Analytic Calibration in Andreasen-Huge SABR Model
AAnalytic calibration in Andreasen-Huge SABR model.
Konstantin Feldman ∗ MUFG Securities EMEA plc.
Abstract
We derive analytic formulae which link α , ν and ρ parameters in Andreasen-Huge styleSABR model to the ATM price and option prices at four equally spaced strikes on the sides ofATM. We give two applications. First, we give a characterisation for the SABR parameters interms of the swap rate forward probability density function. Second, we show how Andreasen-Huge SABR can be combined with other well know analytic SABR formulae to allow consistentuse in the non-arbitrage region.
20 years ago Patrik Hagan, Deep Kumar, Andrew Lesniewski and Diana Woodward published apaper [1] which revolutionised interest rate volatility modelling. They derived a closed form analyticapproximation for the implied volatility of a forward swap rate assuming it satisfies a system ofstochastic differential equations: dF t = α t F βt dW t ; dα t = να t dZ t ; ρ = < W t , Z t > . (1)The analytic approximation gave interest rate traders a simple tool to manage the swap rate prob-ability density distributions which are skewed and smiled over the traditional Gaussian normal dis-tribution. The model got a name - SABR model.Various improvements were made to the analytical formula from [1] in works [2, 3] and even morerecently in [7].One of the issues which became transparent in the industrial use of SABR model was its approx-imation character. As it was an approximation for the forwards behavior around the ATM level, itcouldn’t possibly give the terminal swap rate probability density at every point. Polynomial expan-sions used in the formula were unbounded at infinity and, thus, led to negative values of the implieddensity function.A striking approach to address the problem of constructing terminal swap rate probability densityvalid along the whole real line appeared 10 years later in another seminal paper [8]. Jasper Andreasen ∗ This paper is a personal view and does not represent the views of MUFG Securities EMA plc (MUSE). This paperis not advice. MUSE shall not be liable in any manner whatsoever for any consequences or loss (including but notlimited to any direct, indirect or consequential loss, loss of profits and damages) arising from any reliance on or usageof this presentation and accepts no legal responsibility to any party who directly or indirectly receives this material. a r X i v : . [ q -f i n . C P ] A ug nd Brian Huge showed how within SABR model assumptions one can reduce the construction ofthe forward swap rate probability density function to a solution of a single tridiagonal system. Theconstruction not only resolved the issue of the validity of the probability density function at everypoint, but also prompted more widespread use of the density functions for modelling light exoticslike CMSs and CMS spreads, which were often done by a slower replication based methods. Whilesome market participants went for the approach [8], other developments also made into productionsystems of many banks [4, 5, 6].An important technical problem which accompanies the use of SABR model is the parameters’calibration. The parameter β can be estimated from the log-log plot of the forward variance withrespect to the forward [1] and α could be found as a root of a cubic equation [9]. The skew parameters ν and ρ were often manually calibrated. Multi dimensional solvers were introduced to addressparameter mapping while migrating to the shifted SABR model which became a necessity as therates entered negative territory. Extensive studies have been made on how to chose the initial guessto optimise solver convergence [10].Andreasen-Huge construction of the non-arbitrage SABR did not contain explicit analytics toinvert, and, thus, required use of solvers to imply not only the skew parameters, but the parameteralpha as well. As Andreasen-Huge construction gives considerable computational speed