Back-of-the-envelope swaptions in a very parsimonious multicurve interest rate model
BBack-of-the-envelope swaptions in a very parsimoniousmulticurve interest rate model
Roberto Baviera † December 19, 2017 ( † ) Politecnico di Milano, Department of Mathematics, 32 p.zza L. da Vinci, Milano Abstract
We propose an elementary model to price European physical delivery swaptions in multicurvesetting with a simple exact closed formula. The proposed model is very parsimonious: it isa three-parameter multicurve extension of the two-parameter Hull and White (1990) model.The model allows also to obtain simple formulas for all other plain vanilla Interest Ratederivatives. Calibration issues are discussed in detail.
Keywords : Multicurve interest rates, parsimonious modeling, calibration cascade.
JEL Classification : C51, G12.
Address for correspondence:
Roberto BavieraDepartment of MathematicsPolitecnico di Milano32 p.zza Leonardo da VinciI-20133 Milano, ItalyTel. +39-02-2399 4630Fax. +39-02-2399 [email protected] 1 a r X i v : . [ q -f i n . P R ] D ec ack-of-the-envelope swaptionsin a very parsimoniousmulticurve interest rate model1 Introduction The financial crisis of 2007 has had a significant impact also on Interest Rate (hereinafter IR)modeling perspective. On the one hand, multicurve dynamics have been observed in main inter-bank markets (e.g. EUR and USD), on the other volumes on exotic derivatives have considerablydecreased and liquidity has significantly declined even on plain vanilla instruments.While on the first issue there exist nowadays excellent textbooks (see, e.g. Henrard 2014, Grbac andRunggaldier 2015), the main consequence of the second issue, i.e. the need of very parsimoniousmodels, has been largely forgotten in current financial literature where the additional complexityof today financial markets is often faced with parameter-rich models. In this paper the focus ison the two relevant issues of parsimony and calibration.First, the parsimony feature is crucial: in today (less liquid) markets one often needs to handlemodels with very few parameters both from a calibration and from a risk management perspective.In this paper we focus on a three-parameter multicurve extension of the well known two-parametersHull and White (1990) model. This choice is very parsimonious: one of the most parsimoniousMulticurve HJM model in the existing literature is the one introduced by Moreni and Pallavicini(2014) that, in the simplest WG2++ case, requires ten free parameters. Another one has beenrecently proposed by Grbac et al. (2016), that in the simplest model parametrization involves atleast seven parameters.Second, the model should allow for a calibration cascade, the methodology followed by practi-tioners, that consists in calibrating first IR curves via bootstrap techniques and then volatilityparameters. This cascade is crucial and the reason is related again to liquidity. Instruments usedin bootstrap, as FRAs, Short-Term-Interest-Rate (STIR) futures and swaps, are several order ofmagnitude more liquid than the corresponding options on these instruments.The proposed model, besides the calibration of the initial discount and pseudo-discount curves,allows to price with exact and simple closed formulas all plain vanilla IR options: caps/floors,STIR options and European swaptions. While caps/floors and STIR options can be priced withstraightforward modifications of solutions already present in the literature (see, e.g. Henrard 2010,Baviera and Cassaro 2015), in this paper we focus on pricing European physical delivery swaptionderivatives (hereinafter swaptions).We also show in a detailed example the calibration cascade, where the volatility parameters arecalibrated via swaptions.The remainder of the paper is organized as follows. In Section 2, we recall the characteristicsof a swaption derivative contract in a general multicurve setting. In Section 3 we introduce theMulticurve HJM framework and the parsimonious model within this framework; we also provemodel swaption closed formula. In section 4 we show in detail model calibration. Section 5concludes. 2
Interest Rate Swaptions in a multicurve setting
Multicurve setting for interest rates can be found in the two textbooks of Henrard (2014) andGrbac and Runggaldier (2015). In this section we briefly recall interest rate notation and somekey relations, with a focus on swaption pricing in a multicurve setting.Let (Ω , F , P ), with {F t : t ≤ t ≤ T ∗ } , be a complete filtered probability space satisfying the usualhypothesis, where t is the value date and T ∗ a finite time horizon for all market activities. Let usdefine B ( t, T ) the discount curve with t ≤ t < T < T ∗ and D ( t, T ), the stochastic discount, s.t. B ( t, T ) = E [ D ( t, T ) |F t ] . (1)The quantity B ( t, T ) is often called also risk-free zero-coupon bond. For example, market standardin the Euro interbank market is to consider as discount curve the EONIA curve (also called OIScurve). As in standard single curve models, forward discount B ( t ; T, T + ∆) is equal to the ratio B ( t, T + ∆) /B ( t, T ). A consequence of (1) is that B ( t ; T, T + ∆) is a martingale in the T -forwardmeasure. As in Henrard (2014), also a pseudo-discount curve is considered. The following relation holds forLibor rates L ( T, T + ∆) and the corresponding forward rates L ( t ; T, T + ∆) in tB ( t, T + ∆) L ( t ; T, T + ∆) := E [ D ( t, T + ∆) L ( T, T + ∆) |F t ] , (2)where the lag ∆ is the one that characterizes the pseudo-discount curve; e.g. 6-months in theEuribor 6m case.The (foward) pseudo-discounts are defined asˆ B ( t ; T, T + ∆) := 11 + δ ( T, T + ∆) L ( t ; T, T + ∆) (3)with δ ( T, T + ∆) the year-fraction between the two calculation dates for a Libor rate and the spread is defined as β ( t ; T, T + ∆) := B ( t ; T, T + ∆)ˆ B ( t ; T, T + ∆) . From equation (2) one gets B ( t, T ) β ( t ; T, T + ∆) = E [ D ( t, T ) β ( T, T + ∆) |F t ] (4)i.e. β ( t ; T, T + ∆) is a martingale in the T -forward measure. This is the unique property thatprocess β ( t ; T, T + ∆) has to satisfy.Hereinafter, as market standard, all discounts and OIS derivatives refer to the discount curve,while forward forward Libor rates are always related to the corresponding pseudo-discount curvevia (3).
