Optimal Hedging for Fund & Insurance Managers with Partially Observable Investment Flows

0.33 MB·

by  Masaaki Fujii

Optimal Hedging for Fund & Insurance Managers
with Partially Observable Investment Flows ∗
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1 Masaaki Fujii†, Akihiko Takahashi‡
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2
First version: January 10, 2014

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u This version: July 25, 2014
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Abstract
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n
i All the financial practitioners are working in incomplete markets full of unhedge-
f
- able risk-factors. Makingthe situation worse,they areonly equipped with the imper-
q
fect information on the relevant processes. In addition to the market risk, fund and
[
insurance managers have to be prepared for sudden and possibly contagious changes

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in the investment flows from their clients so that they can avoid the over- as well as
v
under-hedging. Inthiswork,thepricesofsecurities,the occurrencesofinsuredevents
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1 and (possibly a network of) the investment flows are used to infer their drifts and
3 intensities by a stochastic filtering technique. We utilize the inferred information to
2 provide the optimal hedging strategy based on the mean-variance (or quadratic) risk
.
1 criterion. A BSDE approach allows a systematic derivation of the optimal strategy,
0 whichisshowntobe implementablebyasetofsimpleODEsandthe standardMonte
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Carlo simulation. The presented framework may also be useful for manufactures and
1
energy firms to install an efficient overlayof dynamic hedging by financial derivatives
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v to minimize the costs.
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X
r
a
Keywords : Mean-variance hedging, BSDE, Filtering, Queueing, Jackson’s network,
Poisson random measure
∗All the contents expressed in this research are solely those of the authors and do not represent any
views or opinions of any institutions. The authors are not responsible or liable in any manner for any
losses and/or damages caused by theuse of any contentsin thisresearch.
†Graduate School of Economics, TheUniversity of Tokyo. e-mail: [email protected]
‡Graduate School of Economics, TheUniversity of Tokyo. e-mail: [email protected]
1
1 Introduction
Inthispaper,wediscusstheoptimalhedgingstrategybasedonthemean-variancecriterion
forthefundandinsurancemanagers inthepresenceofincompleteness aswell asimperfect
information in the market. If an unhedgeable risk-factor exists, the fund and insurance
managers are forced to work in the physical measure and resort to a certain optimization
technique to decide their trading strategies. In the physical measure, however, they soon
encounter the problem of imperfect information which is usually hidden in the traditional
risk-neutral world.
One of the most important factors in the financial optimizations is the drift term in
the price process of a financial security. In fact, many of the financial decisions consist
of taking a careful balance between the expected return, i.e. drift, and the size of risk.
However, the observation of a drift term is always associated with a noise, and we need to
adoptsome statistical inferencemethod. Ina large numberof existing works on themean-
variance hedging problem, which usually adopt the duality method, Pham (2001) [26], for
example, studied the problem in this partially observable drift context. In spite of a great
amount of literature 1, results with explicit solutions which can be directly implementable
by practitioners have been quite rare thus far. When the explicit forms are available,
they usually require various simplifying assumptions on the dependence structure among
the underlying securities and their risk-premium processes, and also on the form of the
hedging target, which make the motivations somewhat obscure from a practical point of
view.
A new approach was proposed by Mania & Tevzadze (2003) [22], where the authors
studied a minimization problem for a convex cost function and showed that the optimal
value function follows a backward stochastic partial differential equation (BSPDE). They
were able to decompose it into three backward stochastic differential equations (BSDEs)
when the cost function has a quadratic form. Although the relevant equations are quite
complicated, their approach allows a systematic derivation for a generic setup in such a
way that it can be linked directly to the dynamic programming approach yielding HJB
equation. InFujii& Takahashi(2013) [9], wehave studiedtheir BSDEs to solve themean-
variance hedging problem with partially observable drifts. In the setup where Kalman-
Bucyfilteringschemeisapplicable, wehaveshownthatasetofsimpleordinarydifferential
equations (ODEs) and the standard Monte Carlo simulation are enough to implement
the optimal strategy. We have also derived its approximate analytical expression by an
asymptotic expansion method, with which we were able to simulate the distribution of the
hedging error.
