SLIDE 1
Polynomial preserving processes and discrete-tenor interest rate models
Zorana Grbac
Universit´ e Paris Diderot Based on joint work with K. Glau and M. Keller-Ressel
Advanced Methods in Mathematical Finance Angers, 1–4 September 2015
SLIDE 2 Introduction and motivation
Developing interest rate models that ensure on the one side nonnegative interest rates and/or spreads, and on the other side analytical pricing of both caplets and swaptions and enough flexibility for calibration, is a challenging problem. Recall: Caplet with strike K and maturity Tk, settled in arrears: Cplk
t = B(t, Tk+1)δkEPTk+1 [(L(Tk, Tk) − K)+ |Ft]
Swaption with swap rate S and exercise date Tk – option to enter an interest rate swap: Swpt = B(t, Tk)EPTk
δiL(Tk, Ti)B(Tk, Ti) −
n
δiSB(Tk, Ti) +
n
ciB(Tk, Ti) +
where ci = δiS, for k + 1 ≤ i < n and cn = 1 + δnS.
SLIDE 3 Interest rate models based on polynomial preserving processes
Polynomial preserving processes seem to be very suitable to tackle these issues... Seminal paper introducing rational interest rate models:
. Hughston (1996). Positive interest Some references from recent literature:
- S. Cheng and M. Tehranchi (2014). Polynomial models for interest rates and
stochastic volatility
c, M. Larsson and A. Trolle (2014). Linear-rational term structure models
epey, A. Macrina, T.M. Nguyen and D. Skovmand (2014). Rational multi-curve models with counterparty-risk valuation adjustments
SLIDE 4 Interest rate models based on polynomial preserving processes
Polynomial preserving processes seem to be very suitable to tackle these issues... Seminal paper introducing rational interest rate models:
. Hughston (1996). Positive interest Some references from recent literature:
- S. Cheng and M. Tehranchi (2014). Polynomial models for interest rates and
stochastic volatility
c, M. Larsson and A. Trolle (2014). Linear-rational term structure models
epey, A. Macrina, T.M. Nguyen and D. Skovmand (2014). Rational multi-curve models with counterparty-risk valuation adjustments In this paper: we work in the discrete-tenor setup and make use of the polynomial property already in the model construction to ensure the main theoretical and practical modeling requirements → we shall limit the presentation for simplicity to a single curve only
SLIDE 5 Introduction - main ingredients
Discrete tenor structure: 0 = T0 < T1 < . . . < Tn = T ∗, with δk = Tk+1 − Tk, for all k T0 T1 T2 t Tk Tk+1 Tn−1 Tn = T ∗ zero coupon bond with maturity Tk: B(t, Tk) forward price process: F(t, Tk, Tk+1) =
B(t,Tk ) B(t,Tk+1)
forward Libor rate for the interval [Tk, Tk+1]: L(t, Tk) Master relation 1 + δkL(t, Tk) = F(t, Tk, Tk+1)
SLIDE 6 Forward measures
forward martingale measure with numeraire B(·, Tk): PTk Density process for the change between two forward measures dPTk dPTk+1
= B(0, Tk+1) B(0, Tk) B(t, Tk) B(t, Tk+1) = F(t, Tk, Tk+1) F(0, Tk, Tk+1) No arbitrage: B(·, Tj) B(·, Tk) ∈ M(PTk ), ∀j, k ⇔ B(·, Tk−1) B(·, Tk) ∈ M(PTk ), ∀k Martingale condition F(·, Tk−1, Tk), L(·, Tk−1) ∈ M(PTk ).
