Delta and Vega Hedging in the SABR and LMM-SABR Models

CUTTING EDGE. INTEREST RATE DERIVATIVES Delta and vega hedging in the SABR and LMM-SABR models

T Riccardo Rebonato, Andrey Pogudin and Richard d t T T T dwt (2) White examine the hedging performance of the SABR t

and LMM-SABR models using real market data. As a Q T E dzT dwT T dt (3) by-product, they gain indirect evidence about how well t t specified the two models are. The results are extremely The LMM-SABR model extension by Rebonato (2007) posits the following joint dynamics for the forward rates and their instanta- encouraging in both respects neous volatilities:

q M i i i i dft idt ft t eijdz j , i 1, N (4) j1

model (Hagan et al, 2002) has become a market i i t gt,Ti kt (5) The SABR standard for quoting the prices of plain vanilla options. Its main claim to being better than other models capable i M dk k i t =μ i + = + × of recovering exactly the smile surface (see, for example, the local t dt ht ∑ eijdz j , i N 1, 2 N (6) k i volatility model of Dupire, 1994, and Derman & Kani, 1996) is t j=1 its ability to provide better hedging (Hagan et al, 2002, Rebon- with: ato, 2004, and Piterbarg, 2005). M = N + N (7) The status of market standard for plain vanilla options enjoyed V F

by the SABR model is similar to the status enjoyed by the Black NV and NF are the number of factors driving the forward rate and formula in the pre-smile days. The Libor market model (LMM) volatility dynamics, respectively: arose out of the desire to provide a dynamic extension of the Black gt,T a bT t exp cT t d (8) model to handle complex products, while recovering the Black i i i prices for the underlying plain vanilla instruments. The desire to ht,T T t exp T t (9) achieve a similar extension has led to the introduction of the i i i SABR-compatible extensions of the LMM by Henry-Labordere M 2 = (2007) and Rebonato (2007). ∑ eij 1 (10) This article explores how successful the delta and vega hedging j=1 suggested by the SABR and LMM-SABR models are in practice. and: Neither aspect has been quantitatively investigated (the discus- Edz dz dt , j, k 1,2,..., M (11) sion in Hagan et al, 2002, is qualitatively convincing, but little j k jk hard evidence is provided). Fortunately, a sufficiently long history Rebonato (2007) and Rebonato & White (2007) show how to deter- of SABR coefficients has by now become available to make an mine the parameters of the LMM-SABR model in order to recover empirical investigation meaningful. The accuracy and simplicity to a high degree of accuracy the SABR prices for caplets and swap- of the formulas to obtain the prices of swaptions given the LMM- tions of all strikes corresponding to a given set of market SABR SABR dynamics of the forward rates presented by Rebonato & parameters. In brief, the parameters of the functions g(.) are deter- White (2007) make the analysis of the effectiveness of vega hedg- mined by minimising the sum, H2, of the squared discrepancies: ing a practical proposition. N T 2 2 i gTˆ (12) 0 i Description of the models i ^ To make the presentation self-contained, we briefly present the where the sum over i runs over the N caplet expiries, g(Ti) denotes . equations of motion of the SABR and LMM-SABR (Rebonato, the root-mean-squared of the function g() to time Ti and:

2007) and the approximate swaption formulas in Rebonato & T Ti E i 0 t T (13) White (2007). 0 t 0 i

Ti In the SABR model, the underlying (a forward or a swap rate, The initial valuesk 0 are then chosen to provide exact recovery of T T Ti f t, of expiry T) follows, under the T-terminal measure Q , the the SABR market quantities X0 : dynamics: Ti Ti 0 k0 (14) T gTˆ T T T T i dft ft t dzt (1) As for the function h(.), its parameters can be chosen from the