up in com-parison with the other arbitrage free SABR methods, the use of solvers did not represent a limitationfor building infrastructure around their version of SABR model. The goal of the present paper is toshow that Andreasen-Huge construction of SABR model, in fact, allows one to have exact analyticexpressions of all three model parameters: alpha, nu and rho, in terms of option prices close to ATM.In the paper, we show that if call prices c ( F − h ) , c ( F − h ) , c ( F ) , c ( F + h ) , c ( F + 2 h ) are given atfive consecutive grid points of an equally spaced Andresen-Huge shifted SABR grid with the step h and c ( F ) = AT M - the ATM option price, then: α = h ( F + shif t ) β (cid:115) T (cid:115) AT Mc ( F − h ) + c ( F + h ) − AT M , (2) ν = z − h y ( F − h )( F − h + shift ) β − z + h y ( F + h )( F + h + shift ) β T κ h α ( y ( F − h ) − y ( F + h )) − y ( F − h ) y ( F + h ) , (3) ρ = 12 ν (cid:32) ν y ( F − h ) − y ( F − h ) (cid:32) z − h T κ h α ( F − h + shif t ) β − (cid:33)(cid:33) , (4)where y ( k ) is the shifted SABR local volatility diffusion distance from the forward to the strike, κ h is the one half of the value of the Andreasen-Huge local volatility one time step ODE adjustment atthe strikes F ± h and z − = c ( F − h ) − hc ( F − h ) + AT M − c ( F − h ) , z + = c ( F + h ) c ( F + 2 h ) + AT M − c ( F + h ) . (5)We discuss two applications of the analytic expressions. 1) We study the limiting behavior when h →
0, and derive a characterisation for SABR parameters in terms of probability density functionof the forward swap rate distribution. 2) We show how these formulae can be used for migratingfrom/between analytic expansions [1] and the Andreasen-Huge construction.The structure of the paper is as follows. In Section 1 we remind the Andreasen-Huge constructionfor SABR model. In Section 2 we derive analytic formulae for (shifted) SABR parameters in theAndreasen-Huge construction. In Section 3 we study the limiting behavior of the analytic formulae.In Section 4 we show how the Andreasen-Huge construction can be calibrated to the Hagan etl.original construction. We summarise our findings in the conclusion.2
Andreasen-Huge one-time-step method
In this section we remind the Andreasen-Huge one time step ODE reduction for SABR SDE [8]. Itrests on their other remarkable paper [11] where they showed that the option pricing heat equation: c t ( t, k ) = 12 ϑ ( k ) c kk ( t, k ) c ( T, k ) = ( F − k ) + , (6)with the local volatility coefficient ϑ ( k ) dependent only on the strike k and the terminal conditiongiven by the intrinsic value of a call option, can be replaced by an ODE: c ( t, k ) −
12 ( T − t ) θ ( k ) c kk ( t, k ) = ( F − k ) + . (7)Based on this reduction, Andreasen and Huge derived [8] the relation between θ ( k ) and ϑ ( k ) in termsof the standard normal distribution: θ ( k ) = ϑ ( k ) c ( t, k ) − ( F − k ) + ( T − t ) c t ( t, k ) , = 2 ϑ ( k ) (cid:32) − ξ Φ( − ξ ) φ ( ξ ) (cid:33) , (8)where Φ( ξ ) is the CDF of the standard normal random variable, φ ( ξ ) is the PDF of the standardnormal random variable and ξ = | F − k | /σ , with σ being ATM normal volatility. In what follows weshall be calling the coefficient κ ( k ) = 2 (cid:32) − ξ Φ( − ξ ) φ ( ξ ) (cid:33) (9)as Andreasen-Huge local volatility one time step adjustment.The ODE (7) is solved by inverting a tridiagonal matrix corresponding to the finite differenceequation [11]: (cid:32) T θ ( k ) h (cid:33) c (0 , k ) − T θ ( k ) h [ c (0 , k + h ) + c (0 , k − h )] = ( F − k ) + . (10)In the identical way we can solve the option pricing problem in terms of puts: (cid:32) T θ ( k ) h (cid:33) p (0 , k ) − T θ ( k ) h [ p (0 , k + h ) + p (0 , k − h )] = ( k − F ) + . (11)Note, that when reducing (10) or (11) to solving the linear system with a tridiagonal matrix we needto set an absorbing boundary condition [11]: c kk (0 , k −∞ ) = c kk (0 , k ∞ ) = 0 , (12) p kk (0 , k −∞ ) = p kk (0 , k ∞ ) = 0 . (13)In