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A new approach to LIBOR modeling application of affine processes Antonis Papapantoleon FAM TU Vienna Special Semester on Stochastics with Emphasis on Finance RICAM, Linz, Austria, 4 December 2008 Antonis Papapantoleon (TU Vienna) Affine

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A new approach to LIBOR modeling application of affine processes

Antonis Papapantoleon

FAM – TU Vienna Special Semester on Stochastics with Emphasis on Finance RICAM, Linz, Austria, 4 December 2008

Antonis Papapantoleon (TU Vienna) Affine LIBOR model 1 / 27

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1 Interest rate markets 2 LIBOR model: Axioms 3 LIBOR and Forward price model 4 Affine processes 5 Affine LIBOR model 6 Example: Γ-OU Martignales 7 Summary and Outlook

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Interest rate markets

Interest rates – Notation

B(t, T): time-t price of a zero coupon bond for T, i.e. B(T, T) = 1; L(t, T): time-t forward LIBOR for [T, T + δ]: L(t, T) = 1 δ

  • B(t, T)

B(t, T + δ) − 1

  • F(t, T, U): time-t forward price for T and U: F(t, T, U) = B(t,T)

B(t,U)

“Master” relationship F(t, T, T + δ) = B(t, T) B(t, T + δ) = 1 + δL(t, T) (1)

2 / 31

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Interest rate markets

Interest rates evolution

Evolution of interest rate term structure, 2003 – 2004 (picture: Th. Steiner)

3 / 31

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Interest rate markets

Calibration problems

2 4 6 8 10 2.5 4.0 6.0 8.0 10.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 Maturity (in years) Strike rate (in %)

1

Implied volatilities are constant neither across strike nor across maturity

2

Variance scales non-linearly over time (see e.g. Skovmand)

4 / 31

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LIBOR model: Axioms

LIBOR model: Axioms

Economic thought dictates that LIBOR rates should satisfy: Axiom 1 The LIBOR rate should be positive, i.e. L(t, T) > 0 for all t. Axiom 2 The LIBOR rate process should be a martingale under the (corresponding) forward measure, i.e. L(·, T) ∈ M(PT+δ). Practical applications require: Models should describe the empirical evidence adequately. Models should be calibrated to liquid products (caps, atm swaptions). What axioms do the existing models satisfy?

5 / 31

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LIBOR and Forward price model

LIBOR models I (Sandmann et al, Brace et al, . . . , Eberlein & ¨

Ozkan)

Ansatz: model the LIBOR rate as the exponential of a semimartingale H: L(t, Tk) = L(0, Tk) exp t b(s, Tk)ds + t λ(s, Tk)dHTk+1

s

  • ,

(2) where b(s, Tk) ensures that L(·, Tk) ∈ M(PTk+1). H has the PTk+1-canonical decomposition HTk+1

t

= t √csdW Tk+1

s

+ t

  • R

x(µH − νTk+1)(ds, dx), (3) where the PTk+1-Brownian motion is W Tk+1

t

= W T∗

t

− t

  • N
  • l=k+1

δlL(t−, Tl) 1 + δlL(t−, Tl)λ(t, Tl)

  • √csds,

(4)

6 / 31

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LIBOR and Forward price model

LIBOR models II

and the PTk+1-compensator of µH is νTk+1(ds, dx) =

  • N
  • l=k+1

δlL(t−, Tl) 1 + δlL(t−, Tl)

  • eλ(t,Tl)x − 1
  • + 1
  • νT∗(ds, dx).

7 / 31

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LIBOR and Forward price model

LIBOR models II

and the PTk+1-compensator of µH is νTk+1(ds, dx) =

  • N
  • l=k+1

δlL(t−, Tl) 1 + δlL(t−, Tl)

  • eλ(t,Tl)x − 1
  • + 1
  • νT∗(ds, dx).

Consequences for continuous semimartingales:

1 caplets can be priced in closed form; 2 swaptions and multi-LIBOR products cannot be priced in closed form; 3 Monte-Carlo pricing is very time consuming coupled high

dimensional SDEs!

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LIBOR and Forward price model

LIBOR models II

and the PTk+1-compensator of µH is νTk+1(ds, dx) =

  • N
  • l=k+1

δlL(t−, Tl) 1 + δlL(t−, Tl)

  • eλ(t,Tl)x − 1
  • + 1
  • νT∗(ds, dx).

