A flexible matrix Libor model with smiles

Transcription

1 A flexible matrix Libor moel with smiles José Da Fonseca Alessanro Gnoatto Martino Grasselli December 3, 2011 Abstract We present a flexible approach for the valuation of interest rate erivatives base on Affine Processes. We exten the methoology propose in Keller-Ressel et al. 2009) by changing the choice of the state space. We provie semi-close-form solutions for the pricing of caps an floors. We then show that it is possible to price swaptions in a multifactor setting with a goo egree of analytical tractability. This is one via the Egeworth expansion approach evelope in Collin-Dufresne an Golstein 2002). A numerical exercise illustrates the flexibility of Wishart Libor moel in escribing the movements of the implie volatility surface. Keywors: Affine processes, Wishart process, Libor market moel, Fast Fourier Transform, Caps, Floors, Swaptions. JEL coes: G13, C51. Aucklan University of Technology, Department of Finance, Private Bag 92006, 1142 Aucklan, New Zealan. jose.afonseca@aut.ac.nz. Università egli Stui i Paova, Dipartimento i Matematica Pura e Applicata, Via Trieste 63, Paova, Italy. alessanro@alessanrognoatto.com Università egli Stui i Paova, Dipartimento i Matematica Pura e Applicata, Via Trieste 63, Paova, Italy. grassell@math.unip.it an Ecole Supérieure Ingénieurs Léonar e Vinci, Département Mathématiques et Ingénierie Financière, Paris La Défense, France. 1 Electronic copy available at:

2 1 Introuction In this paper we present a unifie framework for the valuation of caps, floors an swaptions. These instruments are the most common erivative securities which are trae in a fixe income esk of a financial institution see e.g. Brigo an Mercurio 2006)). Practitioners usually price these proucts by relying on a Black-Scholes like formula, which was first presente in Black 1976). The market convention of pricing caps an swaptions using the Black formula is base on an application of the Black an Scholes 1973) formula for stock options by assuming that the unerlying interest rates are lognormally istribute. Remarkably, the use of this kin of formulae ha no theoretical justification, since they involve a proceure in which the iscount factor an the Libor rates were assume to be inepenent so as to write the pricing formula as a prouct of a bon price an the expecte payoff. The systematic use of this market practice ignite the interest of acaemics aiming at proviing a coherent theoretical backgroun. In a series of articles, Miltersen et al. 1997), Brace et al. 1997), Jamshiian 1997) an Musiela an Rutkowski 1997), provie these theoretical founations, introucing the Libor an Swap Market Moel. Following these papers a stream of contributions appeare, trying to exten the basic moel to the case where the volatility of the unerlying factor is stochastic. The most famous proposals on this sie can be foun e.g. in Anersen an Brotherton-Ratcliffe 2001), Wu an Zhang 2006), Joshi an Rebonato 2003), Anersen an Anreasen 2002), Piterbarg 2005a), Piterbarg 2005b). Other approaches explore ifferent ynamics for the riving process with respect to the CEV or isplace-iffusion consiere before for the Libor rate: for example Glasserman an Kou 2003), Eberlein an Özkan 2005) introuce jump an more general Lévy processes, allowing for iscontinuous sample paths of the riving process. Another interesting approach is the one of Brigo an Mercurio 2003) base on a mixture of lognormals. A typical problem in the previous approaches is that once the close form solution for cap prices is foun, to obtain an analogous result for swaptions it is customary to assume that the unerlying which is a coupon bon) behaves like a scalar process typically again geometric Brownian motion). This results in inconsistencies between the so-calle Libor an Swap Market Moels. Even more important, by assuming that the coupon bon is riven by a scalar process, we o not take into account the correlation effects among the ifferent coupons, a key feature of a swaption which may be viewe also as a correlation prouct. This last remark is of paramount importance for practitioners see e.g. the introuction of Collin-Dufresne an Golstein 2002)). In this paper we consier a new approach base on the stochastic iscount factor methoology, where instea of moeling irectly the Libor rate, one concentrates on quotients of trae assets i.e. bons). It has been first introuce by Constantinies 1992) an then evelope by Gouriéroux an Sufana 2011) in a spot interest rate framework an by Keller-Ressel et al. 2 Electronic copy available at:

3 2009) in a Libor perspective. In this latter work they use affine processes on the state space R 0 as riving processes an provie a full characterization of the moel, which allows them to provie close form solutions for caps an swaptions up to Fourier integrals. This approach is very interesting an easily overcomes many ifficulties which are to be face in the computation of Raon-Nikoym erivatives. We provie an extension of this approach, by consiering affine processes on the state space S ++, the set of positive efinite symmetric matrices. This state space may seem awkwar at first sight, but the processes belonging to this family amit a characterization in terms of ODE s which resembles the one foun for stanar affine moels, an example being given by the famous Duffie an Kan 1996) moel. In fact, in Cuchiero et al. 2011) the authors exten to the set S + the set of positive semiefinite symmetric matrices) the classification of affine processes performe by Duffie et al. 2003) for the state space R 0 Rn introuce by Duffie an Kan 1996). What is more, the state space S ++ leas to stochastic factors which are non trivially correlate. The most famous example of process efine in the set S ++ is the Wishart process, originally efine by Bru 1991), introuce in finance by Gouriéroux an Sufana 2003) an then extensively applie in Gouriéroux an Sufana 2010), Gouriéroux an Sufana 2011), Da Fonseca et al. 2008), Da Fonseca et al. 2007b), Da Fonseca et al. 2011), Da Fonseca an Grasselli 2011), among others. The interesting feature of our framework is the possibility to obtain semi-close form solutions for the pricing of swaptions in a multifactor setting, which is a well known challenging problem. In fact the exercise probability involves a multi-imensional inequality. There have been many approaches to simplify the problem: for example, Singleton an Umantsev 2002) suggest an approximation of the exercise bounary with a linear function of the state variables. However, the most efficient approach seems to be the one of Collin-Dufresne an Golstein 2002) which heavily uses the affine structure of the moel an is base on the Egeworth expansion for the characteristic function in terms of the cumulants. Since the cumulants ecay very quickly the Egeworth expansion for the exercise probability turns out to be very accurate an fast. The paper is organize as follows. In Section 2 we introuce our framework by recalling some useful efinitions an results on affine processes. Section 3 investigates the case of the state space S ++ an presents the technical results. In Section 4 we focus on the pricing problem of the relevant erivatives. Caps an Floors are briefly treate since their pricing is now quite stanar within the FFT methoology, while we evote more attention to the pricing of swaptions by aopting the approach of Collin-Dufresne an Golstein 2002). Section 5 illustrates the flexibility of our framework through a numerical exercise. Section 6 conclues the paper, an we gather in the technical Appenices proofs an some remarks useful for implementation. 3

