Term structure models driven by
Wiener processes and Poisson measures:
Existence and positivity
Stefan Tappe
ETH Zürich
[email protected]
(joint work with Damir Filipović and Josef Teichmann)
6th World Congress of the Bachelier Finance Society
June 25th, 2010 (Toronto)
Stefan Tappe
2010/06/25
Introduction
• Zero Coupon Bonds P (t, T ).
• The Heath-Jarrow-Morton-Musiela (HJMM) equation:
Z
d
rt + αHJM(rt) dt + σ(rt)dWt +
γ(rt−, x)(µ(dt, dx) − F (dx)dt).
drt =
dξ
E
• Establish existence and positivity.
• The Brody-Hughston equation:
Z
d
dρt =
ρt + ρt(0)ρt dt + σ(ρt)dWt +
γ(ρt−, x)(µ(dt, dx) − F (dx)dt).
dξ
E
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
Zero Coupon Bonds
• Zero Coupon Bonds P (t, T ).
• Financial assets paying the holder one unit of cash at T .
1
0.8
0.6
0.4
0.2
0
2
4
6
8
10
Figure 1: Price process of a T -bond with date T = 10.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
The HJM model with jumps
• Björk, Kabanov, Runggaldier, Di Masi 1997 [1]: For T ≥ 0 we have
f (t, T ) = f ∗(0, T ) +
Z
t
Z
t
σ(s, T )dWs
α(s, T )ds +
0
0
Z tZ
γ(s, x, T )(µ(ds, dx) − F (dx)ds),
+
0
t ∈ [0, T ].
E
• Implied bond market:
P (t, T ) = exp
Z
−
T
f (t, s)ds .
t
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
From HJM to Stochastic Equations
• Drift and volatilities depend on the current forward curve:
α(t, T, ω) = α(t, T, f (t, ·, ω)),
σ(t, T, ω) = σ(t, T, f (t, ·, ω)),
γ(t, x, T, ω) = γ(t, x, T, f (t, ·, ω)).
• Infinite dimensional stochastic equation:


 df (t, T ) = α(t,
R T, f (t, ·))dt + σ(t, T, f (t, ·))dWt
+ E γ(t, x, T, f (t, ·))(µ(dt, dx) − F (dx)dt)


f (0, T ) = f ∗(0, T ).
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
The transformed equation
• Musiela parametrization of forward rates:
rt(ξ) := f (t, t + ξ),
• Making the transformation f (t, T )
rt(ξ) we obtain
t
Z
Z
0
t
St−sσ(rs)dWs
St−sα(rs)ds +
rt = Sth0 +
ξ ≥ 0.
0
Z tZ
St−sγ(rs−, x)(µ(ds, dx) − F (dx)ds),
+
0
t≥0
E
• where (St)t≥0 denotes the shift-semigroup Sth := h(t + ·) on H.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
From HJMM to SPDEs
• Thus, (rt)t≥0 is a mild solution of the SPDE
drt =
Z
d
rt + α(rt) dt + σ(rt)dWt +
γ(rt−, x)(µ(dt, dx) − F (dx)dt),
dξ
E
• with given vector fields
α : H → H,
• where
d
dξ
σ : H → L02(H),
γ : H × E → H,
is the infinitesimal generator of (St)t≥0.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
The HJMM equation
• The bond market P (t, T ) should be free of arbitrage.
• Under a martingale measure Q ∼ P we have
αHJM(h) =
X
j
j
Z
•
j
Z
σ (h)(η)dη −
σ (h)
0
γ(h, x)
R•
− 0 γ(h,x)(η)dη
e
− 1 F (dx).
E
• This leads to the HJMM equation
drt =
Z
d
rt + αHJM(rt) dt + σ(rt)dWt +
γ(rt−, x)(µ(dt, dx) − F (dx)dt).
dξ
E
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
Stochastic partial differential equations
• Consider the SPDE
(
drt = (Art + α(rt))dt + σ(rt)dWt +
r0 = h0,
R
E
γ(rt−, x)(µ(dt, dx) − F (dx)dt)
• with given vector fields
α : H → H,
σ : H → L02(H),
γ : H × E → H,
• where A : D(A) ⊂ H → H is the generator of a C0-semigroup on H.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
Assumptions for the existence result
• Lipschitz continuity: For all h1, h2 ∈ H we have
kα(h1) − α(h2)k + kσ(h1) − σ(h2)kL0(H)
2
Z
1/2
≤ Lkh1 − h2k.