Consequences for continuous semimartingales:

1 caplets can be priced in closed form; 2 swaptions and multi-LIBOR products cannot be priced in closed form; 3 Monte-Carlo pricing is very time consuming coupled high

dimensional SDEs! Consequences for general semimartingales:

1 even caplets cannot be priced in closed form! 2 ditto for Monte-Carlo pricing. 9 / 31

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LIBOR and Forward price model

LIBOR models III: Remedies

1 “Frozen drift” approximation

Brace et al, Schl¨

  • gl, Glassermann et al . . .

replace the random terms by their deterministic initial values: δlL(t−, Tl) 1 + δlL(t−, Tl) ≈ δlL(0, Tl) 1 + δlL(0, Tl) (5) (+) deterministic characteristics closed form pricing (−) “ad hoc” approximation, no error estimates, compounded error . . .

2 Strong Taylor approximation

approximate the LIBOR rates in the drift/compensator by L(t, Tl) ≈ L(0, Tl) + Y (t, Tl)+ (6) where Y is the (scaled) exponential transform of H (Y = LogeH) theoretical foundation, error estimates, simpler equations for MC Siopacha and Teichmann; Hubalek, Papapantoleon & Siopacha

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LIBOR and Forward price model

Forward price model I (Eberlein & ¨

Ozkan, Kluge)

Ansatz: model the forward price as the exponential of a semimartingale H: F(t, Tk) = F(0, Tk) exp t b(s, Tk)ds + t λ(s, Tk)dHTk+1

s

  • ,

(7) where b(s, Tk) ensures that F(·, Tk) = 1 + δL(·, Tk) ∈ M(PTk+1). H has the PTk+1-canonical decomposition HTk+1

t

= t √csdW Tk+1

s

+ t

  • R

x(µH − νTk+1)(ds, dx), (8) where the PTk+1-Brownian motion is W Tk+1

t

= W T∗

t

− t

  • N
  • l=k+1

λ(t, Tl)

  • √csds,

(9)

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LIBOR and Forward price model

Forward price model II

and the PTk+1-compensator of µH is νTk+1(ds, dx) = exp

  • x

N

  • l=k+1

λ(t, Tl)

  • νT∗(ds, dx).

Consequences:

1 the model structure is preserved; 2 caps, swaptions and multi-LIBOR products priced in closed form.

So what is wrong?

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LIBOR and Forward price model

Forward price model II

and the PTk+1-compensator of µH is νTk+1(ds, dx) = exp

  • x

N

  • l=k+1

λ(t, Tl)

  • νT∗(ds, dx).

Consequences:

1 the model structure is preserved; 2 caps, swaptions and multi-LIBOR products priced in closed form.

So what is wrong? Negative LIBOR rates can occur!

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LIBOR and Forward price model

Forward price model II

and the PTk+1-compensator of µH is νTk+1(ds, dx) = exp

  • x

N

  • l=k+1

λ(t, Tl)

  • νT∗(ds, dx).

Consequences:

1 the model structure is preserved; 2 caps, swaptions and multi-LIBOR products priced in closed form.

So what is wrong? Negative LIBOR rates can occur! Aim: design a model where the model structure is preserved and LIBOR rates are positive. Tool: Affine processes

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Affine processes

Affine processes I

An affine process in the spirit of Duffie et al (henceforth DFS) is a time-homogeneous, stochastically continuous, Markov process X with state space D = Rm

0 × Rn ⊆ Rd, with X0 = x ∈ D, such that the

moment generating function has the form Ex[expu, Xt] = exp

  • φt(u) + ψt(u), x
  • ;

(10) for all u ∈ Cd or Rd where the expectation is finite. We assume that X is regular (and conservative). Lemma (Flow property) The functions φ and ψ satisfy the flow equations: φt+s(u) = φt(u) + φs(ψt(u)) ψt+s(u) = ψs(ψt(u)) (11) for all suitable u and t, s > 0.

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Affine processes

Affine processes II

Theorem (DFS, Main characterization) If X is a regular affine process, then φ and ψ solve the generalized Riccati equations ∂ ∂t φt(u) = F(ψt(u)), φ0(u) = 0 (12) ∂ ∂t ψt(u) = R(ψt(u)), ψ0(u) = u, (13) where F and R are defined via F(u) = ∂ ∂t

  • t=0φt(u)

(14) R(u) = ∂ ∂t

  • t=0ψt(u)

(15) Moreover, . . .