Andreasen-Huge SABR construction [8] we capitalise on the analytic form of the local volatility ϑ ( k ) for SABR SDE (1). Using shifted version of the SABR SDE (1) we can write [8, 12]: ϑ ( k ) = αJ ( y )( k + shif t ) β , (14) y ( k ) = 1 α (cid:90) Fk ( u + shif t ) − β du, (15) J ( y ) = (cid:113) − ρνy + ν y . (16)3hus, in order to solve for option prices in (shifted) SABR SDE (1) we can solve linear tridiagonalsystem (10) or (11), whose coefficients are populated using reduction (8) and analytic formulae (14-16).In further sections, we will need explicitly the values of the local volatility one time step adjust-ments for k = F − h, F, F + h respectively, where F is the forward and h is the step of the uniformgrid in (10) (or (11)). They can be calculated as below: κ ( F ) = 2 , (17) κ ( F ± h ) = 2 (cid:32) − hσφ ( h/σ ) (cid:32) − (cid:90) h/σ φ ( s ) ds (cid:33)(cid:33) (18)= 2 (cid:32) − hσφ ( h/σ ) (cid:32) − hσ φ ( ζ ) (cid:33)(cid:33) , ζ ∈ [0; h/σ ] . (19)In what follows we shall be using a notation κ h = κ ( F ± h ) / In this section we use Andreasen-Huge one time step ODE reduction of the (shifted) SABR SDE toderive exact formulae which relate SABR SDE coefficients α , ν and ρ to the prices of five optionsclose to ATM.First we write the the one time step approach of the previous section explicitly in terms of putoption prices (11). The values of all OTM puts p , . . . p n − , ATM put p n and the first ITM put p n +1 satisfy the following tridiagonal system: · · · − z z − z · · · − z z − z · · · · · · − z n − z n − − z n − · · · − z n − z n − − z n −
00 0 0 0 · · · − z n z n − z n p p p ... p n − p n p n +1 = , where the coefficients z j , j = 0 , . . . n , are given by (11): z j = T θ ( k ) h . (20)The value AT M = p n is (typically) observed from the market and is AT M = p n = (cid:115) T π σ AT M , whith σ AT M - the implied normal volatility of the ATM option (put or call). We can reduce the4ridiagoanl system by one dimension to · · · − z z − z · · · − z z − z · · · · · · − z n − z n − − z n −
00 0 0 0 · · · − z n − z n −
00 0 0 0 · · · − z n − z n . p p p ... p n − p n +1 == z n − AT M − (1 + 2 z n ) AT M . The last equation of the system: z n ( p n − + p n +1 ) = (1 + 2 z n ) AT M (21)can be solved separately after solving · · · − z z − z · · · − z z − z · · · · · · − z n − z n − − z n − · · · − z n − z n − p p p ... p n − = z n − AT M . The last row of the later gives a further condition: − z n − p n − + (1 + 2 z n − ) p n − = z n − AT M. (22)The same approach can be repeated for all OTM calls c n +1 , . . . c n + m together with ATM call c n = AT M and the first ITM call c n − . Working in the same vein with (10) we use the first equation ofthe (reduced) call based system to get: − z n +1 c n +2 + (1 + 2 z n +1 ) c n +1 = z n +1 AT M. (23)5he ATM term z n of the put (11) or the call (10) systems, is particularly simple in the shifted SABRapproach: z n = T θ ( F ) h , = T ϑ ( F ) κ ( F )2 h , = Th α ( F + shif t ) β . (24)We use it together with (21) to derive an approximation for αα = h ( F + shif t ) β (cid:115) T (cid:115) AT Mp n − + p n +1 − AT M . (25)Two equations (22) and (23) can be used to imply parameters ρ and ν from option prices close ATM.More explicitly, we have: z n − = p n − p n − + AT M − p n − ,z n +1 = c n +1 c n +2 + AT M − c n +1 . (26)with z i = T ϑ ( k i ) κ ( k i )2 h ,ϑ ( k ) = αJ ( y )( k + shif t ) β , (27) y ( k ) = 1 α (cid:90) Fk ( u + shif t ) − β du, (28) J ( y ) = (cid:113) − ρνy + ν y . (29)Moving all the terms in (27) independent of ν and ρ to one side we obtain a system: ν y ( k n − ) − ρν = 1 y ( k n − ) (cid:32) z n − h T κ ( k n − ) α ( k n − + shif t ) β − (cid:33) , (30) ν y ( k n +1 ) − ρν = 1 y ( k n +1 ) (cid:32) z n +1 h T κ ( k n +1 ) α ( k n +1 + shif t ) β − (cid:33) . (31)This system can be solved analytically for ν and ρ . We obtain the following result: Theorem 1