+
kγ(h1, x) − γ(h2, x)k2F (dx)
E
• Linear growth: We have
R
2
kγ(0,
x)k
F (dx) < ∞.
E
• We assume that (St)t≥0 is pseudo-contractive, that is
kStk ≤ eωt,
t ≥ 0.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
Existence- and uniqueness result
• Unique mild solutions for the SPDE
(
R
drt = (Art + α(rt))dt + σ(rt)dWt + E γ(rt−, x)(µ(dt, dx) − F (dx)dt)
r0 = h0,
• i.e., the ”Variation of constants formula” is satisfied:
t
Z
rt = Sth0 +
Z
St−sα(rs)ds +
0
t
St−sσ(rs)dWs
0
Z tZ
St−sγ(rs−, x)(µ(ds, dx) − F (dx)ds),
+
0
t ≥ 0.
E
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
The HJMM equation
• The HJMM equation is an SPDE
Z
γ(rt−, x)(µ(dt, dx) − F (dx)dt),
drt = (Art + α(rt))dt + σ(rt)dWt +
E
• for which we have
H = Hβ ,
d
A= ,
dξ
α = αHJM,
• where αHJM is given by
Z •
Z
R•
X
αHJM(h) =
σ j (h)
σ j (h)(η)dη −
γ(h, x) e− 0 γ(h,x)(η)dη − 1 F (dx).
j
0
E
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
The space of forward curves
• For β > 0 we define the space
Hβ := {h : R+ → R : h is absolutely continuous with khkβ < ∞},
• where the norm is defined by
khkβ :=
|h(0)|2 +
Z
|h0(ξ)|2eβξ dξ
1/2
.
R+
• The shift semigroup (St)t≥0 on Hβ has the generator
d
d
= {h ∈ Hβ : h0 ∈ Hβ }.
A= , D
dξ
dξ
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
Assumptions on the vector fields
• Lipschitz continuity: For all h1, h2 ∈ Hβ we have
kσ(h1) − σ(h2)kL0(Hβ ) ≤ Lkh1 − h2kβ ,
2
Z
1/2
eΦ(x)kγ(h1, x) − γ(h2, x)k2β F (dx)
≤ Lkh1 − h2kβ .
E
• Boundedness: For all h ∈ Hβ we have
kσ(h)kL0(Hβ ) ≤ M,
2
Z
e
Φ(x)
kγ(h, x)k2β
∨
kγ(h, x)k4β
F (dx) ≤ M.
E
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
Solution of the HJMM equation
• The HJM drift term αHJM : Hβ → Hβ is Lipschitz continuous:
kαHJM(h1) − αHJM(h2)kβ ≤ Kkh1 − h2kβ .
• Unique mild solutions for the HJMM equation
(
drt =
d
rt
( dξ
+ αHJM(rt))dt + σ(rt)dWt +
R
E
γ(rt−, x)(µ(dt, dx) − F (dx)dt)
r0 = h0.
• The interest rates rt(ξ) should not be negative.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
Positivity preserving models
• Let P ⊂ Hβ be the convex cone
P = {h ∈ Hβ : h ≥ 0} =
\
{h ∈ Hβ : h(ξ) ≥ 0}.
ξ∈R+
• The HJMM equation is positivity preserving if for all h0 ∈ P we have
P(rt ∈ P ) = 1,
t ≥ 0.
• Stochastic invariance problem.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
A general invariance result
• Consider an SPDE on the space Hβ of forward curves
drt =
Z
d
rt + α(rt) dt + σ(rt)dWt +
γ(rt−, x)(µ(dt, dx) − F (dx)dt),
dξ
E
• with given vector fields
α : Hβ → Hβ ,
σ : Hβ → L02(Hβ ),
γ : Hβ × E → Hβ .
• This SPDE is positivity preserving if and only if we have (1)–(4).
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
The volatility and the jumps
• At the boundary, the volatility σ is parallel to the edge:
σ j (h)(ξ) = 0,
h ≥ 0 with h(ξ) = 0.
(1)
• The convex cone P captures all jumps:
h + γ(h, x) ∈ P,
h ∈ P and F -almost all x ∈ E.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
(2)
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Stefan Tappe
2010/06/25
Small jumps at the boundary
• In general, we have:
Z
kγ(h, x)k2β F (dx) < ∞,
Z
kγ(h, x)kβ F (dx) = ∞.
but
E
E
• Small jumps, which are not parallel to boundary, are of finite variation:
Z
|γ(h, x)(ξ)|F (dx) < ∞,
h ≥ 0 with h(ξ) = 0.