4 of strictly positive efinite sym- 2 Affine Processes on the set S ++ metric matrices 2.1 General results an notations To outline the setup we will consier affine processes taking values in the interior of the cone S +. We will use the notations ψ tu) = φt, u) an φ t u) = φt, u) so as to be consistent with Keller-Ressel et al. 2009). We will be employing a property of the functions efining the Laplace transform, that we report after the following Definition 1. Cuchiero et al. 2011), Definition 2.1) Let Ω, F, F t ) t 0, P) be a filtere probability space with the filtration F t ) t 0 satisfying the usual assumptions. A Markov process Σ = Σ t ) t 0 with state space S +, transition probability p tσ 0, A) = PΣ t A) for A S +, an transition semigroup P t ) t 0 acting on boune functions f on S + if: is calle affine process 1. it is stochastically continuous, that is, lim s t p s Σ 0, ) = p t Σ 0, ) weakly on S + t, x S +, an 2. its Laplace transform has exponential-affine epenence on the initial state: ] P t e Tr[uΣ0] = E [e Tr[uΣt] F0 = e Tr[uξ] p t Σ 0, ξ) = e φtu) Tr[ψtu)Σ0], 1) S + t an Σ 0, u S +, for some function φ : R 0 S + R 0 an ψ : R 0 S + S+. Having applications in min, we will consier affine processes which are solvable in the sense of Grasselli an Tebali 2008) who investigate affine processes on the more general symmetric cone state space omain): this means that the state space that we will consier is the S ++ interior of S +, namely the cone of strictly positive efinite symmetric matrices, enote by 1. Solvability is important, in fact it ensures that the Riccati Orinary Differential Equation associate to the Laplace transform 1) through the usual Feynman-Kac argument has a regular globally integrable flow: this will be crucial to outline our methoology see e.g. the proof of Theorem 4 in the sequel). The next property closes our survey on affine processes. It will be neee when we prove that the structure of the moel is preserve uner changes of measure. Lemma 2. Cuchiero et al. 2011) Lemma 3.2) Let Σ be an affine process on S +, then the functions φ an ψ satisfy the following property: S φ t+s u) = φ t u) + φ s ψ t u)), ψ t+s u) = ψ s ψ t u)). 1 By analogy, the set of negative resp. strictly negative) efinite symmetric matrices will be enote by resp. S ). 4

5 2.2 Examples The previous general framework may be quite abstract at a first sight, mostly because of the high technical level require to properly introuce the notion of amissibility an existence for affine processes, see Cuchiero et al. 2011). In this subsection we provie some examples which will illustrate some concrete applications. We start with the most important one, which will also constitute our main object of stuy in the numerical illustrations The Wishart process We suppose that the process Σ is governe by the following matrix) SDE: Σ t = ΩΩ + MΣ t + Σ t M )t + Σ t W t Q + Q W t Σt, 2) which was first stuie by Bru 1991) an whose solution is known as Wishart process. We assume M, Q invertible an M negative efinite so as to ensure stationarity of the process. Moreover we require ΩΩ = κq Q for a real parameter κ + 1 to grant solvability or equivalently to grant that DetΣ t ) > 0 with probability 1). Uner the solvability assumption Grasselli an Tebali 2008) showe that the Riccati ODE corresponing to the characteristic function can be linearize an therefore amits a close form solution. This is important in view of possible applications since in this case the functions φ an ψ in efinition 1) are explicitly known: Proposition 3. Consier the process Σ = Σ t ) 0 t T which solves the SDE 2). Then the conitional Laplace transform is given by: [ ] E e Tr[uΣ T ] F t = e φτ u) Tr[ψτ u)σt], 3) where τ := T t. The functions φ τ u) an ψ τ u) satisfy the following system of ODE s: which is solve by where an ψ τ τ = ψ τ u)m + M ψ τ u) 2ψ τ u)q Qψ τ u), ψ 0 u) = u, 4) φ [ ] τ τ = Tr κq Qψ τ u), φ 0 u) = 0 5) ψ τ u) = uψ 12,τ u) + ψ 22,τ u)) 1 uψ 11,τ u) + ψ 21,τ u)), 6) ψ 11,τ u) ψ 21,τ u) ψ 12,τ u) ψ 22,τ u) ) = exp { τ M 2Q Q 0 M )} φ τ u) = κ ] [log 2 Tr uψ 12,τ u) + ψ 22,τ u)) + M τ. 8) 7) Proof. See Grasselli an Tebali 2008). 5

6 The Wishart process constitutes the matrix analogue of the square root Bessel) process. In fact we have that the matrix M can be thought of as a mean reversion parameter: this is evient from the Lyapunov equation efining the long-run matrix Σ, which is given by κq Q = MΣ + Σ M. 9) The secon way to appreciate the analogies w.r.t the square root process is to look at the ynamics of the entries of the matrix process Σ. Concentrating on the main iagonal, in the 2 2 case we have: Σ 11 = κ Q Q 2 21) + 2 M11 Σ 11 + M 12 Σ 12 ) ) t + 2σt 11 Q11 Wt 11 + Q 21 Wt 12 ) + 2σ 12 t Q11 W 21 + Q 21 Wt 22 ) Σ 22 = κ Q Q 2 12) + 2 M21 Σ 12 + M 22 Σ 22 ) ) t + 2σt 12 Q12 Wt 11 + Q 22 Wt 12 ) + 2σ 22 t Q12 W 21 + Q 22 W 22) 11) 10) where we set σ 11 σ 12 σ 12 σ 22 ) := Σ. 12) We refer to Da Fonseca et al. 2007a) for aitional insights on the behavior of the Wishart process when aggregating its parameters The pure jump OU process The proceure we aopt in this paper is general, meaning that we can consier ifferent examples of processes lying in the cone of positive efinite matrices. In particular, we may consier the matrix suborinators propose by Barnorff-Nielsen an Stelzer 2007), or jump-iffusions like in Leippol an Trojani 2010). In what follows we provie some examples with the calculations of the functions φ τ an ψ τ. Let us consier the SDE Σ t = MΣ t + Σ t M + L t, 13) where M GL) is assume as usual to be negative efinite so as to grant stationarity, an L t is a pure jump process compoun Poisson Process) with constant intensity λ an jump istribution ν with support on S ++. The strong solution to this equation is given by: Σ t = e Mt Σ 0 e M t + t 0 e Mt s) L s e M t s). 14) We are intereste in the computation of the Laplace transform of this family of processes: [ ] E e Tr[uΣ T ] F t = e φτ u) Tr[ψτ u)σt]. 15) 6