A swaption is a contract on the right to enter, at option’s expiry date t α , in a payer/receiver swapwith a strike rate K established when the contract is written.The underlying swap at expiry date t α is composed by a floating and a fixed leg; typically paymentsdo not occur with the same frequency in the two legs (and they can have also different daycount)and this fact complicates the notation. Flows end at swap maturity date t ω . We indicate floating The T -forward measure is defined as the probability measure s.t. B ( t, T ) E ( T ) [ • |F t ] = E [ D ( t, T ) • |F t ] (see,e.g. Musiela and Rutkowski 2006). t (cid:48) := { t (cid:48) ι } ι = α (cid:48) +1 ...ω (cid:48) (in the Euro market, typically versus Euribor-6m withsemiannual frequency and Act/360 daycount), and fixed leg payment dates t := { t j } j = α +1 ...ω (inthe Euro market, with annual frequency and 30/360 daycount); we define also t (cid:48) α (cid:48) := t α , t (cid:48) ω (cid:48) := t ω .Let us introduce the following shorthands B α j ( t ) := B ( t ; t α , t j ) B α (cid:48) ι ( t ) := B ( t ; t (cid:48) α (cid:48) , t (cid:48) ι ) β ι ( t ) := β ( t ; t (cid:48) ι , t (cid:48) ι +1 ) δ (cid:48) ι := δ ( t (cid:48) ι , t (cid:48) ι +1 ) δ j := δ ( t j , t j +1 ) c j := δ j K for j = α + 1 , . . . , ω − δ ω K for j = ω . A swap rate forward start in t α and valued in t ∈ [ t , t α ], S αω ( t ), is obtained equating in t theNet-Present-Value of the floating leg and of the fixed leg S αω ( t ) = N αω ( t ) BP V αω ( t )with the forward Basis Point Value BP V αω ( t ) := ω − (cid:88) j = α δ j B α j +1 ( t ) (5)and the numerator equal to the expected value in t of swap’s floating leg flows N αω ( t ) := E (cid:34) ω (cid:48) − (cid:88) ι = α (cid:48) D ( t, t (cid:48) ι +1 ) δ (cid:48) ι L ( t (cid:48) ι , t (cid:48) ι +1 ) (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) = 1 − B ( t, t ω ) + ω (cid:48) − (cid:88) ι = α (cid:48) B ( t, t (cid:48) ι ) [ β ι ( t ) − , (6)where the last equality is obtained using relations (1) and (4). Let us observe that the sum offloating leg flows is composed by two parts: the term [1 − B ( t, t ω )], equal to the single curve case,and the remaining sum of B ( t, t (cid:48) ι ) [ β ι ( t ) −
1] that corresponds to the spread correction present inthe multicurve setting.Receiver swaption payoff at expiry date is R αω ( t α ) := BP V αω ( t α ) [ K − S αω ( t α )] + = [ K BP V αω ( t α ) − N αω ( t α )] + . (7)A receiver swaption is the expected value at value date of the discounted payoff R αω ( t ) := E { D ( t , t α ) R αω ( t α ) |F t } = B ( t , t α ) E ( α ) {R αω ( t α ) |F t } where we have also rewritten the expectation in the t α -forward measure. Lemma 1
The two following two properties holdi) N αω ( t ) and BP V αω ( t ) are martingale processes in the t α -forward measure for t ∈ [ t , t α ] ;ii) Receiver swaption payoff (7) reads R αω ( t α ) = (cid:34) B ( t α , t ω ) + K BP V αω ( t α ) + ω (cid:48) − (cid:88) ι = α (cid:48) B ( t α , t (cid:48) ι ) [1 − β ι ( t α )] − (cid:35) + = (cid:34) ω (cid:88) j = α +1 c j B αj ( t α ) + ω (cid:48) − (cid:88) ι = α (cid:48) +1 B α (cid:48) ι ( t α ) − ω (cid:48) − (cid:88) ι = α (cid:48) β ι ( t α ) B α (cid:48) ι ( t α ) (cid:35) + (8)4 roof . Straightforward given the definitions of discount and pseudo-discount curves ♣ This lemma has some relevant consequences. On the one hand, property i) allows generalizingthe Swap Market Model approach in (Jamshidian 1997) to swaptions in the multicurve case,hence it allows obtaining market swaption formulas choosing properly the volatility structure.One can get the Black, Bachelier or Shifted-Black market formula (see, e.g. Brigo and Mercurio2007) where flows are discounted with the discount curve and forward Libor rates are related to pseudo-discounts via (3), as considered in market formulas. Moreover, property i) implies alsothat put-call parity holds also for swaptions in a multicurve setting.On the other hand, property ii) clarifies that a complete specification of the model for swaptionpricing requires only the dynamics for the forward discount and spread curves as specified in thenext section.