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rebonato.indd 94 1/12/08 14:16:43 relationship from the available market-given SABR volatilities of N Estimation of the unobservable volatility. The swap rate and volatility STi: its volatility are the two state variables of the SABR model. How- 1/2 ever, only the first is directly observable from market values. To k T 2 2 Ti 0 i μ 2 gtht tdt (15) create a time series for the latent volatility state variable, we pro- T 0 0 i ceeded in two different ways. Rebonato (2007) shows that this model gives prices extremely First, we simply recorded the SABR market-fitted values of the

similar to the SABR caplet prices for a very large range of maturi- volatility for each trading day ti. Using the notation above, this is ties and strikes. the time series {X(tiaVi)}. Rebonato & White (2007) then go on to show that the Euro- Second, we estimated a value of the volatility in a manner more pean-style swaption prices implied by a forward-rate-based LMM- consistent with the assumption that the SABR model is a correct SABR model are very closely approximated by the prices obtained speciﬁcation of the market (and its parameters, therefore, should be T from a SABR model for swaptions with initial volatility 80, volatil- constant). To do this, we calibrated again the unknown volatility to T T T T ity of volatility V , exponent B and correlation + , given by: the market prices, CK(ti), of the swaptions at time ti, but this time

keeping the parameters, {Viï = Giï, Siï and Wiï}, at the same values T 1 0 0 i j T i j obtained from the fit to the market prices at time tiï. Using the nota- 0 ijWi W j k0 k0 g g dt (16) tion above, the time series thus obtained is denoted by {X(t aV )}. T 0 i iï i, j Clearly, for a perfectly specified model the parameters should never change ({V} = {V }, ), and the two time series would coin- T 1 0 0 i j T i j ˆ 2 i i+1 i V 2 ijijWi W j k0 k0 g g hij t tdt (17) cide (X(t aV) = X(t aV ), ). In reality, this will never be exactly the T 0 i i i iï i 0 i, j case, and the differences between the two time series contain useful

T information about the hedging performance of the SABR model. B wkk (18) N Tests of the hedging performance of the SABR model. To k 1, n j test how well specified the SABR model is as far as delta hedging is concerned, we proceed as follows. T (19) ijij N Set of tests 1. With the first set of tests, we look at some key i, j statistics of the marginal and joint distributions of the time series i with SRt = 8iwi f t and: X(tiaVi), X(tiaViï), )X(tiaVei), )X(tiaVeiï) and )f(ti) and we test for their consistency with the SABR model. N Set of tests 2. With the second, more direct, set of tests we f i W w (20) compare the realised and predicted changes in swaption prices, i i B T T SR )C K(ti; real) and )C K(ti; pred) and at the realised and predicted ^ T ^ T changes in implied volatilities, )X K(ti; real) and )X K(ti; pred). The first two sets of tests give us information about the quality 0 0 i j T i j ˆ 2 2ijijWi W j k k g g httdt of the SABR delta hedges and of the SABR vega hedges for swap- 0 0 0 (21) ij 2 tions of different strikes but same maturity. To explore the ability V T 0 to hedge across swaptions of different expiries and tails, we must These are the equations used in our study to produce swaption resort to testing a dynamic model such as the LMM-SABR that prices given market SABR parameters for caplets. links the different swaptions together. N Tests of the hedging performance of the LMM-SABR Notation model. To assess the hedging performance of the LMM-SABR

For a set of trading days {ti}, i = 0, 1, 2, ... , N, we denote: model across expiries, we proceed as follows. N the set of SABR parameters {G, W and S} at time ti by {Vi}, i = N Set of tests 3. Given a history of market SABR parameters for 0, 1, 2, ... , N; caplets, {Vi}, we calibrate the volatility and volatility-of-volatility N the same set of SABR parameters at time ti augmented by the functions that appear in the Rebonato (2007) LMM-SABR ti-value of the unobservable state variable X(ti) by {Vei}, where {Vei} model to the market prices of caplets. We then calculate the

= {G, W, SXti}, i = 0, 1, 2, ... , N; changes in the prices of various swaptions using the approxima- N the diﬀerence in the swap rate between day iï and day i by tions reported in equations (16) to (21) (Rebonato & White,

Ƌf(ti) } f(ti ïf(tiï), i = 1, 2, ... , N; 2007). We then compare the predicted changes in swaption prices N the diﬀerence in the volatility between day iï and day i by with the observed market changes.