7 The functions φ an ψ solve the following matrix) ODE s: ψ τ τ = ψ τ u)m + M ψ τ u) ψ 0 u) = u 16) φ ) τ τ = λ e Tr[ψτ u)ξ] 1 νξ) φ 0 u) = 0. 17) S + \{0} The solution for the first ODE is given by: ψ τ u) = e M τ ue Mτ, 18) so we can compute the Laplace transform by quarature: [ ] ) φ τ e τ = λ Tr e M τ ue Mτ ξ 1 νξ). 19) S + \{0} In the following we provie explicit computations by assuming some particular istribution ν ) for the jump size. The proofs of this formulae may be foun in Gupta an Nagar 2000). For the sake of clarity, we specify that the Wishart istribution that we consier in the next sections are the classical istributions arising in the context of multivariate statistics. Wishart Distribution. we have Non-Central Wishart Distribution. Let J be the jump size. Consier the case J W is n, Q). Then τ φ τ u) = λ et I + 2e M s ) n ue Ms 2 Q s + λτ. 20) 0 τ Let be J W is n, Q, M), then we have φ τ u) = λ et Q) n 2 et 2e M s ue Ms + Q 1) n 2 0 { [ exp Tr 1 2 Q 1 MM Q 1 MM Q 1 2e M s ue Ms + Q 1)]} s + λτ. 21) Beta type I istribution. Let be J β I a, b), then φ τ u) = λ Beta type II istribution. τ φ τ u) = λ 0 τ 0 1F 1 a; a + b; e M s ue Ms )s + λτ. 22) Let be J β II a, b), then Γ a + b) Ψa; b + 1 Γ b) 2 + 1); em s ue Ms )s + λτ, 23) where m F n, Γ a), an Ψa; b; R) enote respectively the hypergeometric function of matrix argument, the multivariate Gamma function an the confluent hypergeometric function, see e.g. Gupta an Nagar 2000). 7

8 3 A Libor moel on S ++ To outline the general framework for Libor moels, we start by consiering a filtere measurable space Ω, F, F t ) an a family of probability measures P Tk ) 1 k N. Uner the measure P TN we introuce a stochastic process Σ taking values on the cone state space S ++. At this stage the process may be a iffusion, a pure jump or a jump-iffusion process taking values on S ++. Consier a iscrete tenor structure 0 = T 0 T 1... T N = T. We recall that the Libor rate is efine via quotients of bons: Lt, T k ) := 1 ) Bt, Tk 1 ) 1, 24) T Bt, T k ) where T is assume to be constant an T = T k T k 1. The relation between the Libor rate an the forwar price is given by: F t, T k 1, T k ) = 1 + T Lt, T k ). 25) We procee in full analogy with Keller-Ressel et al. 2009) by extening their results to processes taking values on the cone of positive efinite matrices. The intuition is simple: to buil up a Libor moel with positive rates, quotients of bons shoul be strictly greater than one. On the other han, a no-arbitrage argument see e.g. Geman et al. 1995)) implies that quotients of bons must be martingales uner the forwar risk neutral measure inexe by the maturity of the enominator, so that the key ingreient in the approach of Keller-Ressel et al. 2009) consists in the possibility of constructing a family of martingales that stay greater than one up to a boune time horizon. This will be possible thanks to the affine structure of the moel, since in this framework bon prices are exponentially affine in the positive efinite) factors, as well as their quotients. 3.1 Martingales strictly greater than one Let us first efine the set I T := { u S : E [ e Tr[uΣ T ] ] <, Σ 0 S ++ }. By the affine property of the process Σ we have [ E e Tr[uΣt]] = e φtu) Tr[ψtu)Σ0], φ : [0, T ] I T R, ψ : [0, T ] I T S. 26) Within this setting we are able to construct martingales that stay greater than one up to a boune time horizon T. 8

9 Theorem 4. Let Σ be an affine process, an let u I T S, then the process M u efine by is a martingale an M u t > 1 a.s. t [0, T ]. Proof. See Appenix. M u t = exp { φ T t u) Tr [ψ T t u)σ t ]} 27) Equippe with this positivity result, we can procee by consiering a tenor structure of non negative Libor rates L0, T k ) for k = {1,..., N 1}. Stanar arbitrage arguments see e.g. Geman et al. 1995)) imply that iscounte trae assets, in our case bons, are martingales uner the terminal martingale measure: B, T k ) B, T N ) M P T N ) k {1,..., N 1}, 28) where M P TN ) enotes the set of martingales with respect to the forwar risk neutral probability P TN. The iea in Keller-Ressel et al. 2009) is then to moel quotients of bon prices using the martingales M u efine as follows: Bt, T 1 ) Bt, T N ) = M u 1 t 29) Bt, T N 1 ) Bt, T N ). = M u N 1 t 30) t [0, T 1 ],..., t [0, T N 1 ] respectively. As a consequence, the initial values of the martingales M u k 0 must satisfy the relation M u k 0 = exp { φ T u k ) Tr [ψ T u k )Σ 0 ]} = B0, T k) B0, T N ), 31) for all k {1,..., N 1}, so that it is possible to set u N = 0 as we have M u N 0 = 1. In the following proposition, we show that it is possible to fit basically) any initial term structure of bon rates. The state space we are consiering offers a wie range of possibilities to perform this task. However, since we are intereste in applications, we aopt the simplest choice irectly coming from the scalar case an we focus on the particular but realistic) case where all Libor rates are positive. Proposition 5. Let L0, T 1 ),..., L0, T N ) be a tenor structure of positive initial Libor rates, an let Σ be an affine process on S ++. Define γ Σ := sup u I T S [ ] E e Tr[uΣ T ]. 32) If γ Σ > B0,T 1) B0,T N ) then there exists a strictly increasing sequence of matrices i.e. u k u k+1 if an only if u k u k+1 S ) u 1 u 2... u N 1 0 in I T S an u N = 0 such that M u k 0 = B0, T k), k {1,..., N}. 33) B0, T N ) 9