The problem of imperfect information is not only about the drifts of securities. Fund
andinsurancemanagershavetodealwithstochasticinvestmentflowsfromtheirclients. In
particular,thetimingsofbuy/sellordersareunpredictableandtheirintensitiescanbeonly
statistically inferred. The same is true for loan portfolios and possibly their securitized
products. It is, in fact, a well-known story in the US market that the prepayments
of residential mortgages have a big impact on the residential mortgage-backed security
(RMBS) price, which in turn induces significant hedging demand on interest rate swaps
andswaptions. See[25],forexample,asarecentpracticalreviewontherealestatefinance.
1See Schweizer (2010) [31] as a brief survey.
2
Inthispaper,weextend[9]toincorporatethestochasticinvestmentflowswithpartially
observable intensities 2. In the first half of the paper, where we introduce two counting
processes to describe the in- and outflow of the investment units, we provide the mathe-
matical preparations necessary for the filtering procedures. Then, we explain the solution
technique for the relevant BSDEs in detail, which gives the optimal hedging strategy by
means of a set of simple ODEs and the standard Monte Carlo simulation. In the latter
half of the paper, we further extend the framework so that we can deal with a portfolio of
insurance products. We provide a method to differentiate the effects on the demand for
insurance after the insured events based on their loss severities. Furthermore, we explain
how to utilize Jackson’s network that is often adopted to describe a network of computers
in the Queueing analysis. We show that it is quite useful for the modeling of a general
network of investment flows, such as the one arising from a group of funds within which
investors can switch a fund to invest.
Althoughweareprimarilyinterested inprovidingaflexibleframeworkfortheportfolio
management, the presented framework may be applicable to manufacturers and energy
firms operating multiple lines of production. For example, they can use it to install an
efficientoverlayofdynamichedgingbyfinancialderivatives,suchascommodityandenergy
futures, in order to minimize the stochastic production as well as storage costs.
2 The financial market
Weconsiderthemarketsetupquitesimilartotheoneusedin[9]excepttheintroductionof
the stochastic investment/order flows with partially observable intensities. Let (Ω,F,P)
be a complete probability space with a filtration F = {F ,0 ≤ t ≤ T} where T is a
t
fixed time horizon. We put F = F for simplicity. We assume that F satisfies the usual
T
conditions and is big enough in a sense that it makes all the processes we introduce are
adapted to this filtration.
We consider the financialmarket with onerisk-freeasset, dtradable stocks or any kind
of securities, and m := (n−d) non-tradable indexes or otherwise state variables relevant
for stochastic volatilities, etc. For simplicity of presentation, we assume that the risk-free
interest rate r is zero. Using a vector notation, the dynamics of the securities’ prices
S = {S } and the non-tradable indexes Y = {Y } are assumed to be given
i 1≤i≤d j d+1≤j≤n
by the following diffusion processes:
dS = σ(t,S ,Y ) dW +θ dt
t t t t t
(cid:16) (cid:17)
dY = σ¯(t,S ,Y ) dW +θ dt +ρ(t,S ,Y ) dB +α dt . (2.1)
t t t t t t t t t
(cid:16) (cid:17) (cid:16) (cid:17)
Here, (W,B) are the standard (P,F)-Brownian motions independent of each other and
valued in Rd and Rm, respectively. The known functions σ(t,s,y), σ¯(t,s,y) and ρ(t,s,y)
are measurable and smooth mappings from [0,T]×Rd×Rm into Rd×d, Rm×d and Rm×m,
θ
t
respectively. The risk premium z := is assumed to follow a mean-reverting linear
t
α
t
(cid:18) (cid:19)
2Note that the standard setup with the perfect observation can be treated as a special case of our
framework.
3
Gaussian process:
dz = [µ −F z ]dt+δ dV (2.2)
t t t t t t
where µ, F and δ are continuous and deterministic functions of time taking values in Rn,
Rn×n and Rn×p. V is a p-dimensional standard (P,F)-Brownian motion independentfrom
W as well as B.
Let us now discuss the dynamics of the investment flows. We introduce the two count-
ingprocessesAandD,i.e. right-continuousintegervaluedincreasingprocesseswithjumps
of at most 1. (A ,D ) represent, respectively, the total inflow and outflow of investors or
t t
investment-units3 for an interested fund in the time interval (0,t] with A = D = 0.