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Affine processes

Affine processes III

Theorem (continued) . . . the functions F and R are of L´ evy–Khintchine form: F(u) = b, u − c + a 2u, u

  • +
  • D
  • ez,u − 1 − u, h(z)
  • m(dz)

Ri(u) = βi, u − γi + αi 2 u, u

  • +
  • D
  • ez,u − 1 − u, h(z)
  • µi(dz)

where (a, b, c, m, αi, βi, γi, µi)1≤i≤d are admissible parameters. X is a Feller process, the generator A has affine form, the semimartingale characteristics (conservative case) have affine form, etc. Conversely, to each set of admissible parameters corresponds a regular affine process on D with generator A.

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Affine processes

Affine processes IV

1 Affine processes on R: the admissibility conditions yield

F(u) = −c + bu + a 2u2 +

  • R
  • ezu − 1 − uh(z)
  • m(dz)

R(u) = βu, for a, c ∈ R0 and b, β ∈ R.

Every affine process on R is an Ornstein–Uhlenbeck (OU) process.

2 Affine processes on R0: the admissibility conditions yield

F(u) = −c + bu +

  • D
  • ezu − 1
  • m(dz)

R(u) = −γ + βu + α 2 u2 +

  • D
  • ezu − 1 − uh(z)
  • µ(dz),

for b, c, α, γ ∈ R0 and β ∈ R.

There exist affine process on R0 which are not OU process.

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Affine LIBOR model

Affine LIBOR model: martingales 1

1 Idea: consider an affine process; insert it in its moment generating

function with inverted time. The resulting process is a martingale.

2 If the affine process is positive, the martingale is greater than one.

Theorem The process Mu = (Mu

t )0≤t≤T defined by

Mu

t = exp (φT−t(u) + ψT−t(u), Xt) ,

(16) is a martingale. Moreover, if D = Rd

0, u ∈ I ⊆ Rd 0 then Mt ≥ 1 a.s. for

all t ∈ [0, T], for any X0 ∈ Rd

0.

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Affine LIBOR model

Affine LIBOR model: martingales 1

Proof (martingality). E[Mu

t |Fs] = E[exp (φT−t(u) + ψT−t(u), Xt) |Fs]

= exp(φT−t(u))E[exp (ψT−t(u), Xt) |Fs] = exp (φT−t(u) + φt−s(ψT−t(u)) + ψt−s(ψT−t(u)), Xs) . Now, using the flow properties (11), we get φT−t(u) + φt−s(ψT−t(u)) = φT−s(u) ψt−s(ψT−t(u)) = ψT−s(u). Hence, the statement is proved, since E[Mu

t |Fs] = exp (φT−s(u) + ψT−s(u), Xs) = Ms.

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Affine LIBOR model

Affine LIBOR model: martingales 1

Proof (positivity). Since u, x and X take values in Rd

0, we simply have for any t ≥ 0

Ex[expu, Xt] = exp

  • φt(u) + ψt(u), x
  • ≥ 1.

(17) Example (L´ evy process) Consider a L´ evy subordinator, then Mu

t = exp (φT−t(u) + ψT−t(u), Xt)

= exp ((T − t)κ(u) + u · Xt) = exp(Tκ(u)) exp (u · Xt − tκ(u)) , (18) which is obviously a positive martingale.

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Affine LIBOR model

Affine LIBOR model: Ansatz

Consider a discrete tenor structure 0 = T0 < T1 < T2 < · · · < TN; discounted traded assets (bonds) are martingales with respect to the terminal martingale measure, i.e. B(·, Tk) B(·, TN) ∈ M(PTN), for all k ∈ {1, . . . , N − 1}. (19) We model quotients of bond prices using the martingales M as follows: B(t, T1) B(t, TN) = Mu1

t

(20) . . . B(t, TN−1) B(t, TN) = MuN−1

t

. (21)

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Affine LIBOR model

Affine LIBOR model: forward prices

Consequence: forward prices have the following form B(t, Tk) B(t, Tk+1) = B(t, Tk) B(t, TN) B(t, TN) B(t, Tk+1) = Muk

t

Muk+1

t

= exp

  • φTN−t(uk) − φTN−t(uk+1)

+

  • ψTN−t(uk) − ψTN−t(uk+1), Xt
  • .