In Andreasen-Huge one time step approximation for the shifted SABR SDE, the pa-rameters α , ν and ρ have the following analytic expressions: α = h ( F + shif t ) β (cid:115) T (cid:115) AT Mp n − + p n +1 − AT M , (32) ν = z n − h y ( F − h )( F − h + shift ) β − z n +1 h y ( F + h )( F + h + shift ) β T κ h α ( y ( F − h ) − y ( F + h )) − y ( F − h ) y ( F + h ) , (33) ρ = 12 ν y ( F + h ) z n − h y ( F − h )( F − h + shift ) β − y ( F − h ) z n +1 h y ( F + h )( F + h + shift ) β T κ h α ( y ( F − h ) − y ( F + h )) − y ( F − h ) − y ( F + h ) y ( F − h ) y ( F + h ) , (34)6 here z n − = p n − p n − + AT M − p n − ,z n +1 = c n +1 c n +2 + AT M − c n +1 ,κ h = 1 − hσφ ( h/σ ) (cid:32) − (cid:90) h/σ φ ( s ) ds (cid:33) . (35) In this section we study the limiting behavior of (32-34) as the grid step h goes to zero. First, wederive an analytic expression for α : α = lim h → h ( f wd + shif t ) β (cid:115) T (cid:115) AT Mp n − + p n +1 − AT M , (36)= 1( f wd + shif t ) β (cid:115) T (cid:115) AT Mpdf
AT M . (37)If we assume that the forward swap rate PDF is approximately Gaussian normal we can recover apopular approximation: α ≈ f wd + shif t ) β (cid:115) T (cid:113) σ AT M T = σ AT M ( f wd + shif t ) β . (38)Next, we look at the term ρν in (30,31) . We shall be using the following approximations: z n − h = p n − h p n − + AT M − p n − ≈ p ( F − h ) pdf ( F − h ) , (39) z n +1 h = c n +1 h c n +2 + AT M − c n +1 ≈ p ( F + h ) − hpdf ( F + h ) . (40)(41)The transformation (28) is a change of coordinates with a Jacobian: ∂y ( k ) ∂k = − α ( k + shif t ) β , y ( F ) = 0 . (42)We use this change of coordinates to factor out a derivative with respect to y ( k ) first from (30) byusing (37): ρν = 12 lim h → y ( k n − ) (cid:32) − z n − h T κ ( k n − ) α ( k n − + shif t ) β (cid:33) , (43)= 1 T α lim h → y ( k n − ) (cid:32) p ( F ) κ ( F ) pdf ( F )( F + shif t ) β − p ( F − h ) κ ( F − h ) pdf ( F − h )( F − h + shif t ) β (cid:33) , = − T α ∂ − ∂y p ( k ( y )) κ ( k ( y )) pdf ( k ( y ))( k ( y ) + shif t ) β | k ( y )= F , (44)7here by ∂ − we denoted the left derivative with respect to y . Similarly, using (31) we get ρν = − T α ∂ + ∂y c ( k ( y )) κ ( k ( y )) pdf ( k ( y ))( k ( y ) + shif t ) β | k ( y )= F , (45)with ∂ + being the right derivative with respect to y . We can glue the functions − T α ∂ − ∂y p ( k ( y )) κ ( k ( y )) pdf ( k ( y ))( k ( y ) + shif t ) β − T α ∂ + ∂y c ( k ( y )) κ ( k ( y )) pdf ( k ( y ))( k ( y ) + shif t ) β (46)at k ( y ) = F . Using (33) to see that the result is a differentiable function (as y ( k n − ) and y ( k n +1 ) areindependent), we arrive at: ν = 2 T α ∂ − ∂y p ( k ( y )) κ ( k ( y )) pdf ( k ( y ))( k ( y ) + shif t ) β | k = F (47)= 2 T α ∂ ∂y c ( k ( y )) κ ( k ( y )) pdf ( k ( y ))( k ( y ) + shif t ) β | k = F (48)(49)We summarise our findings in the next theorem: Theorem 2
In the short maturity approximation of Andreasen-Huge SSABR model the followinganalytic expressions for model parameters α , ν and ρ hold: α = 1( f wd + shif t ) β (cid:115) T (cid:115) AT Mpdf
AT M , (50) ν = 1 α (cid:115) T (cid:118)(cid:117)(cid:117)(cid:116) ∂ − ∂y p ( k ) κ ( k ) pdf ( k )( k + shif t ) β | k = F = 1 α (cid:115) T (cid:118)(cid:117)(cid:117)(cid:116) ∂ ∂y c ( k ) κ ( k ) pdf ( k )( k + shif t ) β | k = F , (51) ρ = − α √ T ∂ − ∂y p ( k ) κ ( k ) pdf ( k )( k + shift ) β | k = F (cid:114) ∂ − ∂y p ( k ) κ ( k ) pdf ( k )( k + shift ) β | k = F = − α √ T ∂ + ∂y c ( k ) κ ( k ) pdf ( k )( k + shift ) β | k = F (cid:114) ∂ ∂y c ( k ) κ ( k ) pdf ( k )( k + shift ) β | k = F , (52) where y ( k ) = 1 α (cid:90) Fk ( u + shif t ) − β du. (53) In this section we show how analytic formulae from Section 3 can be used for recalibration of differentversions of SABR to each other.First, we consider an example of EUR 10y 10y Swaption and demonstrate how the analytic skewcalibration improves matching between Hagan and Andreasen-Huge SSABR approximations. We useHagan SSABR parameters: shif t = 3% , β = 40% , α = 2 . , ρ = − . , ν = 26 .