10 Conversely, let the bon prices be given by 29)-30) an satisfy the initial conition 31). Then the Libor rates Lt, T k ) are positive a.s. t [0, T k ] an k {1,..., N 1}. Proof. See Appenix. 3.2 A fully-affine arbitrage-free moel If we look at the efinition of the Libor rate we realize that it is quite natural to require quotients of bons to be riven by an exponentially affine function of the state: in fact, in this case also bon prices as well as forwar prices will be affine functions. This is also in line with the previous approaches of Constantinies 1992) an Gouriéroux an Sufana 2011) base on the stochastic iscount factor. In other wors, the approach of Keller-Ressel et al. 2009) is able to provie a fully affine structure 2. To prove the affine structure or our moel, we first show that uner 29)-30), forwar prices are of exponential-affine form uner any forwar measure. To o this, first we notice that in this framework quotients of bons are exponentially affine in the state factors, so that also forwar prices will be: for k = 1,..., N 1 Bt, T k ) Bt, T k+1 ) = Bt, T k) Bt, T N ) Bt, T N ) Bt, T k+1 ) = M uk t M u k+1 t = exp { φ TN tu k ) + φ TN tu k+1 )} exp {Tr [ ψ TN tu k ) + ψ TN tu k+1 )) Σ t ]} =: exp {A TN tu k, u k+1 ) + Tr [B TN tu k, u k+1 )Σ t ]}. 34) With this result, we are able to show very easily that the moel is arbitrage free, that is forwar prices are martingales with respect to their corresponing forwar measures see Geman et al. 1995)): B, T k ) B, T N ) M P T N ). 35) This comes from the fact that forwar measures are relate one another via the quotients of the martingales M u : P Tk Ft = F t, T k, T k+1 ) P Tk+1 F 0, T k, T k+1 ) = B0, T k+1) B0, T k ) M u k t M u k+1 t, 36) k {1,..., N}. Then L, T k ) is a martingale uner the forwar measure P Tk+1 since the successive ensities from P Tk+1 to P TN yiel a telescoping prouct an a P TN martingale see Keller-Ressel et al. 2009)). More precisely: since 1 + T L, T k ) = B, T k) B, T k+1 ) = M u k M u k+1 N 1 l=k+1 M uk M u M ) P Tk+1 k+1 37) M u l M u l+1 = M u k M P TN ). 38) 2 This is the reason why we will be able to apply the approach by Collin-Dufresne an Golstein 2002), who originally starte from an affine short rate in orer to price swaptions: in fact, also in their framework bon prices are affine functions of the state variables. 10

11 Also, the ensity between the P Tk -forwar measure an the terminal forwar measure P TN is given by the martingale M u k as inicate by 29)-30): P Tk Ft = B0, T N) Bt, T k ) P TN B0, T k ) Bt, T N ) = B0, T N) B0, T k ) M u k t = M u k t M u k 0. 39) In this arbitrage-free moel with positive Libor rates, the affine structure is preserve: that is, it is possible to exten to the state space S ++ the analogous result of Keller-Ressel et al. 2009). Proposition 6. Let the bon structure be efine through 29)-30), where the process M u. is given by 27). Then forwar prices are exponentially affine in the state variable Σ uner any forwar measure. Proof. The result comes irectly from formula 6.23) in Keller-Ressel et al. 2009) once the scalar prouct is replace by the trace operator. 4 Pricing of Derivatives We now focus on the pricing problem for vanilla options like Caps, Floors an for exotic options like swaptions in the affine Libor moel on S ++ introuce in the previous section. We shall see that the pricing of Caps an Floors may be performe using stanar Fourier pricing techniques as in Keller-Ressel et al. 2009), whereas, for the case of swaptions, we will resort to a quasi close form solution. In fact, since the moments of the unerlying affine process are known through its characteristic function, we can expan the exercise probability via an Egeworth evelopment, as shown in Collin-Dufresne an Golstein 2002). This approach will lea to an efficient approximation that will avoi the numerical problems unerlying the computation of the exercise probability in Keller-Ressel et al. 2009). 4.1 Caps an Floors A Cap may be thought of as a portfolio of call options on the successive Libor rates, name Caplets, whereas Floors are portfolios of put options name floorlets. These options are usually settle in arrears, which means that the caplet with maturity T k is settle at time T k+1. The tenor length T is assume to be constant. Since the two proucts are equivalent, we will focus on Caps. A Cap with unit notional has a payoff given by the following: T LT k, T k ) K) + k = 1,..., N 1 40) We rewrite the payoff of caplets as in Keller-Ressel et al. 2009): with K := 1 + T K. T LT k, T k ) K) + = 1 + T LT k, T k ) 1 + T K)) + ) M u k + T = k M u K, 41) k+1 T k 11

12 Thus we see that the caplet is equivalent to an option on the forwar price. In orer to avoi the computation of expectations involving a joint istribution, each single caplet is price uner the corresponing forwar measure: C T k, K) = B0, T k+1 )E P T k+1 [ M u k T k M u k+1 T k K ) + ] = B0, T k+1 )E P T k+1 [ e Y K ) + ], 42) with: Y := log ) M u k T k M u = A k+1 TN T k u k, u k+1 ) + Tr [B TN T k u k, u k+1 )Σ Tk ], 43) T k for A TN T k u k, u k+1 ), B TN T k u k, u k+1 ) efine as in 34). The pricing problem can be solve via Fourier techniques through the Carr an Maan 1999) methoology. Hence we have the following proposition, whose stanar proof is omitte. Proposition 7. Let α > 0. The price of a caplet with strike K an maturity T k is given by the formula: exp { αc} C T k, K) = B0, T k+1 ) 2π [e iv α+1)i)a T N T k u k,u k+1 )+Tr[B TN T k u k,u k+1 )Σ Tk ]) ] where: + e ivc EP Tk+1 α + iv) 1 + α + iv) c = log 1 + T K), A TN T k u k, u k+1 ) = φ TN T k u k ) + φ TN T k u k+1 ), B TN T k u k, u k+1 ) = ψ TN T k u k ) + ψ TN T k u k+1 ). In other wors, pricing a Cap involves the computation of the moment generating function of e.g. the Wishart process, which can be efficiently performe through the linearization of the associate Riccati ODEs as explaine in Proposition 3. The parameter α > 0 represents the amping factor introuce by Carr an Maan 1999). We report in the Appenix B the explicit expression of the characteristic function involve in the pricing proceure. v, 44) 4.2 Swaptions The payoff of a receiver resp. payer) swaption may be seen as a call resp. put) on a coupon bon with strike price equal to one. We consier a receiver swaption that starts at T i with maturity T m, i < m N). The time-t i value is given by: m + S Ti K, T i, T m ) = c k BT i, T k ) 1) 45) k=i+1 12