A Multicurve HJM model (hereinafter MHJM) is specified providing initial conditions for the discount curve B ( t , T ) and the spread curve β ( t ; T, T + ∆), and indicating their dynamics.
Discount and spread curves’ dynamics in the MHJM framework we consider in this paper are (cid:40) dB ( t ; t α , t i ) = − B ( t ; t α , t i ) [ σ ( t, t i ) − σ ( t, t α )] · [ dW t + ρ σ ( t, t α ) dt ] t ∈ [ t , t α ] dβ ( t ; t i , t i +1 ) = β ( t ; t i , t i +1 ) [ η ( t, t i +1 ) − η ( t, t i )] · [ dW t + ρ σ ( t, t i ) dt ] t ∈ [ t , t i ] (9)where σ ( t, T ) and η ( t, T ) are d-dimensional vectors of adapted processes (in particular in theGaussian case they are deterministic functions of time) with σ ( t, t ) = η ( t, t ) = 0, x · y is thecanonical scalar product between x, y ∈ (cid:60) d , and W is a d-dimensional Brownian motion withinstantaneous covariance ρ = ( ρ i j =1 ,..,d ) dW i,t dW j,t = ρ i j dt . Model (9) is the most natural extension of the single-curve Heath et al. (1992) model. The firstequation in (9) corresponds to the usual HJM model for the discount curve (see, e.g. Musielaand Rutkowski 2006). The second equation in (9) is a very general continuous process satisfyingcondition (4) for the spread .We do not impose any other additional condition for curves’ dynamics as the independence hy-pothesis in Henrard (2014) or the orthogonality condition in Baviera and Cassaro (2015).Change of measures are standard in this framework, because they are a straightforward general-ization of single curve modeling approaches (see, e.g. Musiela and Rutkowski 2006). The process dW ( i ) t := dW t + ρ σ ( t, t i ) dt is a d-dimensional Brownian motion in the t i -forward measure. It is immediate to prove that, givendynamics (9), B ( t ; t α , t i ) is martingale in the t α -forward measure and β ( t ; t i , t i +1 ) is martingale inthe t i -forward measure. Remark 1 . Given equations (9), the dynamics for the pseudo-discounts (3) in the t i -forwardmeasure is d ˆ B ( t ; t i , t i +1 ) = − ˆ B ( t ; t i , t i +1 ) [ σ i ( t ) + η i ( t )] · (cid:104) dW ( i ) t − ρ η i ( t ) dt (cid:105) t ∈ [ t , t i ]5here σ i ( t ) := σ ( t, t i +1 ) − σ ( t, t i ) and η i ( t ) := η ( t, t i +1 ) − η ( t, t i ). The pseudo-discount has avolatility which is the sum of discount volatility σ i ( t ) and of spread volatility η i ( t ).In this paper we consider an elementary 1-dimensional Gaussian model within MHJM framework(9). Volatilities for the discount curve σ ( t, T ) and for the spread curve η ( t, T ) are modeled as (cid:40) σ ( t, T ) = (1 − γ ) v ( t, T ) η ( t, T ) = γ v ( t, T ) with v ( t, T ) := σ − e − a ( T − t ) a a ∈ (cid:60) + \ { } σ ( T − t ) a = 0 (10)with a, σ ∈ (cid:60) + and γ ∈ [0 , v ( t, T ) is strictly positive.The selection of this model originates from two facts related to the IR derivatives available forcalibration. On the one hand, in the calibration cascade, “linear” IR derivatives (i.e depos, FRAs,STIR futures and swaps) are used for discount and pseudo-discount initial curve bootstrap, whilethe other parameters are calibrated on IR options. In the market, liquid IR option are STIRoptions, caps/floors and swaptions; unfortunately options on OIS are not liquid in the marketplace (see, e.g. Moreni and Pallavicini 2014, and references therein).On the other hand, in liquid IR options, the key driver is the pseudo-discount curve ˆ B ( t, T ) viaa Libor rate or a swap rate, where the latter can be seen as combinations of Libor rates (see,e.g. eq.