)X(tiaViï) }X(tiaViï ïX(tiïaViï), or by )X(tiaV) }X(tiaVi ïX(tiïaViï), For the sake of brevity, we do not go into detail about the choice i = 1, 2, ... , N, depending on whether the volatility at time ti is esti- of the correlation function. We simply note that: our procedure

mated using information (parameters) available at time ti or tiï; recovers exactly the SABR forward-rate/volatility; as noted in Reb- N the value at time ti of the LMM-SABR call function for strike onato (2006), the model prices of swaptions depend mildly on rea- T K and expiry T by C K(ti); and sonable variations in the forward-rate/forward-rate correlation and N the value at time ti of the Black call function for strike K and more strongly on the shape of the instantaneous volatility function; ^ T ^ T ^ T expiry T with Black implied volatility X K(ti) by C K(tiaX K(ti)). and the model prices of swaptions depend very weakly on reasona- Finally, when there is the possibility of ambiguity, we call ‘true’ ble variations in the volatility/volatility correlation. volatility the volatility that enters the (LMM)-SABR models, to distinguish it from the implied volatility. Results for the SABR model We have at our disposal for the analysis 818 daily records of the N Results for set of tests 1. The SABR model makes the assump-

set {Vei} and of f(ti). tion that the volatility at time t of the quantity:

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A. The SABR quantity X(tiaVi) averaged over all the data 1 Regression of )X(taV ) against )X(taV ): 1y w 10y points in our sample ( = 0.5) i i i i–1 G 1y × 10y 0.0030 Actual 1y 2y 5y 10y 0.0025 y = 0.9882x – 1E-07 2 6m 0.034 0.037 0.038 0.035 0.0020 R = 0.983 0.0015 1y 0.038 0.039 0.039 0.037 0.0010 2y 0.041 0.041 0.040 0.037 0.0005 0 5y 0.040 0.040 0.038 0.036 –0.002 –0.001–0.0005 0.001 0.002 0.003 10y 0.035 0.034 0.033 0.031 –0.0010 Predicted –0.0015 –0.0020 G B. The annualised volatility of the quantity xi = )f(t)/f(t) (daily changes) slope different from one. In reality (see figure 1), for all the swap- 1y 2y 5y 10y tions the slope of the regression is very close to one and the qual- 2 6m 0.038 0.037 0.034 0.031 ity of the linear fit is very high (maximum R = 0.99, minimum R2 = 0.77, average R2 = 0.93). 1y 0.037 0.036 0.033 0.031 We then look at the distributional features of the time series 2y 0.035 0.035 0.032 0.031 {)X(tiaViï)}. Since the volatility of volatility is invariant under a 5y 0.031 0.031 0.031 0.030 change of measure, if the SABR model were correctly specified the 10y 0.032 0.030 0.030 0.030 distribution would be lognormal, and its annualised standard deviation would be equal to the SABR-fitted volatility of volatility C. Market-fitted volatility of volatility (SABR S) averaged parameter, S. Not surprisingly, we can reject the assumption of log- over all the days in our sample (G = 0.5) normality (there is far too much positive skew and the kurtosis is far too high, especially for short expiries and short tails). This obser- 1y 2y 5y 10y vation should be linked to the remarks made about the apparent 6m 0.61 0.60 0.58 0.57 profitability on average of the strategy of selling volatility during 1y 0.47 0.46 0.44 0.43 the period under analysis: the seller would be exposed to rare but 2y 0.39 0.39 0.37 0.37 large losses arising from the large moves that populate the right fat 5y 0.32 0.33 0.32 0.32 tails of the volatility distribution. The standard deviation of the 10y 0.27 0.28 0.27 0.27 distribution, however, is reasonably close to the SABR-fitted volatility of volatility, S. Furthermore, it is encouraging that both the fitted S and the measured standard deviation display the same D. Annualised standard deviation of the distribution of qualitative features of declining both with increasing expiries and the daily changes in volatility, {)X(t aV )} i i–1 with increasing swap tails. A small but noticeable systematic bias is 1y 2y 5y 10y however present, and we discuss its origin below. 6m 0.46 0.35 0.27 0.25 We next move to the analysis of the correlation between the 1y 0.28 0.25 0.20 0.18 changes in volatilities, {)X(tiaViï)}, and the changes in swap rate, 2y 0.20 0.19 0.16 0.15 {)f(ti)}, which should also be invariant under the change of meas- 5y 0.15 0.14 0.13 0.13 ure for a correctly specified model. We find that the agreement is very good, as shown in tables E and F. This again supports the 10y 0.14 0.14 0.13 0.13 view that, with some caveats that we discuss below, the SABR df t model is essentially well specified. xi (22) We can explain as follows the small but noticeable biases noted ft above. Let us assume that, as a first approximation, both the real should be a stochastic quantity with initial value and expectation market smile, X^ mkt(x), and the SABR-produced smile, X^ SABR(x),