13 where c k = { T K if i + 1 k m 1, 1 + T K if k = m. 46) Unfortunately, we face some ifficulties if we try to aopt the Fourier technique that we employe to price a caplet. To see this we look at the proof of Proposition 7.2. in Keller-Ressel et al. 2009), which requires the computation of the Fourier transform of the payoff 3 : fz) = R +1) 2 m e Tr[izΣ T i ] k=i+1 c k e A T N T i u k,u i )+Tr[B TN T i u k,u i )Σ Ti ] 1 ) + vechσ Ti ), 47) where for a symmetric matrix A, vecha) stans for the vector in R +1)/2 consisting in the columns of the upper-iagonal part of A incluing the iagonal. The problem is given by the presence of the positive part in the payoff function. To get ri of it, we shoul be able to fin a value Σ such that m c k e A T N T i u k,u i )+Tr[B TN T i u k,u i ) Σ] = 1, 48) k=i+1 that is we shoul solve a single equation in + 1)/2 unknowns the elements of Σ), which is highly non trivial when > 1. Thus, pricing swaptions is challenging when we consier multiple factor affine moels: this is a well known problem, see e.g. Jamshiian 1989) an Collin-Dufresne an Golstein 2002). Keller-Ressel et al. 2009) investigate the case = 1, that is a Libor moel riven by a univariate) CIR process like in Jamshiian 1987). In that case, solving an equation similar to 48) is simple an the pricing of a swaption is only slightly more numerically complicate than the pricing of a Cap. As our purpose is to exten their methoology to a process with values in the set of strictly positive efinite symmetric matrices we face a numerical ifficulty relate to the imension of the state space. In orer to solve this ifficulty we follow Collin-Dufresne an Golstein 2002) s methoology which strongly epens on the affine property of the process use to moelize the rates. As the processes we use have this affine property we can carry out the approximation for the swaption price propose by these authors. Therefore, we can get aroun the imensional ifficulties pose by the process. We briefly recall the main results of Collin-Dufresne an Golstein 2002) to approximate the exercise probabilities for the swaption. We efine the T i -price of a coupon bon, for i < m N, as follows: CBT i ) = m k=i+1 c k BT i, T k ). 49) Let us erive the general form of the pricing formula for a receiver swaption, for 0 = T 0 = t < 3 BT i, T k ) = BT i,t k ) BT i,t N ) BT i,t N ) BT i,t i ) = M u k T i M u i = exp {A TN T i u k, u i) + Tr [B TN T i u k, u i)σ Ti ]} T i 13

14 T i : S 0 K, T i, T m ) = E Q [ e T i 0 r ss CBT i ) 1) +] [ = E Q e T i 0 r ss ) ] CBT i )1 CBTi )>1) 1 CBTi )>1) m [ = c k E Q e T k ] 0 r ss 1 CBTi )>1) k=i+1 [ E Q e T i ] 0 r ss 1 CBTi )>1). We switch to the forwar measure as follows: [ m S 0 K, T i, T N ) = c k B0, T k )E Q e T k ] 0 r ss B0, T k ) 1 CBT i )>1) k=i+1 [ B0, T i )E Q e T i ] 0 r ss B0, T i ) 1 CBT i )>1) = m k=i+1 c k B0, T k )E P [ ] T k 1CBTi )>1) B0, T i )E P [ ] T i 1CBTi )>1) m = c k B0, T k )P Tk [CBT i ) > 1)] k=i+1 B0, T i )P Ti [CBT i ) > 1)]. The exercise probabilities P Tk [CBT i ) > 1)] an P Ti [CBT i ) > 1)] o not amit in general a close form expression but can be efficiently approximate thanks to an Egeworth expansion proceure. Intuitively, the moments of the coupon bons amit a simple close-form expression in our affine framework, an these moments uniquely ientify the cumulants of the istribution. One can expan the characteristic function in terms of the cumulants an compute the exercise probabilities by Fourier inversion. Using the notation of Collin-Dufresne an Golstein 2002) their formula 5)) for the q th power of a coupon bon we notice that, for i < m N: CBT i )) q = c i+1 BT i, T i+1 ) c m BT i, T m )) q m ) = cj1... c jq BTi, T j1 )... BT i, T jq ) ). 50) j 1,...,j q=i+1 Now in our framework we have see also formula 7.9) in Keller-Ressel et al. 2009)) j 1,...,j q=i+1 BT i, T jl ) = M u jl T i M u 51) i T i for l = 1,..., q, meaning that we can rewrite the q th power of the coupon-bon as follows: u m j1 CBT i )) q ) M T = cj1... c jq i M u... M u jq ) T i i T i M u. 52) i T i 14

15 Recall, from 27), that we have for l = 1,..., q an M u j l T i = exp { φ TN T i u jl ) Tr [ψ TN T i u jl )Σ Ti ]}, 53) M u i T i = exp { φ TN T i u i ) Tr [ψ TN T i u i )Σ Ti ]}. 54) In conclusion, the q th moment uner P Tk has the following expression: E P T k [CBT i ) q ] [ u m j1 ) = cj1... c jq E P M Tk T i M u i j 1,...,j q=i+1 T i m ) = cj1... c jq j 1,...,j q=i+1 E P T k [ exp { q l=1... M u jq )] T i M u i T i ) φ TN T i u jl ) Tr [ψ TN T i u jl )Σ Ti ] ) }] + q φ TN T i u i ) + Tr [ψ TN T i u i )Σ Ti ] = m j 1,...,j q=i+1 E P T k [exp cj1... c jq ) exp { { Tr [ ) } q φ TN T i u jl ) + qφ TN T i u i ) l=1 ) ) ]}] q ψ TN T i u jl ) + qψ TN T i u i ) Σ Ti, 55) l=1 where the functions φ an ψ are as usual the solutions of Riccati ODE s of the form 4), 5). Once the first m moments uner the corresponing forwar measures are exactly etermine, we can estimate the exercise probabilities P Tk [CBT 0 ) > 1)] uner each forwar measure by relying on a cumulant expansion for P Tk [CBT 0 )]. 5 The Wishart Libor Moel The aim of this section is to illustrate a specific choice for the riving process Σ. As in the general setup, we specify the process uner the terminal probability measure P TN. The example we choose is the Wishart process, which was alreay presente in section 2.2.1: Σ t = ΩΩ + MΣ t + Σ t M )t + Σ t W T N t Q + Q W T N t Σt. 56) Here W T N t enotes a matrix Brownian motion, i.e. a matrix of inepenent Brownian motions uner the P N -forwar probability measure. In the sequel we will write W t for notational simplicity. 15

