
Well-posed direct pricing problem Black Scholes equation with measurable, positively bounded volatility σ σ(t, S) Crépey 03 t Π 1 2 σ(t, S)S2 S 2 Π = 0, t < T 2 (1) Π T (S K) + Ill-posed inverse calibration problem σ σ(t, S) Σ Σ T,K Find a stable and accurate algorithm for σ σ(t, K), given π = {π T,K ; (T, K) obs} Regularization methods with prior σ 0 Entropic regularization Avellaneda et al 97 Variational regularization Lagnado & Osher 97, Crépey 03 S. Crépey Page 5
3 Implied delta versus delta Daily delta-neutral rebalancing of a portfolio short one option and long shares in the underlyer. Compare the P&L trajectories obtained by cumulating the following increments: δp &L = δπ + δs with positions in the underlyer at date t given by: the delta of the option, that is BS = S Π BS T,K(t, S; Σ T,K ) where Σ T,K denotes the Black Scholes volatility of the option; S. Crépey Page 6
or, alternatively, the delta of the option, that is loc = S Π T,K (t, S; σ) where σ is a volatility function calibrated with the volatility surface observed at date t. Hedging horizon Dumas et al 98 versus Coleman et al 99 Regimes of volatility Derman 99, Alexander 01, Vähämaa 03 Market regime Stable Trending Jumpy Rule of thumb Sticky strike Sticky delta Sticky tree = BS > BS < BS S. Crépey Page 7
6500 6000 stock 5500 Stock 5000 4500 4000 3months vol 3500 0.45 0.4 0.35 0.3 0.25 0.2 0.15 01/06/20 01/07/20 01/08/20 01/09/20 01/10/20 01/06/20 01/07/20 01/08/20 01/09/20 01/10/20 Date ATM FixedStrike Figure 3: Market regimes on the DAX index, May September 2001. S. Crépey Page 8
4 Theoretical analysis Analysis using a volatility model (Vanilla option) If σ σ(t, S), then δp&l loc = δπ + loc δs = 1 2 S2 Γ loc (σ(t, S) 2 τ ( δs S ) 2 ) + o(τ) where Π, loc and Γ loc are the option s price and its Greeks in the model. So, by positivity of Γ loc (vanilla option) Proposition 4.1 Whether δp&l loc is negative or positive depends on whether the realized volatility δs S is above or below the volatility τ σ(t, S). Proposition 4.2 Asymptotically as τ 0: a. δp&l loc is driven by terms in τ and (δs) 2 ; b. δp&l BS = δp&l loc + ( BS loc ) δs remains directional. S. Crépey Page 9
Moreover, following Coleman et al 01 Π T,K (t, S; σ) = Π T,K (t, S; Σ T,K ) loc = BS + vega BS S Σ S Σ K Σ. So, Proposition 4.3 For negative skews: loc BS δp &L BS δp &L loc iff δs 0. Market regime Slow Fast Rally 0 δp&l loc δp&l BS δp&l loc (δp&l BS ) Sell-Off (δp&l BS ) + δp&l loc δp&l BS δp&l loc 0 Table 1: Vanilla option in a negatively skewed volatility model. S. Crépey Page 10
Proposition 4.4 In a negatively skewed volatility model, given a moderately small rebalancing time interval τ (such as one day): a. the delta provides a better hedge in a slow rally or a fast sell-off, while the delta may provide a better hedge, though to a lesser extent, in a fast rally or a slow sell-off; b. provided we have physical as well as negative skewness, the delta is better on average, as well as on average conditionally on the fact that the market is in a fast regime, or on average conditionally on the fact that the market is in a slow regime. S. Crépey Page 11
Analysis in real markets (Vanilla option) Decompose the P&L increments as δp&l loc = ( δπ loc + loc δs) + (δπ loc δπ) (2) δp&l BS = ( δπ loc + BS δs) + (δπ loc δπ) δπ is the increment of the option s market price between t and t + τ δπ loc is the increment of the price predicted by the volatility model calibrated at date t, given the new value of the underlyer at date t + τ On the right-hand side of (2): the first terms behave as in the previous analysis; the market-makers revise their anticipations between t and t + τ according to the new value of the underlyer observed at t + τ. It is reasonable to expect that δπ loc δπ in fast market regimes whereas δπ δπ loc in slow market regimes. S. Crépey Page 12
Then the situation depicted in Table 1 still holds true in the real market. So, Proposition 4.5 Assuming reversion of volatilities towards realized volatilities, proposition 4.4 applies not only in volatility models, but also in real markets. Overall recommendation Use the delta rather than the delta, in a persistently negatively skewed market By comparison, Derman 99 implies that, in negatively skewed markets, the delta should not be worse or could even be better on average conditionally on the fact that the market is in a slow regime, while the delta should be better on average conditionally on the fact that the market is in a fast regime. In Derman 99 the question of knowing which delta is better on average is left unanswered. S. Crépey Page 13