SLIDE 7 Main modeling requirements
1
Libor rates should be non-negative: L(t, Tk) ≥ 0, for all t, k
2
The model should be arbitrage-free: L(·, Tk) are PTk+1-martingales
3
The model should be analytically tractable: closed or semi-closed formulas for most liquid derivatives (caps and swaptions) or efficient and accurate approximations
4
The model should be flexible, i.e. provide good calibration Post-crisis modeling: Libor rates depend on the tenor and also differ from the discounting rates = ⇒ various other rates have to be modeled in addition to (1), or equivalently their spreads = ⇒ the rates can become negative, whereas the spreads still always remain positive in the current market situation
SLIDE 8 New interest rate models – multiple curve setup
2006 2007 2008 2009 2010 50 100 150 200 250 bp
Euribor - Eoniaswap spreads
spread 1m spread 3m spread 6m spread 12m
SLIDE 9 Modeling of the forward price processes
The forward price process with respect to the terminal tenor date: F(t, Tk, Tn) := B(t, Tk) B(t, Tn), t ∈ [0, Tk], for all 1 ≤ k ≤ n. The modeling requirements now become:
1
For all k = 1, . . . , n − 1 and all t ∈ [0, Tk] 1 ≤ F(t, Tk+1, Tn) ≤ F(t, Tk, Tn)
2
The forward prices F(·, Tk, Tn) should be PTn-martingales
3
Tractability
4
Calibration (flexibility)
SLIDE 10 Comparison of some existing approaches
Libor market model and extensions (Sandmann et al., Brace et al., Musiela and Rutkowski, Jamshidian, Eberlein and ¨ Ozkan, Joshi, Andersen et al., Rebonato, Schoenmakers et al.) L(t, Tk) = L(0, Tk) exp X k
t ,
where X k are semimartingales Forward price models (Musiela and Rutkowski, Eberlein et al.) F(t, Tk, Tk+1) = F(0, Tk, Tk+1) exp X k
t ,
where X k are semimartingales Affine Libor model (Keller-Ressel et al., Da Fonseca et al.) F(t, Tk, Tn) = EPTn [euk ,XTn |Ft] = eφTn−t (uk )+ψTn−t (uk ),Xt where X is an affine process
SLIDE 11 Additive construction of forward price models
Instead of modeling directly the forward prices, we model the forward price spreads: S(t, Tk, Tn) := F(t, Tk, Tn) − F(t, Tk+1, Tn) for all k = 1, . . . , n − 1. Then, requirements (1) and (2) become (S) The forward price spreads S(·, Tk, Tn) are PTn-martingales and S(t, Tk, Tn) ≥ 0 for all k = 1, . . . , n and all t ∈ [0, Tk]. The forward prices are sums of the forward price spreads: F(t, Tk, Tn) =
n
S(t, Tj, Tn) with S(t, Tj, Tn) =
B(t,Tn)
for j < n, 1 for j = n.
SLIDE 12 Additive construction of forward price models
Expressed in terms of bond prices, we have the following decomposition: B(t, Tk) B(t, Tn) = B(t, Tk) − B(t, Tk+1) B(t, Tn)
+ · · · + B(t, Tn−1) − B(t, Tn) B(t, Tn)
+1 and each summand is a PTn-martingale.
SLIDE 13 Additive construction of forward price models
To specify the model, we set S(t, Tj, Tn) := S(0, Tj, Tn)Nj
t
where the initial values S(0, Tj, Tn) are market data and (Nj)1≤j≤n−1 nonnegative PTn-martingales starting at 1. Furthermore, set Nj
t :=
EPTn [f j(Y j
Tn)|Ft]
EPTn [f j(Y j
Tn)]
, where f j are nonnegative functions and Y j are semimartingales such that the conditional expectation above is analytically tractable. Our choice: polynomial functions and polynomial preserving processes
SLIDE 14 Caplets and swaptions – 2
Proposition The price of the caplet at time t ≤ Tk is given by Cplk
t = B(t, Tn)EPTn
µjNj
Tk
+ Ft
- where µk := S(0, Tk, Tn) and µj := −δkKS(0, Tj, Tn), for j > k.
Proposition The price of the swaption at time t ≤ Tk is given by Swpt = B(t, Tn)EPTn
ηjNj
Tk
+ Ft
where ηk := S(0, Tk, Tn) and ηj := (1 − j
i=k+1 ci)S(0, Tj, Tn), for j > k.
SLIDE 15 Polynomial preserving processes
Let X be a time-homogeneous Markov process and a semimartingale on the state space E ⊂ Rd, relative to some filtration (Ft)t≥0 Transition semigroup Ptf(x) :=
f(y)pt(x, dy), where (pt)t≥0 is the transition kernel of X. Then Ex[f(Xt+s)|Fs] = EXs[f(Xt)] = Ptf(Xs) Denote by Pm the vector space of polynomials on E up to degree m ≥ 0: Pm = x →
m
αkxk, x ∈ E, αk ∈ R
SLIDE 16 Polynomial preserving processes
Cuchiero, Keller-Ressel and Teichmann (2012), Filipovi´ c, Gourier, Mancini, Trolle (2012, 2013) The process X is m-polynomial preserving (m-PP) if for all k ≤ m, Pt(Pk) ⊂ Pk. Or equivalently: the generator A of X is m-polynomial preserving: A(Pk) ⊂ Pk, for all k ≤ m. for every k ≤ m, there exists a linear map A on Pk such that Pt
If B = {e1, . . . , eM} denotes a basis of Pk, then A = (Aij)i,j=1,...,M is obtained from Aei = M
j=1 Aijej and
Ptf = (α1, . . . , αM)etA(e1, . . . , eM)⊤, for any f = M
i=1 αiei ∈ Pk
SLIDE 17
Polynomial preserving processes