16 In this section we show the impact of the relevant parameters on the implie volatility surface generate by vanilla options for a Libor moel riven by a Wishart process. With the aim to investigate some complex movements of the implie volatility surface, we first compute the covariation between the Libor rate an its volatility: this covariation is a crucial quantity allowing for the so calle skew effect on the smile, in perfect analogy with the leverage effect for vanilla options in the equity market. 5.1 The skew of vanilla options We want to compute the covariation between the Libor rate an its volatility, so we procee to erive the ynamics of the Libor rate in the Wishart moel. This may be one along the following steps: using the shorthan recall that we have: B k := B TN tu k, u k+1 ) = ψ TN tu k ) + ψ TN tu k+1 ), 57) 1 + T Lt, T k, T k+1 ) = Bt, T k) Bt, T k+1 ) = ea k+tr[b k Σ t]. 58) In ifferential form, after iviing both sies by Lt, T k, T k+1 ) we have Lt, T k, T k+1 ) Lt, T k, T k+1 ) = 1 + T Lt, T k, T k+1 ) Lt, T k, T k+1 ) [...]t + Tr [B k Σ t ]). 59) To preserve analytical tractability, we freeze the coefficients an approximate as follows: 1 + T Lt, T k, T k+1 ) Lt, T k, T k+1 ) 1 + T L0, T k, T k+1 ) L0, T k, T k+1 ) =: C. 60) Proposition 8. Uner the assumption of frozen coefficients 60), the conitional infinitesimal correlation between the Libor rate an its volatility cannot be negative an is given by Lt, T k, T k+1 ), vollt, T k, T k+1 )) Tr [ B k Q QB k Q QB k Σ ] t = Tr [. 61) QB k ΣBk Q ] Tr [ΣB k Q QB k Q QB k Q QB k ] Proof. See Appenix. From the previous formula we realize that the matrix Q is responsible for the shape of the skew. We also have an inirect impact of the mean reversion spee matrix M coming from the term B k which is the ifference of two solutions of the Riccati ODE s 4) an 5). The presence of Σ suggests that in the present framework the skew is stochastic. What is more, it can only have positive sign. 16

17 5.2 Numerical illustration with iagonal parameters The ynamics above show that the Wishart specification provies a very rich structure of the moel. Since we want to get an unerstaning of the impact of ifferent parameters we will look first at the case where all matrices are iagonal, which basically correspons to a moel riven by a two factor square root process see e.g. Da Fonseca an Grasselli 2011)). We use the following set of parameters as a benchmark: ) ) e Σ 0 =, M = e 003, ) Q =, κ = The impact of the Ginikin parameter κ is quite easy to unerstan: the process acts by influencing the overall level of the surface. This is ue to the fact that the higher κ the lower the probability that the process Σ approaches 0. It is interesting to note that there is not only a level impact, but also a curvature effect, as we can see in Figure 1. [Insert Figure 1 here] Let us now look at the parameters along the iagonals of the matrices M an Q. The following claims may be easily checke by looking at the SDE s satisfie by the elements of Σ see also Da Fonseca et al. 2007a)). Note that we assume all eigenvalues of M to lie in the negative real line. For M 11 ) the surface is shifte ownwars upwars). For M 22 ) the surface is shifte ownwars upwars). The impact is more evient for OTM caplets with short maturities. This is ue to the fact that as the process ecreases in matrix sense) the probability that caplets with short maturities are exercise is lowere more than the analogous probability for longer term caplets. [Insert Figure 2 here] We then consier the impact of Q 11, Q 22. We have the following: As Q 11 ) the surface is shifte upwars ownwars). In particular if we multiply Q 22 by a constant c > 1, then the increment in the short term is higher for OTM than for ITM caplets. If c < 1 then the ecrease is higher for short term OTM caplets, which is intuitive, given the iscussion above. The same impacts, with ifferent magnitues, is observe also for Q 22. [Insert Figure 3 here] 17

18 5.3 The term structure of ATM implie volatilities for caplets Diagonal parameters We procee to consier the term structure of caplet implie volatilities. When the matrix Σ 0 is iagonal, the impact of the elements of Q is the same: an increase in the absolute value of any element of Q will result in a steeper term structure of ATM caplet volatilities. [Insert Figure 4 here] Consiering a moel where Σ 0 is a full matrix oes not influence this result in a significant way More complex ajustments: impact of off-iagonal elements To appreciate the flexibility of the Wishart framework, we focus now on the impact of the offiagonal elements. We introuce off-iagonal elements in M an Q an look at the relative change in the short term smile 4 months) an the long term smile 32 months). We introuce a fully populate matrix Σ 0 an look at the impact of M 12 an M 21. Our experiments show that there is a symmetry between the sign of Σ 0,12 the initial value of Σ 12 ) an M 12, M 21. More precisely, the implie volatility changes are as in Table 1. Σ 0,12 > 0 Σ 0,12 < 0 M 12 > 0 Increase Decrease M 12 < 0 Decrease Increase M 21 > 0 Increase Decrease M 21 < 0 Decrease Increase Table 1: Implie volatility changes: relation between Σ 0,12 an M 12, M 21. The reason for this symmetry is to be looke for in the rift part of the ynamics of the single elements of the matrix process Σ. Next we look at the impact of Q 12 an Q 21. To this en we moel Q as a symmetric matrix an set Q 21 = Q 12 = ρ Q 11 Q 22 for a real parameter ρ. Also in this case we recognize two main shapes of the ajustment that we enote by S 1, S 2. We now procee to perform other numerical tests which will show that our moeling framework has a certain egree of flexibility. For these tests we set: ) e M = e 003, 0.02 ρ ) Q 11 Q 22 Q = ρ, κ = 3, Q 11 Q

19 Σ 0,12 > 0 Σ 0,12 < 0 ρ > 0 S 1 S 2 ρ < 0 S 2 S 1 Table 2: Implie volatility changes: relation between Σ 0,12 an ρ. so basically M is parametrize as before but Q is symmetric an equipe with a parameter ρ which summarizes the information on the off-iagonal elements. We require Σ 0 = Σ, where Σ is given by the solution of the Lyapunov equation 9). After that we perturbate Σ 0 in orer to inclue off-iagonal elements an set Σ 0,12 = Σ 0,21 = 2. We have a goo egree of control on the term structure of ATM implie volatilities. In particular, we may have larger percentage shifts in the long-term w.r.t. the short-term ATM implie volatility, or, for ρ = 0.6 we may even reprouce a situation where the short term ATM implie volatility increases whereas the long-term ATM implie volatility ecreases. [Insert Figure 5 here] If we aopt the same kin of parametrization for the matrix M by introucing a secon parameter ρ 2, then we have further flexibility because we can impose many ifferent combinations of ρ an ρ 2. For example, Figure 6 shows that we are able to isolate an effect on the term structure of ATM implie volatility: in fact we have a moerate change for ITM caplets while OTM caplets are practically unchange, but the shape of the term structure of ATM implie volatility is moifie in a significant way. [Insert Figure 6 here] Finally, just for illustrative purposes we report a prototypical Caplet volatility surface generate by the moel. [Insert Figure 7 here] As far as Swaptions are concerne an example of ATM implie volatility surface for ifferent expiries an unerlying swap lengths is given below. [Insert Figure 8 here] 6 The Pure Jump Libor Moel Finally, in this section, we woul like to provie a secon example for the riving process Σ, so as to let the reaer appreciate the egree of generality of this framework. As in the general setup, we specify the process uner the terminal probability measure P TN. The example we choose is a matrix compoun Poisson process, which was alreay presente in section 2.2.2: Σ t = MΣ t + Σ t M + L P T N t. 62) 19