Extensions Same recommendation for hedging an option with mixed Gamma/Vega exposure (reverse barrier) and/or in a persistently positively skewed market. Market regime Slow Fast Rally δp&l BS δp&l loc 0 (δp&l BS ) + δp&l loc Sell-Off δp&l loc (δp&l BS ) 0 δp&l loc δp&l BS Table 2: Negative Gamma/Vega exposure in a negatively skewed market. Market regime Slow Fast Rally (δp&l BS ) + δp&l loc δp&l BS δp&l loc 0 Sell-Off 0 δp&l loc δp&l BS δp&l loc (δp&l BS ) Table 3: Vanilla option in a persistently positively skewed market. S. Crépey Page 14
5 Empirical comparison Stock 5600 5400 5200 5000 4800 4600 4400 4200 4000 3800 3600 current forward 22/08/20 29/08/20 05/09/20 12/09/20 19/09/20 26/09/20 03/10/20 P&LDeltaHedging P&LIncrements 40 20-20 0-40 -60-80 -100-120 -140 22/08/20 29/08/20 05/09/20 12/09/20 19/09/20 26/09/20 03/10/20 40 20-20 0-40 -60-80 -100 22/08/20 29/08/20 05/09/20 12/09/20 19/09/20 26/09/20 03/10/20 Date Figure 4: DAX index 24/08/01 27/09/01 (fast sell-off), P&L. S. Crépey Page 15
Price 450 400 350 300 250 200 150 100 50 0 22/08/20 29/08/20 05/09/20 12/09/20 19/09/20 26/09/20 03/10/20 Delta Gamma 0.6 0.5 0.4 0.3 0.2 0.1 0 22/08/20 29/08/20 05/09/20 12/09/20 19/09/20 26/09/20 03/10/20 0.0005 0.00045 0.0004 0.00035 0.0003 0.00025 0.0002 0.00015 0.0001 5e-05 22/08/20 29/08/20 05/09/20 12/09/20 19/09/20 26/09/20 03/10/20 Date Figure 5: DAX index 24/08/01 27/09/01 (fast sell-off), Greeks. S. Crépey Page 16
Stock 7000 6800 6600 6400 6200 6000 5800 current forward 28/09/19 12/10/19 26/10/19 09/11/19 23/11/19 07/12/19 21/12/19 04/01/20 18/01/20 P&LDeltaHedging P&LIncrements 120 100 80 60 40 20 0-20 28/09/19 12/10/19 26/10/19 09/11/19 23/11/19 07/12/19 21/12/19 04/01/20 18/01/20 50 40 30 20 10-10 0-20 -30-40 28/09/19 12/10/19 26/10/19 09/11/19 23/11/19 07/12/19 21/12/19 04/01/20 18/01/20 Date Figure 6: FTSE index 01/10/99 07/01/00 (slow rally), P&L. S. Crépey Page 17
Price 1100 1000 900 800 700 600 500 400 300 28/09/19 12/10/19 26/10/19 09/11/19 23/11/19 07/12/19 21/12/19 04/01/20 18/01/20 Delta Gamma 1 0.9 0.8 0.7 0.6 0.5 0.4 28/09/19 12/10/19 26/10/19 09/11/19 23/11/19 07/12/19 21/12/19 04/01/20 18/01/20 0.0008 0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0 28/09/19 12/10/19 26/10/19 09/11/19 23/11/19 07/12/19 21/12/19 04/01/20 18/01/20 Date Figure 7: FTSE index 01/10/99 07/01/00 (slow rally), Greeks. S. Crépey Page 18
Barrier options Model risk inherent to barrier options Hull & Suo 2002 Price Delta Gamma 230 220 210 200 190 180 170 160 150 140 0.6 0.5 0.4 0.3 0.2 0.1 0-0.1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.0018 0.0016 0.0014 0.0012 0.001 0.0008 0.0006 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 barriersmoneyness Figure 8: ATM European double-knock-out call with 5%-rebate and one month time-to-maturity. S. Crépey Page 19
Price Delta Gamma 480 460 440 420 400 380 360 340 320 300-0.05 0-0.1-0.15-0.2-0.25-0.3-0.35-0.4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 barriersmoneyness Figure 9: ATM American down-and-in put with 5%-rebate and half-year time-to-maturity. S. Crépey Page 20
Hedging the barrier Stock 5600 5400 5200 5000 4800 4600 4400 4200 4000 3800 3600 current forward 22/08/20 29/08/20 05/09/20 12/09/20 19/09/20 26/09/20 03/10/20 P&LDeltaHedging P&LIncrements 400 350 300 250 200 150 100 50-50 0 22/08/20 29/08/20 05/09/20 12/09/20 19/09/20 26/09/20 03/10/20 300 250 200 150 100 50-50 0-100 -150-200 22/08/20 29/08/20 05/09/20 12/09/20 19/09/20 26/09/20 03/10/20 Date Figure 10: Down-and-out put option on the DAX index 24/08/01 27/09/01 (fast sell-off), P&L. S. Crépey Page 21
Price 550 500 450 400 350 300 250 200 150 100 50 22/08/20 29/08/20 05/09/20 12/09/20 19/09/20 26/09/20 03/10/20 Delta Gamma 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6 22/08/20 29/08/20 05/09/20 12/09/20 19/09/20 26/09/20 03/10/20 0.0004 0.0002-0.0002 0-0.0004-0.0006-0.0008-0.001-0.0012-0.0014 22/08/20 29/08/20 05/09/20 12/09/20 19/09/20 26/09/20 03/10/20 Date Figure 11: Down-and-out put option on the DAX index 24/08/01 27/09/01 (fast sell-off), Greeks. S. Crépey Page 22