0 0
For simplicity, we assume that they do not jump simultaneously. The total number of
investment-units for the fund at time t is denoted by Q , which is given by
t
Q = Q +A −D . (2.3)
t 0 t t
In this way, we model the change of the investment-units by a simple Queueing system
with a single server. Later, we shall make use of a special type of Queueing network to
allow investors to switch within a group of funds,which typically bundles Money-Reserve,
Bond, Equity, Bull-Bear, or regional equity indexes. See [2] as a standard textbook on
Queueing systems.
We assume that the counting processes have (P,F)-compensators, i.e.
t
Aˇ := A − λA(s,X )ds
t t s−
Z0
t
Dˇ := D − λD(s,X )1 ds (2.4)
t t s− {Qs−>0}
Z0
are (P,F)-martingales. Here, the intensity processes are modulated by a finite-state
Markov-chainprocessX whichtakesitsvalueinoneoftheN unit-vectors,E = {~e ,··· ,~e }.
1 N
The dynamics of X is assumed to be given by
t
X =X + R X ds+U . (2.5)
t 0 s s− t
Z0
Here {R ,0 ≤ t ≤ T} is a deterministic RN×N-valued continuous function with [R ]
t t i,j
denoting the rate of transition from state j to state i. U is a bounded RN-valued (P,F)-
martingale independent of W, B, V, A and D.
We assumethat the fundmanager can continuously observe S, {Y}obs ⊂ {Y } ,
j d+1≤j≤n
and the flows of investments, i.e. A and D. Q , which is the initial number of investment-
0
units, is known for the manager at t = 0. We introduce G = {G ,0 ≤ t ≤ T} that is the
t
P-augmented filtration generated by the observable processes (S,{Y}obs, A, D). Q (∈ R)
0
is assumed to be G -measurable. As one can see from the definition of (A,D), the timing
0
of an each investment flow is totally inaccessible for the fund manager. For the fixed-term
3For practical use, one may need the appropriate rescaling to make Q havetractable size.
4
contracts, the manager can know exactly the timing of expiries given the knowledge of
the initiation dates of the contracts. However, we think that it is rather unrealistic to
seek the optimal control based on the knowledge of a specific date of expiry of an each
investment-unit. In our setup, the manager partially knows (i.e. statistically infer) the
rate of the investment flow but cannot tell its timing at all.
{Y}obs are intended to be any index processes continuously observable in the market
butnontradableforthemanager,whichpossiblyincludefinancialindexesbutnon-tradable
for the manager by regulatory or some other reasons. {Y}obs can also represent various
characteristics of investors which affect the dynamics of the investment flows. They can
be very important non-financial factors for the modeling of residential mortgages and
life/health insurance, for example. Various aggregations of individual data at a portfolio
level can be used to construct (approximately) real-time composite indexes, which then
can be used as non-tradable indexes included in {Y}obs. If the process turns out to be
rather stable, then, it can be simply added as a deterministic function.
Remark 1 : It is straightforward to introduce a stochastic interest rate if we assume that
the short-rate process r is perfectly observable. In particular, if r follows a (quadratic)
Gaussian process, we lose no analytical tractability for BSDEs relevant for the mean-
variance hedging. The contracts of Futures written on interest rates, commodities, ener-
gies etc., which have the cycles of enlists and delists, can also be embedded into exactly
the same framework. Full details are available in the extended version of our previous
work [10]. (cid:4)
Assumption (A1)
(i) The stochastic differential equations (SDEs) given in (2.1) have the unique strong so-
lutions for S and Y.
(ii) Every Y (d+1 ≤ j ≤ n) is adapted to the observable filtration G.
j
(iii) The matrices σ and ρ are always invertible.
Let us make a comment on the assumption (ii). Through the observation of the quadratic
(co)variations of (S,{Y}obs), we can recover the values of σ σ⊤, σ¯obsσ⊤ and (σ¯ σ¯⊤ +
t t t t t t
ρ ρ⊤)obs. We can satisfy (ii) by assuming the maps (σ,σ¯,ρ) are constructed in such a way
t t
that they allow to fix the values of all the remaining Y ∈ {Y} \{Y}obs uniquely
k d+1≤j≤n
from these quantities at any time t ∈[0,T] 4.