20 All assumptions presente in section are in orer. More specifically, we assume that L P T N t is a compoun Poisson process with constant intensity λ an jump istribution taking values in S ++. As a specific example of jump istribution we choose the stanar Wishart istribution. By recalling the results in section we have that the solution for the function ψ τ u) is whereas for φ τ u) we have φ τ u) = λ τ 0 ψ τ u) = e M τ ue Mτ, 63) et I + 2e M s ) n ue Ms 2 Q s + λτ. 64) In concrete pricing applications, the computation of the solution for φ τ u) implies a numerical integration with respect to the time imension. This numerical integration has an impact on the performance of the moel which turns out to be slower than the Wishart Libor moel. For illustrative purposes, we report an example for an implie volatility surface for caplets generate by the compoun Poisson Libor moel with central Wishart istribute jumps. The mean reversion matrix M an the jump intensity λ are given by: ) M =, λ = 0.1. As far as the jump size istribution is concerne, the parameters are the following: ) Q =, n = 3.1, an the initial state of the process is Σ 0 = ). [Insert Figure 9 here] 7 Conclusions In this paper we presente an extension of the approach of Keller-Ressel et al. 2009) to the more general setting of affine processes on positive efinite matrices. We showe that their methoology may be aapte to this state space in a straightforwar way. What is more, it is possible to efficiently price European swaptions in this multi-factor setting by means of a cumulant expansion ue to Collin-Dufresne an Golstein 2002). In oing so we are in front of a setting which is potentially able to capture correlation effects which can not be escribe by a single-factor framework. We provie numerical examples for the Wishart Libor moel, 20

21 where the introuction of off-iagonal elements gives rise to new possibilities in the control of the shape of the implie volatility surface. Our contribution may be seen as a starting point for a escription of market moels in this state space, in consequence we believe that there are many possible irections for future research. An example is given by the problem of calibrating this family of moels to real market ata. As the structure of the proucts in the fixe-income market suggests, even in the plain vanilla case, we expect the objective function that shoul be minimize in the calibration proceure to be quite involve. Yet, some calibration results were obtaine on equity erivatives in Da Fonseca an Grasselli 2011) for Wishart base moels so a calibration using interest rates erivatives might be feasible. Certainly, it will be a elicate issue an may constitute an interesting contribution by its own. Once the moel is calibrate on vanillas, one coul then further investigate the performance of the moel on more exotic structures, like e.g. Bermuan swaptions an barrier options. Theses issues are left for future work. 21

22 Appenix A: proofs Proof of Theorem 4 For all u I T we have [ ] E [MT u ] = E e Tr[uΣ T ] <, an by the affine property we obtain E [M u T F t ] = E [exp { φ T T u) Tr [ψ T T u)σ T ]} F t ] = E [exp { Tr [uσ T ]} F t ] = exp { φ T t u) Tr [ψ T t u)σ t ]} = M u t, hence the process is a martingale. Now we show that Mt u > 1. Recall that by assumption u I T S an Mt u = E [exp { Tr [uσ T ]} F t ], so that if Tr [uσ T ] > 0 a.s. then we are one. We procee as in Gouriéroux an Sufana 2003) an apply the singular value ecomposition to the negative efinite matrix u, i.e. u may be written as: u = n λ i u i u i i=1 where λ i are the eigenvalues of u an u i are the eigenvectors. By assumption Σ T takes values in S ++, hence [ n as we wante. Tr [uσ T ] = Tr = = i=1 λ i u i u i Σ T ] n ] λ i Tr [u i u i Σ T i=1 n λ i u i Σ T u i > 0 65) i=1 Proof of Proposition 5 We follow closely the proof in Keller-Ressel et al. 2009). By assumption, initial Libor rates are strictly positive, then B0, T 1 ) B0, T N ) > B0, T 2) B0, T N ) >... > B0, T N) = 1. 66) B0, T N ) Recall that we have [ ] E e Tr[u 1Σ T ] = M u 1 0 = exp { φ T u 1 ) Tr [ψ T u 1 )Σ 0 ]} = B0, T 1) B0, T N ). 67) 22

23 By the efinition of γ Σ in 32), we have that if γ Σ = then we are one, else we can claim that there exists an ɛ > 0 such that γ Σ ɛ > B0,T 1) B0,T N ). Then we can fin a matrix ũ s.t. [ ] E e Tr[ũΣ T ] > γ Σ ɛ > B0, T 1) B0, T N ). 68) In analogy with Keller-Ressel et al. 2009) we introuce the function f : [0, 1] R 0 [ ] ξ E e Tr[ξũΣ T ] 69) an we want to show that f is continuous. First, since Σ S ++ an u S we have that if u v then Tr [uσ T ] > Tr [vσ T ], hence by monotone convergence we can conclue that f is increasing. We now introuce an increasing sequence a n ) n N 1 an apply Fatou s lemma to obtain [ ] [ ] [ ] lim inf E e Tr[anũΣ T ] E lim inf n n e Tr[anũΣ T ] = E e Tr[ũΣ T ], implying that f is lower semi-continuous. Since f is also increasing we have that f is continuous. Now f0) = 1 an f1) > B0,T 1) B0,T N ), hence there exist some numbers 0 = ξ N < ξ N 1 <... < ξ 1 < 1 such that f ξ k ) = M ξ kũ 0 = B0, T k), k {1,..., N}. B0, T N ) By setting u k = ξ k ũ for k = 1,..., N 1) we obtain a sequence of matrices u k u k+1, u k u k+1 S which allows us to fit the initial tenor structure of Libor rates as esire. Finally, we apply Proposition 1 an Lemma 3.2 ii) in Cuchiero et al. 2011) in orer to obtain the last sentence of the Proposition 5. Proof of Proposition 8 In this section we procee as in the proof of Proposition 4.1 in Da Fonseca et al. 2008). Recall that W t is a shorthan for W T N t. From 59) it follows that [ Lt, T k, T k+1 ) Lt, T k, T k+1 ) = C...)t + 2 Tr [ QB k ΣB k Q ] Tr ] QB k ΣWt Tr [ QB k ΣB k Q ] := C...)t + 2 Tr [ ) QB k ΣB k Q ] W t, 70) where C was efine in 60) an the scalar noise riving the factor process may be erive as follows: [ Tr QB k Σ t B k Q ] [ = Tr QB k βq QB k Q ] [ + 2Tr QB k MΣ t B k Q ]) t + 2Tr [QB k Σt W t QB k Q ] =...)t + 2 Tr [Σ t B k Q QB k Q QB k Q Tr [ QB k Q ] QB k Σt W t QB k ] Tr [ΣBk Q QB k Q QB k Q QB k ] :=...)t + 2 Tr [Σ t B k Q QB k Q QB k Q QB k ]Z t. 71) 23