FTSE 01/10 07/01 DAX 24/08 27/09 DAX 24/08 27/09 Barrier final P &L BS (% init. opt. premium) 20.89% 22.09% 90.53% final P &L loc (% init. opt. premium) 9.80% 1.38% 36.64% Enclosed abs. area (log-diff. ) -0.62-1.44-0.93 stdev δp&l BS 8.63 23.26 87.89 stdev δp&l loc 7.48 17.46 46.88 relative difference -13.34% -24.95% -46.66% diff. signif. (F-Test p-values) 87.62% 90.11% 99.77% diff. signif. (Shuffling p-values) 65.98% 70.60% 95.92% Trans. volumes impl. 2.83 1.39 2.35 Trans. volumes loc. 2.93 1.20 2.24 Trans. costs (% init. opt. premium) 5.74% 2.62% 7.45% Trans. costs (% init. opt. premium) 5.94% 2.27% 7.12% Table 4: Performance measures. S. Crépey Page 23
FTSE 01/10 07/01 DAX 24/08 27/09 DAX 24/08 27/09 Barrier Realized vol. (annualized) 17.31% 53.99% Init. vol. 28.65% 22.67% Av. vol. 26.22% 28.64% Vol of vol 31.21% 66.33% Corr. impl. vol changes / index returns -44.43% -85.11% Av. daily return 0.13% -1.15% Market regime Slow rally Fast sell-off better delta Local Local Local Consistency with the theoretical analysis Yes Yes Yes Table 5: Volatility analysis. Note that the transaction costs do not blur the results. S. Crépey Page 24
6 Markets & Models Assuming reversion of volatilities towards realized volatilities, the delta should be preferred to the Black Scholes delta in negatively skewed markets, provided that the physical underlying process as well as the risk-neutral process are negatively skewed. The fact that fast markets may exhibit more physical negative skewness than slow markets might be an explanation for the results in Vähämaa 03 according to which the outperformance of the delta compared to the delta is more significant in fast markets. So Derman s intuition may be right. Moreover, we draw the same conclusions in the case of positively skewed markets, and we show that our conclusions are true when transaction costs are taken into account. S. Crépey Page 25
When barrier options are considered we find that the delta outperforms the delta but there is a need to resort to more elaborate multi-instrument hedging schemes in multi-factor models in order to obtain acceptable absolute hedging performances. The tree captures the anticipations by the agents today of the marginal laws of S (unless American options are used) Fitting the joint dynamics of the smile and the underlyer Bergomi 04 δp&l mod = ( δπ mod + mod δs) + (δπ mod δπ) ATM volatility and skew dynamics Correlation between volatility changes and returns in the underlyer Implied trees are appropriate in markets where the negative correlation (both risk-neutral and physical) between returns in the underlyer and volatility changes is the dominant effect Highly volatile or jumpy markets S. Crépey Page 26
Low credit stocks, Interest or exchange rates with economically significant (such as central-bank) support levels Limitations of volatility Flattening forward volatility surfaces at least in SV Prices Generating Processes, such as the Heston model. In equity index markets jumps are important in the short term Case of symmetric smiles FX markets One factor models are misspecified in general Bakshi Cao Chen 2000 One should allow for independant moves between volatility and the underlyer Stochastic volatility Overdetermined VolOfVol in Heston model One parameter to account for the market skew and volatility moves Bad skew dynamics in (pure, at least) jump models S. Crépey Page 27
Use a good basic model (beyond Black Scholes?) with corrections depending on the market and/or instrument under study, or go for more elaborate models Market models of volatility surfaces Schönbucher 99, Brace et al 01-02 but beware of the realism of the underlyer s dynamics Stochastic trees Derman Kani 98 Local volatility + stochastic volatility SABR model Hagan et al 02 or an improvement that would incorporate some mean reversion of volatility, Dupire s Reech Capital Model Local vol + Stoch vol + Jumps Albanese and Kuznetsov 04 S. Crépey Page 28
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