(22) Forward measures are related via: dPTk−1 dPTk

  • Ft = F(t, Tk−1, Tk)

F(0, Tk−1, Tk) = B(0, Tk) B(0, Tk−1) × Muk−1

t

Muk

t

(23)

  • r equivalently:

dPTk−1 dPTN

  • Ft = B(0, TN)

B(0, Tk−1) × B(t, Tk−1) B(t, TN) = B(0, TN) B(0, Tk−1) × Muk−1

t

. (24)

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Affine LIBOR model

Affine LIBOR model: dynamics under forward measures

The moment generating function of Xt under a forward measure is EPTk+1[evXt] = EPTN [Muk+1

t

evXt] = exp

  • φT−t(uk+1) + φt
  • ψT−t(uk+1) + v
  • +
  • ψt
  • ψT−t(uk+1) + v
  • , x
  • .

(25) Let us denote by

B(t,Tk) B(t,Tk+1) = M

uk t

M

uk+1 t

= eAk+Bk·Xt; then the moment generating function is EPTk+1[ev(Ak+Bk·Xt)] = exp

  • vφTN−t(uk) + (1 − v)φTN−t(uk+1)

+ φt

  • vψTN−t(uk) + (1 − v)ψTN−t(uk+1)
  • +
  • ψt
  • vψTN−t(uk) + (1 − v)ψTN−t(uk+1)
  • , x
  • .

(26)

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Affine LIBOR model

Affine LIBOR model: consequences

1 The model structure is preserved! 2 More precisely, with respect to any forward measure all LIBOR rates

are of exponential-affine form.

3 Caps, swaptions and other multi-LIBOR products can be priced in

closed form.

4 Pricing (always) takes place under the terminal measure! 25 / 31

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Affine LIBOR model

Affine LIBOR model: caplet pricing

We can re-write the payoff of a caplet as follows (here K := 1 + δK): δ(L(Tk, Tk) − K)+ = (1 + δL(Tk, Tk) − 1 + δK)+ = Muk

Tk

Muk+1

Tk

− K + =

  • eAk+Bk·XTk − K

+ . (27) Then we can price caplets by Fourier-transform methods: C(Tk, K) = B(0, Tk+1)EPTk+1

  • δ(L(Tk, Tk) − K)+

= KB(0, Tk+1) 2π

  • R

Kiv−R ϕAk+BkXTk (R − iv) (R − iv)(R − 1 − iv)dv (28) where ϕAk+BkXTk is given by (26).

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Example: Γ-OU Martignales

Example: Γ-OU martingales I

Let H be a Gamma subordinator, i.e. H1 ∼ Γ(α, β), α, β > 0. The cumulant is κΓ(v) = −β log

  • 1 − v

α

  • .

(29) The Gamma-OU process is an affine process X with state space D = R0, that satisfies the SDE dXt = −λ(Xt − θ)dt + dHt, X0 = x ∈ R0, (30) where λ, θ > 0. The moment generating function of the Gamma-OU process is Ex[evXt] = exp

  • θ(1 − e−λt)v +

t κΓ(e−λsv)ds

  • φt(v)

+x · e−λtv

ψt(v)

  • .

(31)

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Example: Γ-OU Martignales

Example: Γ-OU martingales II

The Γ-OU martingales have the form Mu

t = exp (φT−t(u) + ψT−t(u), Xt)

= exp

  • θ(1 − e−λ(T−t))u +

T−t κΓ(e−λsu)ds

  • =:A

+ e−λ(T−t)u

  • =:B

·Xt

  • .

The moment generating function is ϕA+BXTN−1(v) = evAEx[evBXTN−1] = exp

  • vuθ(1 − e−λTN) + v

δ κΓ(e−λsu)ds + TN

δ

κΓ(vue−λs) + vue−λTNX0

  • .

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Example: Γ-OU Martignales

Numerical illustration

Examples of caplet implied volatilities for the Γ–OU martingales

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Summary and Outlook

Summary and Outlook

1 We have presented a LIBOR model that

is very simple (Axiom 0 !), and yet . . . captures all the important features . . . especially positivity and analytical tractability.

2 Future work:

thorough empirical analysis extensions: multiple currencies, default risk

3 M. Keller-Ressel, A. Papapantoleon, J. Teichmann (2008)

A new approach to LIBOR modeling, Preprint.

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Summary and Outlook

Summary and Outlook

1 We have presented a LIBOR model that

is very simple (Axiom 0 !), and yet . . . captures all the important features . . . especially positivity and analytical tractability.

2 Future work:

thorough empirical analysis extensions: multiple currencies, default risk

3 M. Keller-Ressel, A. Papapantoleon, J. Teichmann (2008)

A new approach to LIBOR modeling, Preprint. Thank you for your attention!

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