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 94 Risk December 2008 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.