On ratchet-cap pricing problems under the Levy LIBOR model Hsuan Ku - - PowerPoint PPT Presentation
On ratchet-cap pricing problems under the Levy LIBOR model Hsuan Ku Liu Department of Mathematics and Information Education National Taipei University of Education Outline Outline Simple forwards 1 Modelling jump 2 Arbitrage-free dynamics
Hsuan Ku Liu
Department of Mathematics and Information Education National Taipei University of Education
Outline
1
Simple forwards
2
Modelling jump
3
Arbitrage-free dynamics
4
The partial integro-differential equations for pricing ratchet caps
Simple forwards
1
Simple forwards
2
Modelling jump
3
Arbitrage-free dynamics
4
The partial integro-differential equations for pricing ratchet caps
Simple forwards
δ > 0fixed. L(t, T) : the forward rate for the interval from T to T + δ as of time t ≤ T P(t, T) : the time-t price of a ZCB maturity at time T, the forward rate satisfies L(t, T) = 1 δ ( P(t, T) P(t, T + δ) − 1) f(t, T) : the instantaneous forwards of HJM framework satisfies L(t, T) = 1 δ (exp{ T+δ
T
f(t, s)ds} − 1)
Modelling jump
1
Simple forwards
2
Modelling jump
3
Arbitrage-free dynamics
4
The partial integro-differential equations for pricing ratchet caps
Modelling jump
J(t) =
r
N(i)
t
Hi(X (i)
n , τ (i) n ) : the jump process
{(τ (i)
n , X (i) n )|n = 1, 2, . . .}, i = 1, 2, . . . , r : the marked point
process N(i)
t
is the counting process associated with the i-th marked process Hi, i = 1, 2, . . . , r : the jump-size function
Modelling jump
A marked point process {(τn, Xn)} can be described by a random measure µ on the product of the time axis and the mark space: the measure µ assigns unit mass to each point (τn, Xn). This makes it possible to write
N(i)
t
Hi(X (i)
n , τ (i) n ) =
t ∞ δi(x, s)µ(dx, ds). For a marked point process in which the points follow a Poission process and marks are i.i.d. random variable.
Modelling jump
The intensity has the property that, for all suitably integrable δi t ∞ δi(x, s)µi(dx, s)ds − t ∞ δi(x, s)λ(i)(dx, s)ds is a martingale in t
Modelling jump
The intensity takes the form λ(i)(dx, t) = λifi(x)dx For the logarithm of the jump sizes to have an asymmetric double distribution, we take the density of the jump size to be fi(x) = 1 x piη1,ie−η1,i log(x)1x≥1 + 1 x piη2,ieη2,i log(x)10<x<1
Arbitrage-free dynamics
1
Simple forwards
2
Modelling jump
3
Arbitrage-free dynamics
4
The partial integro-differential equations for pricing ratchet caps
Arbitrage-free dynamics
Define a model of term structure of simple forwards L(t, T) to be arbitrage free if it can be embedded in an arbitrage-free model of instantaneous forwards f(t, T)
Arbitrage-free dynamics
Suppose that a bond price is specified through forward rate, ie. for T ∈ R+ P(t, T) = exp
T
t
f(t, s)ds
t ≤ T where f(t, T) is the forward rates of which the dynamics is given by df(t, T) = α(t, T)dt + σT(t, T) · dWt +
r
δi(t, x, T)µi(dt, dx) Here, Wt is a standard wiener process in Rn, µi is a random jump measure with the compensator λi(dt, dx) = Fi(dx)dt, i = 1, 2, ..., r
Arbitrage-free dynamics
For all finite t and t ≤ T T T
t |α(u, s)| dsdu < ∞,
T T
t |σ(u, s)|2 dsdu < ∞
and T
T
t
|δi(u, x, s)|2 ds νi(du, dx) < ∞, i = 1, 2, ..., r
Arbitrage-free dynamics
It is convenient to extend the definitions of the coefficients by putting them equal to zero for t > T Put A(t, T) = − T
t
α(t, s)ds σ∗
i (t, T) = −
T
t
σi(t, s)ds, i = 1, 2, ..., m σ∗(t, T) = (σ∗
1(t, T), ..., σ∗ m(t, T))
Di(t, T, α) = − T
t
δi(t, x, s)ds, i = 1, 2, ..., r
Arbitrage-free dynamics
Assume that
bond (ZCB) price process P(t, T) on [0, T] has the form
dP(t−,T) P(t,T)
=
2 σ∗(t, T)2
dt + σ∗(t, T)dwt +
r
Arbitrage-free dynamics
Let Yt = ln P(t, T) = − T
0 f(t, s)ds
= ln P(0, T) + t
0 A(u, T)du +
t
0 σ∗(u, T)dWu
+ r
i=1
t
t
0 rudu.