(1.28) in Grbac and Runggaldier 2015). Hence, when IR curves move, the main driver is pseudo-discount curve, directly related to option underlyings; the discount curve appears only inweights or discount factors, and swaption sensitivities w.r.t. the discount curve are less than thecorresponding sensitivities w.r.t. the pseudo-discount curve.These facts lead to the conclusion that is much more difficult to calibrate volatility parametersspecific to the discount curve. Thus Remark 1 plays a crucial role when selecting the most par-simonious model within framework (9): ˆ B ( t, T ) dynamics has volatility equal to v ( t, T ) in MHWmodel (10). A parsimonious choice should associated a fraction 1 − γ of volatility v ( t, T ) to the discount curve and the remaining fraction γ to the spread dynamics; in fact, as previously dis-cussed, options on OIS are not liquid enough and then a separate calibration of σ ( t, T ) and η ( t, T )in a generic MHJM is not feasible in practice.Moreover, MHW model (10) allows pricing IR options in an elementary way. STIR options andcaps/floors Black-like formulas can be obtained via a straightforward generalization of the solutionsin Henrard (2010) and Baviera and Cassaro (2015). In this section we show that it is possible toprice also swaptions via a simple closed formula. To the best of our knowledge, MHW model (10)is the first Multicurve HJM where all plain vanilla derivatives can be written with simple exactclosed formulas that are extensions of Black (1976) formulas.The remaining part of this section is divided as follows. We first show in Lemma 2 how to write,within MHW model (10), each element in receiver swaption payoff (8) as a simple function of onesingle Gaussian r.v. ξ . Then, (technical) Lemma 3 shows that swaption payoff can be rewrittenas a function of ξ and this function presents interesting properties. Finally in Proposition 1 weprove the key result of this section: the exact closed formula for swaptions according to model(10). 6t is useful to introduce the following shorthands v α (cid:48) ι := v ( t α , t (cid:48) ι ) ι = α (cid:48) , . . . , ω (cid:48) ς α (cid:48) ι := (1 − γ ) v α (cid:48) ι ι = α (cid:48) , . . . , ων α (cid:48) ι := ς α (cid:48) ι − (cid:0) η ( t α , t (cid:48) ι +1 ) − η ( t α , t (cid:48) ι ) (cid:1) ι = α (cid:48) , . . . , ω (cid:48) − . Remark 2 . Volatilities { v α (cid:48) ι } ι = α (cid:48) +1 ...ω (cid:48) are always positive and are strictly increasing with ι . Thequantities { ν α (cid:48) ι } ι = α (cid:48) +1 ...ω (cid:48) can change sign depending on the value of γ . In fact ν α (cid:48) ι = v α (cid:48) ι − γ v α (cid:48) ι +1 = v α (cid:48) ι +1 (˜ γ ι − γ )with ˜ γ ι := v α (cid:48) ι /v α (cid:48) ι +1 ∈ (0 , γ = 0 all { ν α (cid:48) ι } ι = α (cid:48) +1 ...ω (cid:48) − are positive and ν α (cid:48) α (cid:48) isnegative, while for larger values of γ some ν α (cid:48) ι become negative. For γ equal or close to 1 all { ν α (cid:48) ι } ι = α (cid:48) ...ω (cid:48) − are negative. Due to these possible negative values, { ν α (cid:48) ι } ι are not volatilities; wecall them extended volatilities. Lemma 2
Discount and spread curves in t α can be written, according to the MHW model (10) inthe t α -forward measure, as B α (cid:48) ι ( t α ) = B α (cid:48) ι ( t ) exp (cid:26) − ς α (cid:48) ι ξ − ς α (cid:48) ι ζ (cid:27) ι = α (cid:48) + 1 , . . . , ω (cid:48) β ι ( t α ) B α (cid:48) ι ( t α ) = β ι ( t ) B α (cid:48) ι ( t ) exp (cid:26) − ν α (cid:48) ι ξ − ν α (cid:48) ι ζ (cid:27) ι = α (cid:48) , . . . , ω (cid:48) − where ξ := (cid:90) t α t dW ( α ) u e − a ( t α − u ) (12) a zero mean Gaussian r.v. whose variance is ζ := − e − a ( t α − t ) a a ∈ (cid:60) + \ { } t α − t a = 0 . Proof . A straightforward application of Itˆo calculus, given dynamics (9) and deterministic volatil-ities (10) ♣ A consequence of previous lemma is that receiver swaption payoff (8) in the t α -forward measurecan be written as a function of a unique r.v. ξ as R αω ( t α ) =: [ f ( ξ )] + . (13)In the following lemma we show that f ( ξ ) is equal to a finite sum of exponential functions of ξ ,i.e. f ( ξ ) = (cid:88) i w i e λ i ξ with w i , λ i ∈ (cid:60) where some w i < λ i ≥
