In what follows, we study one-factor interest rate models where the short-term interest raterconstitutes the only stochastic factor determining bond prices. We will see that the question of option pricing leads to an integro-differential equation. The reasoning here is analogous to the case of stock price models treated in Chapter 1.
Letrbe given as the solution of a stochastic differential equation r(t) =r(0) +
Xd α=1
Z
fα(t−, r(t−))dLαt, (4.16)
driven by a vector (L1, . . . , Ld) of independent Lévy processes. Assume that each fα is a Lipschitz function in the sense of the definition given in Protter (1992), Chapter V.3, p. 194:
Definition 4.9. A functionf : IR+×IRn→ IRis Lipschitz if there exists a (finite) constantksuch that (i) |f(t, x)−f(t, y)| ≤k|x−y|, eacht∈IR+;
(ii) t7→f(t, x)is right-continuous with left limits, eachx∈IRn. f is autonomous iff(t, x) =f(x), allt∈IR+.
As a consequence of Protter (1992), Theorem V.7, equation (4.16) has a unique solution.4 Protter (1992), Theorem V.32, shows thatr is a Markov process. For this, the independence of the increments ofLα is essential.
Remark: The above considerations are still valid for stochastic differential equations with an additional drift term “. . . dt”. This is because one can take account of the drift term by considering the deterministic Lévy processLd+1t =t. This yields an equation of the form (4.16).
4The cited theorem is valid for so-called functional Lipschitz coefficient functions. According to a remark in Protter (1992), p. 195, the functional Lipschitz functionsfαinduce functional Lipschitz coefficients.
Assumption 4.10. The short rate follows the diffusion (with jumps) given by drt=k(rt−, t)dt+f(rt−, t)dLt,
(4.17)
with Lipschitz coefficient functionsk(r, t)andf(r, t).
For example, the Lévy version of the extended Vasicek model (4.1) is of this form. In this case, the functionskandf are given by
k(r, t) = (ρ(t)−r)a f(r, t) =−bσ, (4.18)
with a deterministic functionρ(t)and positive constantsbσ,a.
Proposition 4.11. Assume that we are given a European option with a price processV(t)satisfying the following conditions.
(i) The payoff at the maturity dateT isV(T) =v r(T)
, with a deterministic functionv(x).
(ii) exp −Rt
0r(s)ds
V(t)is aQ-martingale. That is,Qis a martingale measure for this option.
(iii) exp −RT
t r(s)ds
V(T)isQ-integrable for allt∈[0, T].
Then there is a functiong(x, t)such that the option price at timet∈[0, T]is given byV(t) =g r(t), t . Remark. If the term structure model possesses an affine term structure, condition (i) is satisfied for all simple European options on a zero bond.5 This is because the zero bond price P(T, S) itself can be written as a deterministic (exponential-affine) function of the short rater(T).
Proof of Proposition 4.11. By assumptions (i) and (ii), the option price at any timet ∈ [0, T]can be obtained by taking conditional expectations.
exp −
Z t
0
r(s)ds
V(t) =EQ
exp −
Z T
0
r(s)ds
v r(T)Ft
,
hence V(t) = 1
exp −Rt
0r(s)dsEQ
exp
− Z T
0
r(s)ds
v r(T)Ft
=EQ
exp
− Z T
t
r(s)ds
v r(T)Ft
.
For the last equality, we have used condition (iii) from above. The last conditional expectation only depends on the conditional distribution of(r(s))t≤s≤T givenFt. But because of the Markov property of r, this is equal to the conditional distribution givenr(t), and hence
V(t) =EQ
exp
− Z T
t
r(s)ds
v r(T)r(t)
. For eacht∈[0, T], this conditional expectation can be factorized. V(t) =g r(t), t
.
If we impose additional differentiability assumptions on the function g(x, t), we can deduce that this function satisfies a linear integro-differential equation.
5Simple here means that the value of the option at its expiration dateT can be written as a deterministic function, of the bond price.
Proposition 4.12. We make the following assumptions.
(i) The value process V(t) of a security can be represented in the formV(t) = g r(t), t with a deterministic functiong(x, t).
(ii) The functiong(x, t)from (i) is of classC2,1(IR×IR+), that is, it is twice continuously differentiable in the first variable and once continuously differentiable in the second variable.
(iii) The short-rate process r satisfies Assumption 4.10. For each t > 0, the distribution ofrt− has supportI, whereI is a bounded or unbounded interval.
Then the functiong(r, t)satisfies the integro-differential equation 0 =−g(r, t)r+∂2g(r, t) + (∂1g)(r, t) f(r, t)b+k(r, t)
+1
2(∂11g)(r, t)ãcãf(r, t)2 (4.19)
+ Z
g r+f(r, t)x, t
−g r, t
−(∂1g) r, t
f(r, t)x
F(dx) r∈I, t∈(0, T) v(r) =g(r, T),
where(b, c, F(dx))is the Lévy-Khintchine triplet of the Lévy process driving the stochastic differential equation (4.10).
Remark. We are not aware of suitable results about continuity and differentiability of the functiong(x, t) from Proposition 4.11 in the case whererfollows a jump-diffusion. Hence we introduce assumption (ii) in the proposition in order to guarantee that the differential equation (4.19) makes sense.