Ti Ti given by Et[XY ] = X0 YTi. Given the time series xi, we there- are exactly given by a parabolic function of the strike, x (over an fore compare its annualised volatility with the SABR volatility appropriate range): X(t aV) averaged over all the data points in our sample.1 The results i i ˆ mkt x x x 2 (23) are shown in tables A and B. 0 1 2 We note that the two sets of quantities are in close agreement, with SABR 2 ˆ x k k x x% k x x% (24) a small negative bias for the realised volatility. One can interpret the 0 1 1 2 2 small negative bias as a ‘risk premium’: it would have been proﬁtable To a very good degree of accuracy, the SABR quantities X(0), W

on average to sell volatility but the seller would however be exposed and S control the level (k0), the slope (k1) and the curvature (k2) of to rare, large, non-normal moves in volatility, as discussed below. the smile, respectively. The SABR model, however, also prescribes ~ Next, we perform a regression of the time series X(tiaVi) on both the tilt point for the slope (x1) and location of the minimum ~ X(tiaVi–1). If the SABR model were poorly specified, after each time of the parabola (x2). Needless to say, one can always fit the para- step a large change in the parameters {Vi} would be required to fit bolic market smile by choosing: the new market prices and the time series X(t aV) and X(t aV ) fit 2 i i i i–1 k 2 x% x% x% (25) would display a low correlation (low R2) in the regression, and a 0 0 1 2 2 1 2 2 1 For all the results in this section, a value of 0.5 was always used for the exponent G, as this appears to be k fit 2 x% (26) standard market practice 1 1 2 2

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rebonato.indd 96 1/12/08 14:16:49 E. Market-fitted correlation (SABR W) averaged over all 2 Predicted and observed changesinprices for the the days in our sample (G = 0.5) 10y w 10y swaption, strike =ATM 1y 2y 5y 10y Strike 0(ATM) 0.0008 Actual 6m –0.25 –0.24 –0.12 –0.08 0.0006 y = 1.0006x + 2E-07 1y –0.30 –0.29 –0.21 –0.18 0.0004 R2 = 0.994 2y –0.33 –0.33 –0.30 –0.28 0.0002 0 5y –0.30 –0.32 –0.34 –0.33 –0.0008 –0.0004 –0.0002- 0.0004 0.0008 10y –0.30 –0.33 –0.36 –0.35 Predicted –0.0004 –0.0006 F. Statistically measured realised correlation obtained –0.0008 using the procedure in test 2 (daily changes) 1y 2y 5y 10y 3 Actual and predicted changes in implied volatility: 6m –0.27 –0.28 –0.23 –0.18 10y w 10y 1y –0.37 –0.36 –0.33 –0.29 10y × 10y 0.006 Actual 2y –0.41 –0.42 –0.38 –0.36 0.005 y = 0.9474x + 6E-06 R2 = 0.905 5y –0.38 –0.38 –0.39 –0.39 0.004 0.003 10y –0.42 –0.39 –0.41 –0.41 0.002 fit 0.001 k2 2 (27) 0 –0.004 –0.001 0.002 0.004 0.006 However, there is no guarantee that a naive best fit to market prices –0.002 Predicted will locate the pivot point of the tilt and the inflection point of the –0.003 curvature where the SABR model with the statistically determined –0.004 W and S would. If it did, the model would be ‘well specified’. Fortu- nately, the analysis above shows that the fitted model parameters turn out to be remarkably close to their statistically determined 4 Regression of changesinpredicted implied volatilities (‘physical’) values. It would be interesting to explore how much obtained using the change in forward rates only: 5y w 5y worse the fit would be if, instead of leaving it totally unconstrained, 5y × 5y 0.008 Actual one introduced a Bayesian type of estimation, with the prior given 0.006 y = 0.5267x – 3E-05 R2 = 0.001 by the empirically determined correlation and volatility of volatil- 0.004 ity and the likelihood function based on market information. We 0.002 leave this topic for a separate investigation. 0 –0.0004 –0.00020.0002 0.0004 0.0006 0.0008 To see to what extent the small discrepancies observed above –0.002 can affect in practice the hedging performance, we move to the Predicted –0.004 second set of tests. N Results for set of tests 2. For selected swaptions, we compared –0.006 T the actual changes in swaption prices for a given strike (K), )CK(ti; T real), with the model-predicted changes in swaption prices, )CK(ti; whether the implied volatilities change in reality as a function of pred). The predicted changes in prices were calculated as follows: changes in forward rates and in ‘true’ volatility as the SABR C T C T model predicts. To estimate the model changes in implied volatil- T K K (28) CK ti ; pred fti ti i1 ity we used the expression: f ˆ T ˆ T We then tested how well one can offset, using the model, the vol- ˆ T K K (29) K ti ; pred fti ti i1 atility exposure at one strike with the volatility exposure at f ^ T ^ T another strike for the same expiry (same-expiry vega hedging). where the terms yX K/yf and yX K/yX were obtained from the Hagan