0. Hence, the swaption looks like a non-trivial spread option,with a number of terms equal to ω − α − ω (cid:48) − α (cid:48) ).In Lemma 3 we prove that, even if the function f , for some parameters choices, is not a decreasingfunction of ξ , however there exists a unique value ξ ∗ s.t. f ( ξ ∗ ) = 0, i.e. the equality S αω ( t α ) = K is satisfied for this unique value. 7 emma 3 According to MHW model (10) , the function f ( ξ ) in swaption payoff is equal to f ( ξ ) = ω (cid:88) j = α +1 c j B αj ( t ) e − ς αj ξ − ς αj ζ / (a)+ ω (cid:48) − (cid:88) ι = α (cid:48) +1 B α (cid:48) ι ( t ) e − ς α (cid:48) ι ξ − ς α (cid:48) ι ζ / (b) − ω (cid:48) − (cid:88) ι = α (cid:48) β ι ( t ) B α (cid:48) ι ( t ) e − ν α (cid:48) ι ξ − ν α (cid:48) ι ζ / (c) and ∃ ! ξ ∗ s.t. f ( ξ ∗ ) = 0 for a, σ ∈ (cid:60) + and γ ∈ [0 , . Moreover, the function f is greater than zerofor ξ < ξ ∗ .Proof . See Appendix A ♣ We have proven that, even if function f is not monotonic in its argument, there is a uniquesolution for equation f ( ξ ) = 0. This fact grants the possibility to extend to MHW the approachof Jamshidian (1989). In the following proposition we prove that a closed form solution holds fora receiver swaption for model (10). Proposition 1
A receiver swaption, according to MHW model (10), can be computed with theclosed formula R mhw αω ( t ) = B ( t , t α ) (cid:40) ω (cid:88) j = α +1 c j B αj ( t ) N (cid:18) ξ ∗ ζ + ζ ς αj (cid:19) + ω (cid:48) − (cid:88) ι = α (cid:48) +1 B α (cid:48) ι ( t ) N (cid:18) ξ ∗ ζ + ζ ς α (cid:48) ι (cid:19) − ω (cid:48) − (cid:88) ι = α (cid:48) β ι ( t ) B α (cid:48) ι ( t ) N (cid:18) ξ ∗ ζ + ζ ν α (cid:48) ι (cid:19) (cid:41) (14) where N ( • ) is the standard normal CDF and ξ ∗ is the unique solution of f ( ξ ) = 0 .Proof . See Appendix A ♣ Let us comment above proposition, which is the most relevant analytical result of this paper. Itgeneralizes the celebrated result of Jamshidian (1989) to this Multicurve HJM model. The maindifference is that also negative addends appear in the receiver swaption R mhw αω ( t ) and there are extended volatilities instead of standard volatilities. It is straightforward to prove that, mutatismutandis , a similar solution holds for a payer swaption. In this section we show in detail model calibration of market parameters in the Euro marketconsidering European ATM swaptions vs Euribor 6m with the end-of-day market conditions ofSeptember 10, 2010 (value date).As discussed in the introduction, the calibration cascade is divided in two steps. First, we boot-strap the discount and the pseudo-discount curves from 6m-Depo, three FRAs (1 × , × ×
9) and swaps (both OIS and vs Euribor 6m). Then, we calibrate the three MHW parameters p := ( a, σ, γ ) with European ATM swaptions vs Euribor 6m on the 10y-diagonal (i.e. consideringthe M = 9 ATM swaptions 1y9y, 2y8y, . . . , 9y1y).8IS rate (%) swap rate vs
6m (%)1w -0.132 -2w -0.132 -1m -0.132 -2m -0.133 -3m -0.136 -6m -0.139 -1y -0.147 0.0442y -0.135 0.0803y -0.083 0.1544y 0.008 0.2595y 0.122 0.3776y 0.254 0.5127y 0.392 0.6528y 0.529 0.7869y 0.655 0.90910y 0.766 1.01611y 0.866 1.10912y 0.957 1.19515y 1.160 1.383Table 1:
OIS rates and swap rates vs Euribor 6m in percentages: end-of-day mid quotes (annual 30/360day-count convention for swaps vs 6m, Act/360 day-count for OIS) on 10 September 2015.
