models Dipartimento di studi per l economia e l impresa University of Piemonte Orientale Faculty of Finance Cass Business School

Transcription

1 A.M. Gambaro R. G. Fusai Dipartimento di studi per l economia e l impresa University of Piemonte Orientale Faculty of Finance Cass Business School Dipartimento di Statistica e Metodi Quantitativi University of Milano Bicocca January 30, 2015

2 Outline

3 Motivations Why do financial institutions need a fast and accu method for pricing swaption? Calibration of stochastic to real market data: volatility quotations are very liquid for all the most important currencies (Euribor, USD Libor, GBP Libor...); Calibration involves the a large number of (different option maturity, swap lengths and strikes), so Monte Carlo method is useless because too time consuming. Pricing and hedging of derivatives; Calculus of the Credit Value Adjustment (Basel III): CVA of an Interest Rate Swap can be calculated as a portfolio of s.

4 Existing literature Weak points in existing literature. 1 Most studies are limited to simple, because they have a restricted scope of application, in particular the Gaussian case. 2 Literature methods are generic approximation, so they do not specify the direction of the error (instead our method is a Lower Bound); 3 None of the existing method is able to control the approximation error a part from: Nunes and Prazeres (2013) method that is applicable only for Gaussian ; Collin-Dufresne and Goldstein (2002) method that requires the existence of the moment generating function and that is very time consuming. 4 The most precise and general method is due to Don H. Kim (2012), which extends the method of Singleton and Umantsev.

5 Existing literature. Weak points in Kim s method Also Kim s method has few limitations: 1 It requires as many Fourier inversion of the characteristic function as the number of cash flows in the underlying swap contract; 2 It is presented as a generic approximation by Kim, but we demonst that his method is a Lower Bound (it is a particular case of our method); 3 Kim is not able to control the approximation error.

6 Our main contributions. We produce a general formula to obtain Lower bounds of swaption price, based on approximation of exercise region. It has a wide scope of application: it applies whenever the joint characteristic function of the state variables is known (in closed form or via numerical procedures); As a particular application it includes Kim s method, but it is implemented with a more efficient algorithm that performs only one Fourier inversion independently from the number of cash flows of the underlying swap; So our procedure gives a new perspective with respect to existing method, such as S&U and Kim. Indeed, we prove that their approximations are also lower bounds to the swaption price. This has gone completely unnoticed up to now. For affine it is possible to control the error of the method and calculate an Upper Bound of the price; In addition, it can be used as a control variable to improve the accuracy of the Monte Carlo simulation method; Finally, we test the bounds for different and we find very accu ;

7 Problem definition European Receiver An European is a contract that gives the right (but not the obligation) to enter an Interest Rate Swap (IRS) at a given future time (maturity of the option). IRS(T ) = n P(T, T h )(T h T h 1 )(R F (T, T h 1, T h )) [ Price(t) = E t e ] T r(x(u))du t max(irs(t ), 0) where t is the today date, T is the option expiration date, T 1,..., T n are the payment dates, R is the fixed of the the underlying IRS and P(T, T h ) is the discount factor. can be seen as an option on a portfolio of ZCBs: Price(t) = E t [ = P(t, T ) E T t e T t r(x(u))du ( ] n w h P(T, T h ) 1) + ( n ) + w h P(T, T h ) 1 where w h = R (T h T h 1 ), w n = R (T n T n 1 ) and R is the fixed of the IRS.

8 . General formula [( n ) ] Price(t) = P(t, T ) E T t w h P(T, T h ) 1 I(A) where A is the exercise region seen as a subset of the space events Ω: n n A = {ω Ω : w h P(X(ω); T, T h ) 1} = { w h P(T, T h ) 1}. Then G Ω: [( n ) + ] [( n + ] E T t w h P(T, T h ) 1 E T t w h P(T, T h ) 1) I(G) [( n ) ] E T t w h P(T, T h ) 1 I(G)

9 Lower Bound: Affine framework. Affine In affine the short is a linear combination of the model factors: r(x(t)) = a r + b r X(t) as a consequence the ZCB price is an affine function of the model factors: [ P(t, T ) = E t e ] T t r(x(s))ds = e B(T t) X(t)+A(T t) where functions A(τ) and B(τ) are the solution of a system of d + 1 ordinary differential equations (d is the number of model factors).