The regression of the predicted against observed changes in prices formula, )f(ti) was taken as market-given, )X(tiaViï) was estimated shows for all swaptions and across all strikes a very strong linear rela- as explained above, and the SABR parameters (G, S and W) were

tionship, with the slope virtually identical to one, intercept almost kept at their tiï level. Figure 3 shows the excellent quality of a lin- exactly zero, and R2 ranging from 0.952 to 0.996 (see ﬁgure 2). ear regression of predicted against observed implied volatilities: the These ﬁndings should be compared with the poor performance slope of the regression is very close to one and the R2 is very high for of the Black model, discussed in Rebonato (2003). These tests indi- all swaptions (average 0.95, minimum 0.89, maximum 0.99). cate that changes in the state variables of the SABR model correctly To understand better what drives these correct predictions we account for the bulk of the observed changes in prices and therefore look at the regression of the changes in realised implied volatili- provide on average correct and unbiased ‘hedge ratios’. ties versus the changes in predicted implied volatilities obtained The good results above have been obtained using changes in using the changes in forward rates only. More precisely, we keep

prices, which are naturally dominated by changes in forward the volatility at its tiï level, we use the market change in forward rates. If we look at implied volatilities instead, the first-order rate as an input to the Hagan formula, and we convert the result- dependence on the forward rate disappears, and we can focus on ant change in price into a change in implied volatility. We find vega hedging more clearly. Specifically, we would like to know that the dependence between the predicted and experienced

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5 Comparison with (virtual) reality of model changesin 7 Prediction of the 5y w 5y swaption implied volatility implied volatility when only the forward rate changes: curve for all strikes using only a ﬁt to caplets during the 10y w 10y, strike =ATM ‘normal’period 0.004 Actual 32 Strike 0(ATM) 30 Market 0.003 28 Model (flat curve) y = 0.971x + 3E-06 Model (fitted curve) 0.002 2 26 R = 0.2977 24

0.001 % 22 0 20 –0.001 –0.000 –0.000 0.000 0.000 0.001 0.001 –0.001 18 16 –0.002 14 Predicted 2 3 4 5 6 7 8 9 10 11 –0.003 %