12% (54)8igure 1:
EUR 10y 10y Swaption Skew.
Using formulae from Section 3 we find calibrated skew parameters to use in Andreasen-Huge SSABRas shif t = 3% , β = 40% , α = 2 . , ρ = − . , ν = 27 . . (55)The comparison is plotted in Figure 1.Next we show how to recalibrate Andreasen Huge SSABR following a change of the parameterbeta. We use the same Hagan data as in the previous example, but change beta from 40% to 60%.The recalibrated skew parameters for Andreasen-Huge SSABR are: shif t = 3% , β = 60% , α = 4 . , ρ = − . , ν = 29 .
50% (56)The comparison is plotted in Figure 2.Figure 2:
EUR 10y 10y Swaption Skew, Hagan beta 40% vs Andreasen-Huge beta 60%. shif t = 3% , β = 20% , α = 1 . , ρ = − . , ν = 25 .
84% (57)The comparison is plotted in Figure 3.Figure 3:
EUR 10y 10y Swaption Skew, Hagan beta 40% vs Andreasen-Huge beta 20%.
We shall now consider a change of shift from 3% to 3.25%. The recalibrated skew parameters forAndreasen-Huge SSABR are: shif t = 3 . , β = 40% , α = 2 . , ρ = − . , ν = 27 .
17% (58)The comparison is plotted in Figure 4.Figure 4:
EUR 10y 10y Swaption Skew, Hagan shift 3% vs Andreasen-Huge shift 3.25%.
If we also change Andreasen-Huge beta from 40% to 50% we obtain recalibrated skew parametersfor Andreasen-Huge SSABR as: shif t = 3 . , β = 50% , α = 2 . , ρ = − . , ν = 28 .
01% (59)10he comparison is plotted in Figure 5.Figure 5:
EUR 10y 10y Swaption Skew, Hagan shift 3%, beta 40% vs Andreasen-Huge shift 3.25%, beta 50%.
While analytic skew calibration allows us to make a closer match between Hagan and Andreasen-Huge SSABR implementations, we cannot achieve a perfect match between the two due to numericallimitations. Firstly, the calibration only operates in a small neighborhood of ATM and there is anunavoidable divergence at the wings. Secondly, the use of various numerical procedures to interpolateand extrapolate Andreasen-Huge SSABR in to non-grid points leads to numerical noise that preventsus from having an exact comparison.
Conclusion
We derived analytic formulae which express SABR parameters α , ν and ρ in Andreasen-Huge 1time step framework in terms of five option prices close ATM. We used these formulae to give acharacterisation for SABR parameters in terms of terminal swap rate distribution. We showed howthe formulae can be used for calibrating Andreasen-Huge SSABR model to analytical approximationfor the solution of the SSABR SDE, as well as how Andreasen-Huge SSABR model can be recalibratedafter changes of shift or beta parameters. The later procedure can be used for constructing consistentrisk in the multi parametric SSABR model where different regions of the strike space are governedby different SABR regimes. References [1] Hagan, P. S., Kumar, D., Lesniewski, A.S., Woodward, D.E.,
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