Then, applying Ito Lemma to P(t, T) = P(0, T)eYt, we get the theorem.
Arbitrage-free dynamics
Let Γ be an m-dimensional predictable process and Ψ = Ψ(w, t, x) be a strictly positive measurable function such that for finite t t ||Γs||2ds < ∞, t
|Ψ(s, x)|λ(s, dx)ds < ∞. Define dLt = LtΓtdWt+Lt−
(Ψ(t, x)−1){µ(dt, dx)−ν(dt, dx)}, L0 = 1, and suppose that for all finite t EP[Lt] = 1.
Arbitrage-free dynamics
Then there exists a probability measure Q on F locally equivalent to P with dQt = LtdPt such that we have dWt = Γtdt + dW Q
t ,
where W Q lsi a Q-Wiener process. The point process µ has a Q-intensity, given by λQ(t, dx) = Ψ(t, x)λ(t, dx).
Arbitrage-free dynamics
Under the martingale measure Q, the dynamics of bond prices are follows. dP(t−, T) = P(t, T)
t
+
r
i )(dt, dx)
Arbitrage-free dynamics
A measure Q is a martingale measure if and only if the bond dynamics under Q is of the form dP(t, T) = rtP(t, T)dt + dMQ
P ,
where MQ
P is a Q-local martingale. Thus, using Girsonov
theorem to dP(t, T) and comparing with the above equation, we get the theorem.
Arbitrage-free dynamics
The forward rate dynamics, under the martingale measure Q, are follows: df(t, T) = −σ(t, T)σ∗(t, T)dt + σ(t, T)T · dwQ
t
+
r
i (dt, dx))
Arbitrage-free dynamics
1 The maturity T is restricted to a finite set of dates
0 = T0 < T1 < · · · < TM < TM+1.
2 Let Ln(t) = L(t, Tn) so that Ln is the forward rate for the
accrual period [Tn, Tn+1].
3 Let Pn(t) = P(t, Tn) denote the price of a ZCB maturing at
Tn.
Arbitrage-free dynamics
For each n = 1, 2, . . . , M let γn(.) be a bounded, adapted, Rd-valued process. Let Hni, i = 1, 2, . . . , r be deterministic functions from [0, ∞) to [−1, ∞). The model dLn(t) Ln(t) = αn(t)dt + γn(t)dw(t) + dJn(t) is arbitrage free if
Arbitrage-free dynamics
αn(t) = γn(t)ψ0(t) + γn(t)
n
δγk(t)T Lk(t) 1+δLk(t)
− ∞
r
Hni(x)
n
1+δLk(t−) (1+δLk(t−))(1+Hki(x))ψi(x, t)λ(i)(dx, t)
for some Rd-valued process ψ0 and strictly positive scalar process ψi(x, .) x ∈ (0, ∞), i = 1, 2, . . . , r satisfying t
0 ||ψ0(s)||2ds < ∞ and
t
0 ψi(s)λ(i)(dx, s)ds < ∞,
i = 1, 2, . . . , r
Arbitrage-free dynamics
For two maturities T and U with T < U, the forward process is defined by FB(t, T, U) = P(t,T)
P(t,U). Let δ > 0, the δ-LIBOR rates
are defined by L(t, Tn) = 1 δ (FB(t, Tn, Tn+1) − 1); that is dFB(t, Tn, Tn+1) = δdL(t, Tn). Applying Ito Lemma to the above equation, we get the theorem.
Arbitrage-free dynamics
Under the forward measure PM+1, if for each n = 1, 2, . . . , M we have dLn(t) Ln(t) = αn(t)dt + γn(t)dwn+1(t) + dJn(t) is arbitrage-free if αn(t) =
M
δγn(t)γk(t)T Lk(t) 1+δLk(t)
− ∞
r
Hni(x)
M
1+δLk(t−)(1+Hki(x)) 1+δLk(t−)
λ(i)
M+1(dx, t)
Arbitrage-free dynamics
Define Z(t, T) =
B(t,T) exp( t
0 r(s)ds), the the dynamics
dZ(t,T+δ) Z(t,T+δ)
= σ∗(t, T + δ)dW Q + r
i=1
∞
0 (eD(t,T+δ) − 1)(µi(dx, dt) − λQ i (dx, t)dt).
is martingale. Define the measure PT+δ through the likelihood ratio dPT+δ dQ
P(0, T + δ). Applying Girsanov’s theorem, the intensity of µ(i)(dx, dt) is given by λ(i)
T+δ(dx, t) = eD(t,T+δ)λ(i) Q (dx, t) and the process
WT+δ = dWQ(t) + σ∗(t, T + δ) is a standard Brownian motion.