10 Lower Bound: Affine framework. Problem formulation The payoff of a receiver swaption can be written as an option on a portfolio of Z.C.B.: ( n + Payoff = R (T h T h 1 )P(T, T h ) + P(T, T n) 1), where the ZCB price P(T, T h ) in affine framework is given by: P(T, T h ) = e B(T h T ) X(t)+A(T h T ) dj=1 b = e h,j X j (T )+a h. The swaption price at time t in T-forward measure: [( n ) ] Price(t) = P(t, T ) E T t w h e b h X(T )+a h 1 I(A), where A = { n w h e b h X(T )+a h > 1} is the exercise region.

11 Lower Bound: Affine framework. Approximating set We approximate the option exercise region via an event set defined through a linear combination of the state variables. n G k = {g(x(t )) > k} = {β X + α > k} vs A = { w h e b h X(T )+a h > 1}. A family (i.e. k) of Lower Bounds is given by : [( n ) ] LB(t, k) = P(t, T )E T t w h e b h X(T )+a h 1 I(G k ).

12 Lower Bound: Affine framework. Fourier transform Let us introduce the characteristic function of X calculated under the T-forward measure: [ ] Φ(λ) = E T t e iλ X, and we assume that it is known. The FT of the Lower Bound, ψ δ (γ), can be written as: ( n ) ψ δ (γ) = w h e a h Φ ( ib h + (γ iδ)β) Φ ((γ iδ)β) e (iγ+δ)α iγ + δ.

13 Lower Bound: Affine framework. Lower bound computation Finally the lower bound is the inverse transform of ψ δ (γ): LB(t, k) = P(t, T ) e δk π + e iγk ψ δ (γ)dγ. 0 The price approximation is obtained through maximization of the lower bound respect to k: LB(t) = max LB(t, k). k R

14 The approximating set Tangent hyperplane approximating set We test our Lower Bound formula using the approximated exercise set proposed in Kim (2012, approximation "A") and Singleton and Umantsev (2002). The set G is defined in the following way: where G = {ω Ω : β X + α 0}, α = B(X ) X and β = B(X ), k = 0 B(X) = n w he b h X+a h ; X is the point on the exact exercise boundary where the density function of the model factors is largest. The equation β X + α = B(X ) (X X ) = 0 defines the tangent hyperplane to the true exercise boundary, {B(X) = 1}, in X.

15 The approximating set Figure: Histogram of the probability density function. Figure: Light blue and red lines are respectively the boundary of the true region and of the approximate set. Blue circle is the region where the probability is highest years, with the 2-factors CIR model.

16 Upper Bound The Lower Bound approximation error where Price(t) = P(t, T ) + LB(t) = E T t [(B(X) 1)+ ] E T t [(B(X) 1) I(G)] = E T t [(B(X) 1)+ I(G c )] + E T t [(1 B(X))+ I(G)] henceforth X(T ) = X, B(X) = n w h P(X; T, T h ) is the portfolio of ZCBs, G is the approximated exercise region.

17 Upper Bound Proposition In affine case, if G is the tangent hyperplane exercise set, then E T t [(1 B(X))+ I(G)] = E T t [(1 B(X))+ I(β X + α 0)] = 0 and so: = E T t [(B(X) 1)+ I(β X + α < 0)]

18 Upper Bound Figure: Light Blue line represents the true exercise boundary for a 2 10 years swaption with 2-factors CIR model. The blue star indicates the point X. The approximate exercise region G is the half-space below the red line. Since the sub-level {B(X) 1} is convex, then G {B(X) 1} = by the hyperplane separation theorem.

19 Upper Bound Upper Bound Formula In general, is not explicitly computable. So we provide un upper bound ɛ to it and therefore a upper swaption price: where ɛ. An upper bound of the error is simply: ɛ = UB(t) = LB(t) + P(t, T ) ɛ n E T t [(w hp(x, T, T h ) K h ) + I(β X + α < 0)]. (1) The inequality holds for every (K 1,..., K n) such that n K h = 1. However, without a proper choice of the strikes (K 1,..., K n), the approximation is very rough. So we want to find the values of (K 1,..., K n) that makes small the upper bound without performing a multidimensional numerical minimization that should be too time consuming.