The discount curve is bootstrapped from OIS quoted rates with the same methodology describedin Baviera and Cassaro (2015). Their quotes at value date are reported in Table 1 (with marketconventions, i.e. annual payments and Act/360 day-count); in the same table we report also theswap rates (annual fixed leg with 30/360 day-count). In Table 2 we show the relevant FRA ratesand the Euribor 6m fixing on the same value date (both with Act/360 day-count). All market dataare provided by Bloomberg. Convexity adjustments for FRAs, present in the MHW model, areneglected because they do not impact the nodes relevant for the diagonal swaptions co-terminal10y considered in this calibration and they are very small in any case. In figure 1 we show the discount and pseudo-discount curves obtained via the bootstrapping technique.rate (%)Euribor 6m 0.038FRA 1 × × × Euribor 6m fixing rate and FRA in percentages (day-count Act/360). FRA rates are end-of-daymid quotes at value date.
We show the swaption ATM volatilities in basis points (bps) in Table 3; the swaption marketprices are obtained according to the standard normal market model; a model choice that allowsfor negative interest rates. 9igure 1:
Discount
OIS curve (in red) and pseudo-discount
Euribor-6m curve (in blue) on September10, 2010, starting from the settlement date and up to a 12y time horizon. expiry tenor volatility (bps)1y 9y 64.702y 8y 66.783y 7y 68.534y 6y 70.915y 5y 72.366y 4y 73.077y 3y 73.218y 2y 73.519y 1y 73.45Table 3:
Normal volatilities for ATM diagonal swaptions co-terminal 10y in bps on 10 September 2015.
We minimize the square distance between swaption model and market prices
Err ( p ) = M (cid:88) i =1 [ R mhw i ( p ; t ) − R mkt i ( t )] where market ATM swaption pricing formula according to the multicurve normal model is reportedin Appendix B.We obtain the parameter estimations minimizing the Err function w.r.t. a, γ and ˜ σ := σ/a ;the solution is stable for a large class of starting points. As estimations we obtain a = 13 . σ = 1 .
27% and γ = 0 . Err function w.r.t. γ is less pronouncedcompared to the one w.r.t. a and σ ; even if the minimum values for the Err function are achievedfor very low values of γ , however, differences in terms of mean squared error are very smallincreasing, even significantly, γ : another evidence that the most relevant dynamics for swaption10igure 2: Market prices for ATM diagonal swaptions co-terminal 10y in percentages (squares in red)and the corresponding ones obtained via the MHW calibration (diamonds in blue) for the 9 expiriesconsidered. valuation is the one related to the pseudo-discount curve, where the corresponding volatility doesnot depend on γ parameter. Is it possible to consider a parsimonious multicurve IR model without assuming constant spreads ?In this paper we introduce a three parameter generalization of the two parameters Hull andWhite (1990) model, where the additional parameter γ lies in the interval [0 , γ = 0 correspondsto the S0 hypothesis in Henrard (2010), where the spread curve is constant over time, while γ = 1corresponds to the S1 assumption in Baviera and Cassaro (2015).We have proven that the model allows a very simple closed formula for European physical deliveryswaptions (14) with a formula, very similar to the one of Jamshidian (1989), with the presence of extended volatilities, that can assume negative values. Model calibration is immediate: we haveshown in detail how to implement the calibration cascade on the September 10, 2010 end-of-daymarket conditions.The proposed model allows also Black-like formulas for the other liquid IR options (caps/floorsand STIR options) and simple analytical convexity adjustments for FRAs and STIR futures;furthermore numerical techniques similar to the HW model can be applied.This very parsimonious model is justified by the good calibration properties on ATM swaptionprices and by the observation that the pseudo-discount dynamics is the relevant one in the valuationof liquid IR options. Furthermore a very parsimonious model, as the proposed MHW model (10),can be the choice of election in challenging tasks where the multicurve IR dynamics is just oneof the modeling elements: two significant examples are the pricing and the risk management ofilliquid corporate bonds, and the XVA valuations including all contracts between two counterpartswithin a netting set at bank level. 11 cknowledgments We would like to thank Aldo Nassigh, Andrea Pallavicini and Wolfgang Runggaldier for some nicediscussions on the subject. The usual disclaimers apply.