24 The covariation between the noise of the Libor rate an its volatility is then given by W [ ] [ Tr QBk Σt W t Tr QBk Q ] QB k Σt W t t, Z t = Tr [ QB k Σ t Bk Q ], Tr [Σt B k Q QB k Q QB k Q QB k ] p,q,r,s Q ) pqb qr Σrs W sp a,b,c,,e,f,g Q abb bc Q c Q ) eb ef Σfg W ga = = Tr [ QB k ΣBk Q ] Tr [ΣB k Q QB k Q QB k Q QB k ] a,b,c,,e,f,g,q,r B feq e B cbq ba Q aqb qr Σrg Σgf t Tr [ QB k ΣBk Q ] Tr [ΣB k Q QB k Q QB k Q QB k ] Tr [ B k Q QB k Q ] QB k Σ t t = Tr [. QB k Σ t Bk Q ] Tr [Σ t B k Q QB k Q QB k Q QB k ] Now we turn on the positivity of the skew. With the notation in the proof of Proposition 5, from ξ k > ξ k+1 we have u k u k+1 an then B k S +. In all terms in the numerator an the enominator we recognize congruent transformations of matrices in S + which leave the signs of the eigenvalues unchange. The self-uality of S + allows us to claim that all traces are positive, hence we are one. Appenix B: the characteristic function With the purpose of pricing caplets, we nee to have a more explicit form for the characteristic function appearing in Proposition 7. Once we have this expression we can plug in the functions φ τ u) an ψ τ u) to obtain a close form solution. The pricing problem will be then solve via FFT. Recall that we are consiering the following expectation: [ ϕv) = E P T k+1 e iv α+1)i)a k+tr[b k Σ Tk ]) ] = e iv α+1)i)a k E P T k+1 exp Tr i v α + 1) i) B }{{ k } u where A k := φ TN T k u k ) + φ TN T k u k+1 ); Σ Tk 72) B k := ψ TN T k u k ) + ψ TN T k u k+1 ). 73) As we compute the shape of the function φ τ u) an ψ τ u) uner the P TN -forwar measure, we nee to switch from the P Tk+1 to the P TN -forwar measure: e iv α+1)i)a k E P T k+1 [ e Tr[uΣ T k ] ] [ ] = e iv α+1)i)a k E P PTk+1 T N e Tr[uΣ T k ] P TN [ u M k+1 = e iv α+1)i)a k E P T T k N M u k+1 0 ] e Tr[uΣ T k ], 74) 24

25 where the last equation follows from 39). Let us focus on the expectation which becomes: [ { E P T N exp φ TN T k u k+1 ) Tr [ψ TN T k u k+1 )Σ Tk ] } ] + φ TN u k+1 ) + Tr [ψ TN u k+1 )Σ 0 ] + Tr [uσ Tk ] { = exp } φ TN T k u k+1 ) + φ TN u k+1 ) + Tr [ψ TN u k+1 )Σ 0 ] ] E P T N [e Tr[ ψ T N T k u k+1 )+u)σ Tk ] { = exp φ TN T k u k+1 ) + φ TN u k+1 ) + Tr [ψ TN u k+1 )Σ 0 ] ) ] } φ Tk ψ TN T k u k+1 ) + u Tr [ψ Tk ψ TN T k u k+1 ) + u )Σ 0. 75) Now, recalling the previous terms in front of the expectation in 74), we obtain the final expression which is { A k {}}{ ) exp iv α + 1)i) φ TN T k u k ) + φ TN T k u k+1 ) φ TN T k u k+1 ) + φ TN u k+1 ) {}}{ ) ) φ Tk ψ TN T k u k+1 ) + iv α + 1)i) ψ TN T k u k ) + ψ TN T k u k+1 ) }{{} u +Tr [ψ TN u k+1 )Σ 0 ] {}}{ ]} ) Tr [ψ Tk ψ TN T k u k+1 ) + iv α + 1)i) ψ TN T k u k ) + ψ TN T k u k+1 ) )Σ 0. }{{} u B k B k 25

26 Figures Figure 1: Doubling κ with respect to the basic case causes an upwar shift of the surface. The plot represents the two smiles 4 months an 32 months) for the basic κ = 3) an the moifie case κ = 6). Figure 2: Impact of M 11. M 11 is negative an the present image shows the effects on the two smiles 4 months an 32 months) we obtain when we multiply it by a constant c =

27 Figure 3: Impact of Q 11. Q 11 is positive an the present image shows the effects on the two smiles 4 months an 32 months) we obtain when we multiply it by a constant c = 2 Figure 4: Impact of Q on the term structures of ATM implie volatilities. Here we consier Q 11 an multiply its value by a constant c = 1, 1.5, 2 so as to get the values in the legen. 27

28 Figure 5: The images above highlight the flexibility of the Wishart Libor moel. We are able to impose ifferent patterns to the term structure of ATM implie volatility. On the top we have the smiles an on the bottom we observe the relative changes ) of the smiles, i.e. for every point of the smiles we calculate the quantity σ imp final σimp initial /σ imp initial. Notice in particular the situation on the left sie, where we observe aroun 5% ATM) an increase of the short term smile an a ecrease on the long term. 28

29 Figure 6: Impact on the implie volatility surface when both M an Q are parametrize as symmetric matrices. Notice the level aroun 5%, corresponing to ATM. This shows that if we parametrize both M an Q via ρ, ρ 2 we have a flexible setting which is controlle just by two parameters that allow us to perform ifferent combinations. In particular ρ an ρ 2 have opposite impacts in the present example ρ > 0 whereas ρ 2 < 0), meaning that we have a goo egree of control. 29

30 Figure 7: Caplet Implie Volatility Surface generate by the Wishart Libor moel Figure 8: ATM Swaption Implie Volatility Surface generate by the Wishart Libor moel 30

31 Figure 9: Caplet Implie Volatility Surface generate by the compoun Poisson Libor moel with Wishart istribute jumps 31

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