20 Upper Bound Useful equality We note that the following equality holds where Strikes choice = n E T t [(w hp(x, T, T h ) K h (X)) + I(β X + α < 0)] If we choose strikes (K 1,..., K n) such that: K h (X) = w hp(x, T, T h ). B(X) K h = K h (X ) = w h P(X, T, T h ) B(X ) = w h P(X, T, T h ), (2) then in formula (1) the equality holds for X = X, the point on the true exercise boundary where the density function of the model factors is largest.

21 Upper Bound Upper Bound computation For affine Upper Bound can be computed performing a double integration: UB(t) = LB(t) + P(t, T ) ɛ(0) ɛ(k) = n E T t [(w hp(x, T, T h ) K h ) + I(β X + α < k)] + + n = dγ dγ h 0 w h e a h 2 π 2 e ( iγ+δ)k e ( iγ h δ h )k h ψ δ,δh (γ, γ h ) ψ δ,δh (γ, γ h ) = e(iγ δ)α Φ((γ + iδ)β + (γ h iδ h i)b h ) (iγ δ)(δ 2 h + δ h γ 2 h + i(2δ h + 1)γ h ) where - ψ δ,δh (γ, γ h ) is the double FT of respect to k and k h, - k h = log(k h ) log(w h ) a h and (K 1,..., K n) are defined in equation (2), - Φ(λ) is the joint T-forward characteristic function of X.

22 Example of numerical 2-factors Gaussian model with jumps dx(t) = K (θ X(t)) dt + Σ dw(t) + dz + (t) dz (t) X(0) = x 0 r = φ + X 1 + X 2, where K is a d d diagonal matrix, Σ is a d d triangular matrix and Z ±. Z ± are compounded Poisson processes with exponential Jump size: N ± (t) Z ± l = Y ± j,l j=1 where N ± (t) are Poisson variables of intensity µ± and Y ± d j,l, for a fixed l, are i.i.d.: ( ) Y ± j,l ν(m ± l ) = 1 y m ± exp l m ± l

23 : 2-factors Gaussian model with jumps. Payer s (ATMF) Opt. Mat. 1 2 Swap MC LB UB MC Kim MC LB UB MC length (HP) (CV) (HP) (CV) Opt. Mat. 5 Swap MC LB UB MC length (HP) (CV) Kim Kim

24 : 2-factors Gaussian model with jumps. Overall LB MC Kim MC UB time (HP) (CV) (sec) Swap length LB (HP) (sec) Kim (sec) Swap length LB (HP) Kim 2 0% 20% 5 0% 55% 10 34% 132% 15 69% 225% % 305% Table: For each swaption we report in the first table the run time in seconds and in the second table the percentage variation of the run time respect to the first row. The maturity of the is 2 years.

25 THANKS FOR YOUR ATTENTION.

26 References I Appendix P. Balduzzi, S.R. Das, S. Foresi and R. Sundaram (1996). A simple approach to three-factor affine term structure. The Journal of Fixed Income. R., G. A. Gnoatto and M. Grasselli (2014). General closed-form basket option pricing bounds. working paper, posted at ssrn.com in January P. Collin-Dufresne and R.S. Goldstein (2002). Pricing s Within an Affine Framework. THE JOURNAL OF DERIVATIVES, D.F.. Schrager and A.J. Pelsser (2006). Pricing s and Coupon Bond Options in Affine Term Structure Models. Mathematical Finance 16, K.J. Singleton and L. Umantsev (2002) Pricing Coupon-Bond Options and s in Affine Term Structure Models. Mathematical Finance 12, D.H. Kim (2012). Pricing in Affine and other. Mathematical Finance (2012), 1-31.

27 References II Appendix C. Munk (1999) Stochastic Duration and Fast Coupon Bond Option Pricing in Multi-Factor Models. Rev. Derivat 3, J.P.V. Nunes and P.M.S. Prazeres (2013). Pricing s under Multifactor Gaussian HJM Models. Mathematical Finance (2013), 1-28.