Appendix A
Proof of Lemma 3 . Function f ( ξ ) is obtained from direct substitution of swaption payoff com-ponents (11) in Receiver payoff (8). f ( ξ ) is a sum of exponentials exp( λ i ξ ) multiplied by somecoefficients ω i , where both λ i , ω i ∈ (cid:60) . Function f ( ξ ) is composed by different parts: positiveaddends with negative exponentials (terms a and b ) and a negative term with a positive exponen-tial (first addend in c for ι = α (cid:48) ), which becomes a negative constant for γ = 0. The remainingcoefficients in ( c ) are always negative and they can be divided into three parts; one with negativeexponentials ( ν αι > ν αι <
0) and a third part constantwhen at least one ν αι is equal to 0.Let us study f ( ξ ) as a function of ξ ∈ (cid:60) . It is a very regular function ( C ∞ ), a finite sum ofexponentials and constants. We divide the addends of function f in two parts. In the first one f + ( ξ ) we consider the sum of all positive addends (i.e. terms a and b) and in the second one f − ( ξ )the sum of all negative addends (i.e. term c) in absolute value, i.e. f ( ξ ) =: f + ( ξ ) − f − ( ξ )where both f + ( ξ ) and f − ( ξ ) are positive functions of their argument: f + ( ξ ) is the sum of negativeexponentials while f − ( ξ ) can be the sum of both positive, negative exponentials and a constant(only for a finite set of values for γ , for the values of γ equal to one of the { ˜ γ ι } ι = α (cid:48) +1 ,...,ω (cid:48) ).First, let us observe that a positive addend is leading for small ξ . This fact is a consequence ofthe following inequalities that hold ∀ ι = α (cid:48) + 1 , . . . , ω (cid:48) v α (cid:48) ι − < v α (cid:48) ι , ν α (cid:48) ι ≤ (1 − γ ) v α (cid:48) ι where the equality holds only for γ = 0 , (15)immediate consequences of volatility definitions (10). For all values of γ the leading term of f ( ξ )for small ξ is c ω B αω ( t ) e − (1 − γ ) v αω ξ + ··· because, due to inequalities (15), − (1 − γ ) v αω is the lowest exponent coefficient that multiplies ξ among the exponentials in f ( ξ ); i.e. there exists always a ˆ ξ s.t. ∀ ξ < ˆ ξ f + ( ξ ) > f − ( ξ ).Then, let us define ˜ γ := max ι ˜ γ ι and let us distinguish three cases depending on γ value:1. When ˜ γ ≤ γ ≤ f − ( ξ ), due to Remark 2 , is a positive linear combination of positiveexponentials (and a positive constant when γ = ˜ γ ). Also this case admits one uniqueintersection with f + ( ξ ), which is a sum of negative exponentials for γ <
1, as mentionedabove, while is a constant for γ = 1.2. When 0 < γ < ˜ γ , f − ( ξ ) is a u-shaped positive function since it is a positive linear combinationof positive and negative exponentials (and a constant for some values of γ ). Moreover f + ( ξ )and f − ( ξ ) present one unique intersection, because f + ( ξ ) goes to + ∞ for ξ → −∞ fasterthan f − ( ξ ) and to 0 for ξ → + ∞ . 12. The case with γ = 0 should be treated separately. In this case f ( ξ ) = ω (cid:88) j = α +1 c j B αj ( t ) e − v αj ξ − v αj ζ / − β α (cid:48) ( t ) − ω (cid:48) − (cid:88) ι = α (cid:48) +1 ( β ι ( t ) − B α (cid:48) ι ( t ) e − v α (cid:48) ι ξ − v α (cid:48) ι ζ / all addends are negative exponentials and constants, and then the limit for ξ → + ∞ isequal to − β α (cid:48) ( t ). Moreover, due to inequalities (15), − v α (cid:48) α (cid:48) +1 (always lower than zero) isthe largest exponent coefficient that multiplies ξ among the exponentials in f ( ξ ), the leadingterm for large ξ is − ( β α (cid:48) +1 ( t ) − B α (cid:48) α (cid:48) +1 ( t ) e − v α (cid:48) α (cid:48) +1 ξ − v α (cid:48) α (cid:48) +1 ζ / hence f ( ξ ) tends to − β α (cid:48) ( t ) < ξ → ∞ . With similar arguments applied tothe first derivative of f ( ξ ), one can show that the function has one minimum. Summarizing,for γ = 0 the function f ( ξ ) is a decreasing function up to its minimum ξ min (reaching a valuelower than − β α (cid:48) ( t ) <