^ T The term yX K/yX )X(tiaViï) in equation (29) is missing because in 6 Predicted and realised ATM volatility for the 5y w 5y the virtual change the volatility remains fixed. We can now esti- swaption, with 3m and 10y caplets used as inputs mate and compare ‘virtual’ and predicted market changes in implied volatilities when only the forward rate changes. 30 Model When this is done the relationship between the changes in pre- 25 Market 3m caplet dicted and experienced (virtual) implied volatilities as the forward 20 10y caplet rate changes becomes much clearer. This means the changes in ^ T implied volatility caused by yX K/yf )f(ti), while much smaller ^ % 15 T than the contribution from yX K/yX )Xi, are sufficiently more sig- ^ T ^ T 10 nificant than the ‘noise’ given by the terms yX K/yf )Si and yX K/yf )Wi (that should be zero for a perfectly specified model) to give a 5 clearly detectable signal (see figure 5). These findings are robust 0 across expiries and levels of at-the-moneyness. Jan 2005 Jul 2006 Jan 2008 In sum: the changes in implied volatility are well explained by the changes in the true SABR volatility. As the SABR model pre- implied volatility has now all but disappeared (see figure 4). dicts an almost parallel move of the smile as the volatility changes, To understand whether changes in forward rates truly have no this means the observed moves in the market smile (for a fixed impact on changes in implied volatilities, we approximate in the maturity) are usually very close to parallel. In addition, there following way a ‘controlled experiment’ under which only the for- exists a weaker and noisier residual dependence of the smile on ward rate changes. changes in the forward rate. The SABR model again makes pre- Neglecting the difference between state variables and parame- dictions similar to what is observed in reality. ters, consider the Hagan formula as a function of four quantities, The very good results of the regression tests should not be inter- f, X, S and W.2 To the extent that the SABR model always per- preted as an indication that the model is ‘close to perfect’, but give fectly fits the market, we can write for the true realised change in us confidence that the SABR model can be a good building block implied volatility: for the dynamic extension of the LMM-SABR model that we discuss below. Ultimately, SABR is a purely diffusive model, and it ˆ T K ti ;real cannot accommodate jumps either in forward rates or in volatili- T T T T (30) ties. These are rare, but can be economically important. Their ˆ K ˆ K ˆ K ˆ K fti i i i impact on a hedging strategy depends on the time to expiry and f f f level of moneyness of the option when the jump occurs (essentially, We now define ‘virtual’ changes as changes calculated using the on the gamma and ‘vanna’ of the option). This is probably why, Hagan formula with the true observed changes in the forward despite the bias indicated by the analysis above, selling volatility is

rate, )f(ti), and in )S(ti) and )W(ti), but keeping the volatility at its not a ‘free lunch’. So, for most days the SABR model appears well

tiï level. Therefore: specified and the model parameters change very little. When the ˆ T ˆ T ˆ T rare ‘jumps’ (unaccounted for by the model) occur, the SABR ˆ T K K K K ti ;virtual fti i i (31) parameters have to be changed significantly. This is still a very sig- f f f nificant improvement on the Black or on the local volatility mod- This virtual change is our controlled experiment: our best approx- els, which are clearly mis-specified even on ‘normal’ days. imation of what the change in the implied volatility would have been if the volatility had not moved. Results for the SABR-LMM model Finally, the predicted changes in implied volatilities were cal- With the set of tests 3, we try to predict the changes in swaption culated inputting in the Hagan formula the true change in for- prices making use only of caplet information. Since caplets are a

ward rate )f(ti): very special subset of swaptions, this is probably the toughest self- ˆ T consistency test to which the model can be put. ˆ T K K ti ; pred fti (32) Since the transformation from caplet to swaption volatilities f depends on the terminal (as opposed to instantaneous) decorrela- 2 The exponent G is always kept at 0.5 tion among forward rates, and this mainly depends on the time-

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rebonato.indd 98 1/12/08 14:16:57 8 Prediction of the changesinriskreversal:5y w 5y swaption 9 Prediction of the changes in strangle:5yw 5y swaption 14 5.0 Model Model 3m caplet 4.5 12 Market Market 10y caplet 3m caplet 4.0 10 10y caplet 3.5 8 3.0 %

% 2.5 6 2.0 4 1.5 1.0 2 0.5 0 0 Jan 2005 Jul 2006 Jan 2008 Jan 2005 Jul 2006 Jan 2008