Hence: the expected value of any polynomial of (Xt) is again a polynomial in the initial value X0 = x Moments of Xt can be computed explicitly and easily without knowing the probability distribution or characteristic function of Xt: Ex[(Xt)k] = (0, . . . , 0, 1, 0, . . . , 0)etA(x0, x1, . . . , xm)⊤, where we assumed d = 1 for simplicity. The only task is to compute the matrix exponential etA
SLIDE 18 Polynomial preserving processes
Hence: the expected value of any polynomial of (Xt) is again a polynomial in the initial value X0 = x Moments of Xt can be computed explicitly and easily without knowing the probability distribution or characteristic function of Xt: Ex[(Xt)k] = (0, . . . , 0, 1, 0, . . . , 0)etA(x0, x1, . . . , xm)⊤, where we assumed d = 1 for simplicity. The only task is to compute the matrix exponential etA Applications in finance: explicit formulas for polynomial claims, approximations for
- ther claims, variance reduction techniques for Monte Carlo computations in pricing
and hedging
SLIDE 19 Polynomial preserving processes
Semimartingale characteristics (B, C, ν) of X is necessarily of the form: Bi
t =
t bi(Xs)ds Cij
t +
t
t aij(Xs)ds, where bi ∈ P1 and aij ∈ P2 Examples of polynomial preserving processes: affine processes (with finite moments), exponential L´ evy processes, quadratic term structure models Pearson diffusions (Forman and Sørensen (2010)) dXt = −b(Xt − θ)dt +
t dWt,
X0 = 0, for b > 0 and a, a1, a2 such that the square root is well-defined
SLIDE 20 Polynomial specification for the additive model
X = (Xt)t≥0 an m-polynomial preserving process on E ⊂ I Rd and X0 = x0 ∈ E x → pj(x), j = 1, . . . , n − 1, be a family of nonnegative polynomial functions of degree m. the PTn-martingales Nj, j = 1, . . . , n − 1, are defined as follows: Nj
t := EPTn [pj(XTn)
EPTn [pj(XTn)] = PTn−tpj(Xt) PTnpj(x0) . PTn−tpj(Xt) is a polynomial of degree m in Xt, hence each Nj
t is a polynomial in
Xt
SLIDE 21 Caplet and swaption pricing
Revisiting the formulas for the time-0 price of the caplet and the swaption: Caplet Pricing Formula Cplk
0 = B(0, Tn)EPTn
ai
i + , where the coefficients ai are explicitly determined by the infinitesimal generator A of the process X. Swaption Pricing Formula Swp0 = B(0, Tn)EPTn
bi
i + , with the coefficients bi explicitly determined by the infinitesimal generator A of the process X.
SLIDE 22 Tractable examples
quadratic OU Gaussian model (OU Gaussian process as driver combined with quadratic functions) quadratic OU L´ evy model linear model (a positive PP-process as driver and linear functions), e.g. linear CIR model Note: in the first two cases, polynomial functions of higher (even) degree can be used as well if the assumption on the positivity of the interest rates (and/or spreads) is relaxed, in the first two cases, the linear function can be applied, as well as, any
- ther polynomial of odd degree. In the third case, there is no need to take a
positive process
SLIDE 23 Examples: Quadratic L´ evy case
Let (Xt) with values in Rd be given by dXt = (b + VXt)dt + dLt, where L is a L´ evy process with the triplet (0, a, ν). The generator A of X is Af(x) = b⊤∇f(x) + x⊤V∇f(x) + 1 2
d
ajk ∂2f(x) ∂xj∂xk +
f(x + y) − f(x) −
d
∂f(x) ∂xj h(y)j ν(dy) = ⇒ X is a polynomial preserving process Assume V = diag (v1, . . . , vd) and consider the subspace of P2 with basis B = {1, x1, . . . , xd, x2
1, . . . , x2 d}
SLIDE 24 Examples: Quadratic L´ evy case
Then A(1) = 0 A(xi) = bi + xivi +
A(x2
i ) = 2xi
- bi +
- Rd (yi − h(y)i)ν(dy)
- =:γi
- + 2x2
i vi + aii +
i ν(dy)
and A := A
. . . . . . γ1 + b1 v1 . . . . . . . . . . . . ... . . . . . . ... . . . γd + bd . . . vd . . . ξ1 + a11 2(γ1 + b1) . . . 2v1 . . . . . . . . . ... . . . . . . ... . . . ξd + add . . . 2(γd + bd) . . . 2vd
SLIDE 25 Examples: Quadratic L´ evy case
Particular case: V = 0 = ⇒ X is a L´ evy process and A is nilpotent, i.e. An = 0, for every n ≥ n0 We have Ptf(x) = (α1, . . . , α2d+1)etA(1, x1, . . . , xd, x2
1, . . . , x2 d)⊤,
where (αi) are the coefficients of f(x) in terms of the basis B = {1, x1, . . . , xd, x2
1, . . . , x2 d}
Therefore, PTn−tf(Xt) =
d
C2,i(Tn − t)(X i
t )2 + d
C1,i(Tn − t)X i
t + C0(Tn − t)
and the coefficients C2,i(Tn − t), C1,i(Tn − t) and C0(Tn − t) are explicitly given as linear combinations of elements from the matrix C(Tn − t) := e(Tn−t)A
SLIDE 26 Examples: Quadratic L´ evy case
Now in order to obtain the price of all considered interest rate derivatives an expectation of the following form must be computed π0 := EPTn
Tk uXTk + v ⊤XTk + w
+ , for some u ∈ Rd×d, v ∈ Rd and w ∈ R. Then we proceed using the Fourier transform methods: π0 = 1 2π
ˆ g(iR + v)Mh(XTk )(R − iv)dv, where Mh(XTk ) is the moment generating function of h(XTk ) and ˆ g is the Fourier transform of the payoff function g.