0) and then it gradually goes to − β α (cid:48) ( t ) from below for ξ > ξ min .Also in this case the function f ( ξ ) presents a unique intersection with zero.We have then proven that, for all parameters choices, there exists a unique value ξ ∗ s.t f ( ξ ∗ ) = 0.The proof is complete once we observe that, for ξ < ξ ∗ , the function f ( ξ ) is larger than zero inthe three cases described above ♣ Proof of Proposition 1 . Due to
Lemma 3 , swaption receiver is equivalent to R αω ( t ) /B ( t , t α ) = E { f ( ξ ) } + = E { f ( ξ ) | ξ ≤ ξ ∗ } = ω (cid:88) j = α +1 c j E (cid:110)(cid:104) B αj ( t ) e − ς αj ξ − ς αj ζ / (cid:105) ξ ≤ ξ ∗ (cid:111) + ω (cid:48) − (cid:88) ι = α (cid:48) +1 E (cid:110)(cid:104) B αι ( t ) e − ς αι ξ − ς αι ζ / (cid:105) ξ ≤ ξ ∗ (cid:111) − ω (cid:48) − (cid:88) ι = α (cid:48) E (cid:110)(cid:104) β ι ( t ) B αι ( t ) e − ν αι ξ − ν αι ζ / (cid:105) ξ ≤ ξ ∗ (cid:111) and then, after straightforward computations, one proves the proposition ♣ Appendix B
In this appendix we report the Normal-Black formula for a receiver swaption: R mkt αω ( t ) = B ( t , t α ) BP V αω ( t ) (cid:8) [ K − S αω ( t )] N ( − d ) + σ αω √ t α − t φ ( d ) (cid:9) where N ( • ) is the standard normal CDF, φ ( • ) the standard normal density function and σ αω thecorresponding implied normal volatility d := S αω ( t ) − Kσ αω √ t α − t . The ATM formula simplifies to R mkt αω ( t ) = B ( t , t α ) BP V αω ( t ) σ αω (cid:114) t α − t π . eferences Baviera, R. and Cassaro, A., 2015. A note on dual curve construction: Mr. Crab’s bootstrap,
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Symbol Description a, σ, γ
Multicurve Hull and White (10) parameters; a, σ ∈ (cid:60) + and γ ∈ [0 , B ( t, T ) discount curve, zero-coupon bond in t with maturity TB ( t ; T, T + ∆) forward discount in t between T and T + ∆, t ≤ T < T + ∆ˆ B ( t ; T, T + ∆) forward pseudo-discount in t between T and T + ∆, t ≤ T < T + ∆ β ( t ; T, T + ∆) forward spread in t between T and T + ∆, t ≤ T < T + ∆ β ( t, T ) spread curve in t with maturity Tδ ( t j , t j +1 ) year-fraction between two payment dates in swap’s fixed leg δ ( t (cid:48) ι , t (cid:48) ι +1 ) year-fraction between two payment dates in swap’s floating leg∆ the lag that characterizes the pseudo-discounts , e.g. 6-months for Eur6m K strike rate N ( • ) the standard normal CDF ρ correlation matrix in (cid:60) d × d s.t. dW i,t dW j,t = ρ i j dtσ ( t, T ) HJM discount volatility in (cid:60) d between t and Tη ( t, T ) HJM spread volatility in (cid:60) d between t and T R αω ( t α ) receiver swaption payoff at expiry R αω ( t ) receiver swaption price at value date t value date t α swaption expiry date t ω underlying swap maturity date t := { t j } j underlying swap fixed leg payment dates, j = α + 1 , . . . , ω t (cid:48) := { t (cid:48) ι } ι underlying swap floating leg payment dates, ι = α (cid:48) + 1 , . . . , ω (cid:48) W t vector of correlated Brownian motions in (cid:60) d s.t. dW i,t dW j,t = ρ i j dtx · y canonical scalar product in (cid:60) d x scalar product x · ρx with x ∈ (cid:60) d and ρ correlation matrix ξ Gaussian r.v. defined in (12) with zero mean and variance ζ ξ ∗ the unique solution of f ( ξ ) = 0; f ( ξ ) defined in (13) ζ standard deviation of the Gaussian r.v. ξ horthands B α j ( t ) : B ( t ; t α , t j ) B α (cid:48) ι ( t ) : B ( t ; t (cid:48) α (cid:48) , t (cid:48) ι ) β ι ( t ) : β ( t ; t (cid:48) ι , t (cid:48) ι +1 ) δ (cid:48) ι : δ ( t (cid:48) ι , t (cid:48) ι +1 ) δ j : δ ( t j , t j +1 ) c j : δ j K for j = α + 1 , . . . , ω − δ ω K for j = ωv α (cid:48) ι : v ( t α , t (cid:48) ι ) ς α (cid:48) ι : (1 − γ ) v α (cid:48) ι ν α (cid:48) ι : ς α (cid:48) ι − (cid:0) η ( t α , t (cid:48) ι +1 ) − η ( t α , t (cid:48) ι ) (cid:1) IR : Interest RateMHW : Multicurve Hull White model (10)r . v . : random variables . t . : such thatw . r . t . : with respect to ..