dependence of the volatility functions (see Rebonato, 2006, and explanation was provided for the small systematic bias. the discussion of figure 7 below), we are ultimately testing the For the LMM-SABR model, we undertook what is probably functions g(.) and h(.), which are the crucial ingredients of the the toughest realistic test, that is, we tried to predict changes in LMM-SABR extension of Rebonato (2007). the swaption smile in the centre of the swaption matrix (the 5y w N Results for set of tests 3. The predicted and realised at-the- 5y swaption) given only knowledge of the caplet smile. Despite money volatility of a 5y w 5y swaption is shown in figure 6, which this, the results were good, at least during normal market condi- also displays the at-the-money volatility for two of the caplets to tions. The predicted levels of and changes in the at-the-money which the volatility curve had been fitted. The quality of the pre- volatility, risk reversal and strangle were in fair-to-good agree- diction is high, especially considering that the volatilities of the ment with the observed values. During the period of exceptional caplets that were used to obtain the 5y w 5y swaption volatility are turbulence in early 2008, the predictions of the model become very different between themselves, and from the volatility of the poorer, but no diffusive model with a single regime of volatility target swaption (see figures 6, 8 and 9). Yet the model, which only can be expected to handle such circumstances. ‘knows’ about caplets, is able to predict an almost correct value Given its ease of calibration, the availability of closed-form approx- for the at-the-money swaption volatility. imation for swaption prices, its ability to produce realistic future A regression of the model-predicted versus realised at-the- smiles and the overall good quality of its hedging performance even money volatilities over the whole period gives a correlation of under a very severe test, the LMM-SABR model appears to offer a 77%. This overall value stems from two different regimes: a period simple and useful dynamic extension of the SABR model. N of ‘normal’ market conditions (approximately up to August 2007), and the exceptionally excited period that followed. The Riccardo Rebonato is head of market risk and head of quantitative research, predictive power of the model is much better during the normal Andrey Pogudin is a senior quantitative analyst in the quantitative solutions period (correlation of 85%). For instance, during the ‘normal’ group and Richard White worked as a senior quantitative analysts in the period the quality of the prediction of the whole smile (all strikes) quantitative analysis group at the Royal Bank of Scotland. Email: riccardo. for the 5y w 5y swaption using only information from caplets can [email protected], [email protected], [email protected] be as high as shown in figure 7. We stress that the volatility func- . . tions g() and h() were not fitted to this particular swaption. The References same figure also shows the importance of the time-dependence of the volatility function g(.) in influencing the level of the swaption Derman E, I Kani and J Zou, 1996 Rebonato R, 2004 The local volatility surface: unlocking Volatility and correlation price: the curve labelled “Model (flat curve)” shows what the the information in index option prices John Wiley swaption price would be if the caplets were still exactly fitted, but Financial Analysts Journal, July/ Rebonato R, 2006 using flat volatility functions g(.) and h(.). August, pages 25–36 Forward-rate volatilities and the We finally looked at the ability of the LMM-SABR to predict Dupire B, 1994 swaption matrix: why neither time- changes in the risk reversal and strangle for the 5y w 5y swaption, Pricing with a smile homogeneity nor time dependence Risk January, pages 18–20 will do again making use only of caplet information (see figures 8 and 9). Hagan P, D Kumar, A Lesniewski International Journal of Theoretical The quality of the prediction is again impressive, especially if one and D Woodward, 2002 and Applied Finance 9(5), pages notes how different the strangles and risk reversals were in the Managing smile risk 705–746 period considered for the target swaption and for the caplets on Wilmott Magazine 3, pages 84–108 Rebonato R, 2007 A time-homogeneous, SABR- which the model was built. Henry-Labordere P, 2007 Unifying the BGM and SABR models: a consistent extension of the LMM: short ride in hyperbolic geometry calibration and numerical results Conclusions Risk October, pages 88–92 Risk November, pages 102–106 We have examined the hedging performance of the SABR and the Piterbarg V, 2005 Rebonato R and R White, 2007 LMM-SABR models. We found that the SABR model is as close Time to smile Linking caplets and swaption prices in Risk May, pages 87–92 the LMM-SABR model to being well specified as one can expect for a purely diffusive Submitted to Journal of pricing model that must retain analytical tractability. Its predic- Rebonato R, 2003 Computational Finance and available Which process gives rise to the at www.riccardorebonato.co.uk tive power given the observed changes in the underlying state observed dependence of swaption variables was found to be very high. The ‘implied’ and the statisti- implied volatilities on the underlying? cally estimated values for the model parameters (correlation and International Journal of Theoretical and Applied Finance 6(4), pages 419–442 volatility of volatility) were found to be in good agreement. An

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