Arbitrage-free dynamics
In this paper, we assume that each LIBOR is affected by the same rare events, that is the dynamics of LIBOR rate is written as dLn(t) Ln(t) = αn(t)dt + γn(t)dwn+1(t) +
Nt
Hn(Xk, τk)
The partial integro-differential equations for pricing ratchet caps
1
Simple forwards
2
Modelling jump
3
Arbitrage-free dynamics
4
The partial integro-differential equations for pricing ratchet caps
The partial integro-differential equations for pricing ratchet caps
A ratchet cap is a contract that can be decomposed into simpler contracts, called ratchet caplets.
The partial integro-differential equations for pricing ratchet caps
The ratchet caplet payoff, paid at time Ti, is given by (Li
Ti−1 − Ki)+, where the strike Ki is recursively defined as
follows.
Kj+1 = (aLj
Tj−1 − bKj + c)
given 1 < j < i with a, b, c >0. Note that (Li
t)t≤Ti−1 is the value of the i-th
forward rate accounts for the period [Ti−1, Ti]
The partial integro-differential equations for pricing ratchet caps
The discounted price of the i-th ratchet caplet is given by Πi
t = EQi[(Li Ti − Ki)+|Ft],
t ≤ Ti−1 and the absolutely price Ci(t, L) is equal to Πi
tBi t
Let L_i be the real variable corresponding to the i-th forward LIBOR rate, for i = 1, 2, . . . , N Πi(t, L) = Ci(t,L)
Pi+1(t)
= EQi
(Li
Ti −K)+
Pi+1(Ti+1)|Ft
(Li
Ti − K)+|Ft
The partial integro-differential equations for pricing ratchet caps
Let i ∈ {1, 2, . . . , N} be fixed index and assume that for j = i − 1, i. We have Πi
t = ui,j(t, Lj t, Li t),
t ∈ [0, Ti−1] and the function ui,j = ui(t, Lj
t, Li t),
t ∈ [0, Ti−1], Li−1
t
, Li
t > 0
Is uniquely defined by the following backward recursion starting from Ti
The partial integro-differential equations for pricing ratchet caps
(a) ui,i is the uniquely non-negative solution of the PIDE Li,iui,i = 0, ui,i(Ti−1, Li; K) = (Li − K)+ in (Ti−2, Ti−1) × R+ in R+ where Liis the two-dimensional operator Li,iui,i = (σi(t)Li)2
2
∂LiLiui,i + ∂tui,i + ∞
−∞ (ui(t, Liehix) − ui,i(t, Li))F(dx)
The partial integro-differential equations for pricing ratchet caps
(b) ui,i−1(t, Li−1, Li) for t ∈ (0, Ti−2) is the uniquely non-negative solution of the PIDE Li,i−1ui,i−1 = 0, ui,i−1(Ti−2, Li−1, Li) = ui,i(Ti−2, Li) in (Ti−2, Ti−1) × R+ × R+ in R+ where Liis the two-dimensional operator Li,i−1ui,i−1 =
2
(σi−l(t)σi−m(t)Li−lLi−m)2 2
∂Li−lLi−mui,,i−1 +
2
αi−lLi−l∂i−lui,,i−1 + ∂tui,,i−1 + ∞
−∞ (ui,,i−1(t, Li−1ehi−1x, Liehix) − ui,,i−1(t, , Li−1, Li))F(dx)
The partial integro-differential equations for pricing ratchet caps
Applying Ito formula to Πi(Li, Li−1, t) and using the dynamics of Li and Li−1 under the forward measure Qi+1, we obtain dΠi = ∂Πi ∂ t dt + ∂Πi ∂ Li dLi + ∂Πi ∂Li−1 dLi−1 +1 2 ∂2Πi ∂ L2
i
(dLi)2 + 1 2 ∂2Πi ∂L2
i−1
(dLi−1)2 + ∂2Πi ∂Li∂Li−1 (dLi)(dLi−1) +Πi(Lie∆Xi(t), Li−1e∆Xi−1(t), t) − Πi(Li, Li−1, t). Since (Li − aLi−1 − c)+ is a martingale under the forward Qi+1 measure, the drift term in the dynamics of Πi is equal to zero; that is the expected value Πi satisfies the PIDE.
The partial integro-differential equations for pricing ratchet caps
The partial integro-differential equations for pricing ratchet caps