(3)
E
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
The drift vector field
• Subtract the F (dx)dt-part of the stochastic integral to the drift:
drt =
Z
d
rt + α(rt) dt + σ(rt)dWt +
γ(rt−, x)(µ(dt, dx) − F (dx)dt).
dξ
E
• At the boundary, the corrected drift term is inward pointing:
Z
α(h)(ξ) −
γ(h, x)(ξ)F (dx) ≥ 0,
h ≥ 0 with h(ξ) = 0.
(4)
E
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
Remarks concerning the drift vector field
• The convex cone P has particular properties.
• The shift semigroup (St)t≥0 leaves P invariant:
St P ⊂ P
for all t ≥ 0.
• No Stratonovich correction term, because
j
j
Dσ (h)σ (h) (ξ) = 0,
h ≥ 0 with h(ξ) = 0.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
Invariance conditions for the HJMM equation
• This SPDE is positivity preserving if and only if we have (1)–(4).
• Consequence: The HJMM equation
Z
d
rt + αHJM(rt) dt + σ(rt)dWt +
γ(rt−, x)(µ(dt, dx) − F (dx)dt)
drt =
dξ
E
• is positivity preserving if and only if
σ j (h)(ξ) = 0,
h ≥ 0 with h(ξ) = 0
γ(h, x)(ξ) = 0,
h + γ(h, x) ∈ P,
h ≥ 0 with h(ξ) = 0 and F -almost all x ∈ E
h ∈ P and F -almost all x ∈ E.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
(5)
(6)
(7)
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Stefan Tappe
2010/06/25
Another approach to bond price markets
• Following Brody, Hughston 2001 [2] we define the bond prices
Z
∞
ρt(ξ)dξ,
P (t, T ) =
T −t
• where (ρt)t≥0 is a process of probability densities on R+.
• Then we have P (T, T ) = 1 for all T ≥ 0
• and T 7→ P (t, T ) is non-increasing with limit 0 for T → ∞.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
The Brody-Hughston equation
• Consider the Brody-Hughston equation
dρt =
d
ρt + ρt(0)ρt dt + σ(ρt)dWt +
dξ
Z
γ(ρt−, x)(µ(dt, dx) − F (dx)dt),
E
• on the state space Hβ0 with vector fields
σ : Hβ0 → L02(Hβ0 ),
γ : Hβ0 × E → Hβ0 ,
• where Hβ0 = {h ∈ Hβ : limξ→∞ h(ξ) = 0}.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
Stochastic invariance problem
• We observe that Hβ0 ⊂ L1(R+).
• Stochastic invariance of the convex set P ⊂ Hβ0 of probability densities
Z
n
P = h ∈ Hβ0 : h ≥ 0 and
o
h(ξ)dξ = 1
R+
Z
o
n
0
0
h(ξ)dξ = 1
= {h ∈ Hβ : h ≥ 0} ∩ h ∈ Hβ :
{z
}
|
R+
|
{z
}
Use our previous results
Invariance conditions are known
• Unique mild solutions for the Brody-Hughston equation.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
Conclusion
• Zero Coupon Bonds P (t, T ).
• The Heath-Jarrow-Morton-Musiela (HJMM) equation:
Z
d
rt + αHJM(rt) dt + σ(rt)dWt +
γ(rt−, x)(µ(dt, dx) − F (dx)dt).
drt =
dξ
E
• We have established existence and positivity.
• The Brody-Hughston equation:
Z
d
dρt =
ρt + ρt(0)ρt dt + σ(ρt)dWt +
γ(ρt−, x)(µ(dt, dx) − F (dx)dt).
dξ
E
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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Stefan Tappe
2010/06/25
References
[1] Björk, T., Di Masi, G., Kabanov, Y., Runggaldier, W. (1997): Towards a
general theory of bond markets. Finance and Stochastics 1(2), 141–174.
[2] Brody, D. C., Hughston, L. P. (2001): Interest rates and information
geometry, Proceedings of The Royal Society of London. Series A.
Mathematical, Physical and Engineering Sciences, 457, 1343–1363.
[3] Filipović, D., Tappe, S., Teichmann, J. (2010): Term structure models
driven by Wiener processes and Poisson measures: Existence and positivity.
Forthcoming in SIAM Journal on Financial Mathematics.
[4] Heath, D., Jarrow, R., Morton, A. (1992): Bond pricing and the term
structure of interest rates: a new methodology for contingent claims
valuation. Econometrica 60(1), 77–105.
Term structure models driven by Wiener processes and Poisson measures: Existence and positivity
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