SLIDE 27
Examples: Quadratic L´ evy case
We have two possible choices: (1) Setting g(x) = x+, g : I R → I R h(y) = y ⊤uy + v ⊤y + w, h : I Rd → I R = ⇒ Fourier transform of g is easy, mgf Mh(XTk ) has to be computed (2) Setting g(x) = (x⊤ux + v ⊤x + w)+, g : I Rd → I R h(y) = y, h : I R → I R = ⇒ mgf Mh(XTk ) is given in closed form for typical L´ evy processes used in finance, but the Fourier transform of g can be numerically demanding to compute (finding zeros and dimension of integration) = ⇒ If d = 1, (2) is more convenient; if d ≥ 2, (1) is preferred
SLIDE 28 Quadratic L´ evy case: Transform Formula
To apply Fourier pricing we need a new result on computing the mgf of quadratic forms of L´ evy processes. Condition A The characteristic exponent ψ of X can be extended to an analytic function on a domain D ⊂ Cd which contains the set Θ =
π 4 , 3π 4
5π 4 , 7π 4
Moreover, the extended function ψ satisfies the growth bound lim sup
r→∞
ℜ(ψ(yreiθ)) r 2 ≤ 0 for all y ∈ I Rd, θ ∈ π
4 , 3π 4
This is a multivariate version of a similar condition from Keller-Ressel, Muhle-Karbe (2013).
SLIDE 29 Quadratic L´ evy case: Transform Formula
Proposition (Transform Formula for Quadratic L´ evy Processes) Let Xt be an I Rd-valued L´ evy process with characteristic exponent ψ(u) that satisfies Condition A and let Qt = X ⊤
t ΣXt + X ⊤ t µ.
Then the Fourier-Laplace transform of QT is given by E
= exp u 4 µ⊤Σ−1µ
i 2 √ 2uµ⊤Σ−1Z
√ 2uZ
- for all u ∈ C with ℜu ∈ (0, C), where the expectation on the right hand side is taken
with respect to the d-dimensional normal random variable Z with zero mean and covariance Σ. C is a constant depending on µ and the order of finite exponential moments of X. Remark: for the quadratic Gaussian case, Filipovi´ c et al. (2013) provide the expression for the moment generating function
SLIDE 30 Concluding remarks
Additive construction of discrete-tenor interest rate models based on polynomial preserving process Efficient pricing of caplets and swaptions using Fourier methods (no approximations needed) Tractable specification based on quadratic L´ evy Process New transform formula for quadratic forms of L´ evy processes Straightforward extension to post-crisis multiple curve framework Work in progress: calibration and study of implied volatilities in the model Based on:
- K. Glau, Z.G. and M. Keller-Ressel (2015): Construction of Libor models
from polynomial preserving processes. Working paper.
SLIDE 31
Thank you for your attention
SLIDE 32 New interest rate models – multiple curve setup
2006 2007 2008 2009 2010 50 100 150 200 250 bp
Euribor - Eoniaswap spreads
spread 1m spread 3m spread 6m spread 12m
SLIDE 33 New interest rate models – multiple curve setup
In addition to the family (Nj
t ), model another family of nonnegative
PTn-martingales (˜ Nj
t ), j = 1, . . . , n − 1
The OIS-forward prices are modeled in terms of (Nj
t ), which do not have to
necessarily positive given the current market conditions F OIS(t, Tk, Tn) =
n
βjNj
t
The Libor rates are modeled as rational functions of (Nj
t ) and (˜
Nj
t )
The prices of caplets and swaptions are again of the same additive form EPTn
n
ajNj
t + n
˜ ai ˜ Ni
t
+
Polynomial specification produces an equally tractable and flexible multiple curve model as in the classical single curve case