Proof of Proposition 4.12. Consider the discounted option price process Ve(t) := exp
− Z t
0
r(s)ds
V(t) = exp −
Z t
0
r(s)ds
g r(t), t . The discount factor process γ(t) := exp(−Rt
0r(s)ds)is continuous and of finite variation, and so the quadratic co-variation[γ, V]vanishes. (See e. g. Jacod and Shiryaev (1987), Proposition I.4.49 d.) Hence
d(γ V)t=γ(t−)dV(t) +V(t−)dγ(t).
(4.20)
Ito' s formula provides the stochastic differential ofV: dV(t) =gt(rt−, t)dt+gr(rt−, t)drt+1
2grr(rt−, t)dhrc, rcit
+ g(rt−+ ∆rt, t)−g(rt−, t)−gr(rt−, t)∆rt .
The predictable quadratic variation ofrc(which appears as an integrator here) is given bydhrc, rcit = f(rt−, t)2c dt. The processγhas differentiable paths. Therefore it is of bounded variation.
dγ(t) =−rtγ(t)dt.
HenceV is the sum of a local martingale and the following predictable process of finite variation.
gt(rt−, t)dt+gr(rt−, t) f(rt−, t)b+k(rt−, t) dt+1
2grr(rt−, t)ãcãf(rt−, t)2dt (4.21)
+ Z
g(rt−+f(rt−, t)x, t)−g(rt−, t)−gr(rt−, t)f(rt−, t)x
F(dx)
dt.
By (4.20), this means that the processγV is the sum of a local martingale and the following predictable process of finite variation starting in0.
γ(t)
n−g(rt−, t)r(t) +gt(rt−, t) +gr(rt−, t) f(rt−, t)b+k(rt−, t)
+1
2grr(rt−, t)ãcãf(rt−, t)2 (4.22)
+ h Z
g(rt−+f(rt−, t)x, t)−g(rt−, t)−gr(rt−, t)f(rt−, t)x F(dx)
io dt.
This decomposition is the special semimartingale decomposition of γV in the sense of Jacod and Shiryaev (1987), Definition I.4.22. Since the decomposition of a special semimartingale into a local martingale and a predictable process of finite variation is unique, we conclude that the process (4.22) vanishes identically: Otherwise there would be two special semimartingale decompositions, because of courseγV =γV + 0, whereγV is a (local) martingale by assumption. Hence we have
0 =−g(rt−, t)r(t) +gt(rt−, t) +gr(rt−, t) f(rt−, t)b+k(rt−, t) +1
2grr(rt−, t)ãcãf(rt−, t)2 +
Z
g(rt−+f(rt−, t)x, t)−g(rt−, t)−gr(rt−, t)f(rt−, t)x F(dx).
Since the distribution of rt− has support I, we conclude that for every r ∈ I the following equation holds.
0 =−g(r, t)r+gt(r, t) +gr(r, t) f(r, t)b+k(r, t) +1
2grr(r, t)ãcãf(r, t)2 +
Z
g r+f(r, t)x, t
−g r, t
−gr r, t
f(r, t)x
F(dx).
This is the desired integro-differential equation for the option pricing functiong(r, t).
The Fourier Transform of the Option Pricing Equation
We briefly sketch a possible method for solving the integro-differential equation forg(x, t). Of course, further studies in this direction are necessary in order to come up with numerically tractable methods.
However, this would go far beyond the scope of this thesis. We assume thatg(x, t)is sufficiently regular for the Fourier inversion to make sense. Assume that we have coefficient functions (4.18).
We have the following identities for sufficiently regular functionsf: Z
f(x)xexp(iux)dx= 1 i
Z
f(x) ∂
∂uexp(iux)dx
=−i ∂
∂u Z
f(x) exp(iux)dx,
and Z
f0(x) exp(iux)dx=− Z
f(x) ∂
∂xexp(iux)dx
=−iu Z
f(x) exp(iux)dx,
The transform off0(x)xis therefore
f0d(x)x(u) =−ifd0(x)u(u)
=−(ufb(u))u =−fb(u)−ufbu(u).
Thus by Fourier transforming (4.19) with respect to the variablex, we get the following equation for the Fourier transformbg(u, t) =R
exp(iur)g(r, t)dr.
0 =i∂1gb+∂2bg−iu(σbb +aρ(t))gb+agb+au∂1bg−1
2cbσ2u2gb +bg
Z
e−iubσx−1 +iubσx F(dx).
The sum of the third and the last two terms on the right-hand side is readily identified asbglnφ(−bσu), whereφ(u)denotes the exponent in the Lévy-Khintchine representation of the characteristic function of L1. Hence
b
gt=−(au+i)bgu+ (iuaρ(t)−a)bg−bgãlnφ(−bσu) (4.23)
This is a partial differential equation involving only the first derivatives of the Fourier transform bg.
Furthermore, in contrast to the original integro-differential equation, integration with respect to the Lévy measureF(dx)is not required here. This could be an advantage, since the calculation of the density of the Lévy measure often is computationally demanding.
Chapter 5
Bond Price Models: Empirical Facts