As a result, we can see that G is in fact the augmented filtration generated by
(S,Y,A,D), and we express this fact by G = FS,Y,A,D. If necessary, we can extend the
model of (S,Y) in such a way that (σ,σ¯,ρ) can be generic G-predictable processes, and
hence can be dependent on the past history of (A,D), as long as Assumption (A1) is sat-
isfied. This may represent a possible feedback from the investment flows to the financial
market.
4In the case of d=m=1, it is automatically satisfied by many stochastic volatility models where σ2
dependson Y monotonically.
5
Assumption (A2)
(i) For every ~e ∈ E, {λA(s,~e),0 ≤ s ≤ T} and {λD(s,~e),0 ≤ s ≤ T} are strictly positive
G-predictable processes.
(ii) E T λA(s,X )ds +E T λD(s,X )ds <∞.
0 s− 0 s−
h i h i
R R
The assumption (ii) simply guarantees Aˇ and Dˇ are true (P,F)-martingales. Note that
the assumption (i) allows (λA, λD) to be dependent on (S ,Y ,A ,D ) and possibly on
t t t t t− t−
their past history. This flexibility is crucial for the practical use, where the first step to
describe the flow of investments is regressing it by various observable quantities. We are
going to model remaining unobservable effects by the hidden Markov-chain X. Note that
this setup can incorporate the self-exiting jump processes (Cohen & Elliott (2013) [3]),
which may be useful when there exist strong clusterings in the buy/sell orders from the
investors. See also [8] for various techniques and applications of hidden Markov models.
W
t
Let us put w := and introduce the following process:
t
B
t
(cid:18) (cid:19)
t t 1
ξ := 1− ξ z⊤ dw + ξ −1 dAˇ
t s− s− s s− λA(s,X ) s
Z0 Z0 (cid:18) s− (cid:19)
t 1
e + ξ e −e1 dDˇ (2.6)
s− λD(s,X ) s
Z0 (cid:18) s− (cid:19)
e
which yields
t 1 t
ξ = exp − z⊤dw − ||z ||2ds
t s s 2 s
(cid:18) Z0 Z0 (cid:19)
t t
e ×exp (λA(s,X )−1)ds+ (λD(s,X )−1)1 ds
s− s− {Qs−>0}
(cid:18)Z0 Z0 (cid:19)
1 ∆As 1 ∆Ds
× . (2.7)
λA(s,X ) λD(s,X )
s− s−
s∈Y(0,t]h i s∈Y(0,t]h i
We also define
t
ξ := 1− ξ z⊤dw
1,t 1,s s s
Z0
t 1 t
e = exp −e z⊤dw − ||z ||2ds (2.8)
s s 2 s
(cid:18) Z0 Z0 (cid:19)
t 1 t 1
ξ := 1+ ξ −1 dAˇ + ξ −1 dDˇ
2,t 2,s− λA(s,X ) s 2,s− λD(s,X ) s
Z0 (cid:18) s− (cid:19) Z0 (cid:18) s− (cid:19)
t t
e = exp e(λA(s,X )−1)ds+ (λD(s,Xe )−1)1 ds
s− s− {Qs−>0}
(cid:18)Z0 Z0 (cid:19)
1 ∆As 1 ∆Ds
× . (2.9)
λA(s,X ) λD(s,X )
s− s−
s∈Y(0,t]h i s∈Y(0,t]h i
6
We can show that {ξ ,0 ≤ t ≤ T} is a true (P,F)-martingale due to the linear Gaussian
1,t
nature of z and Lemma 3.9 in [1].
e
Assumption (A3)
(i) {ξ ,0 ≤ t ≤ T} is a true (P,F)-martingale.
t
(ii) {ξ ,0 ≤ t ≤ T} is a true (P,F)-martingale.
2,t
e
UndereAssumption (A3), we can definethe three probability measures P, P and P equiv-
1 2
alent to P on (Ω,F):
e e e
dP
= ξ , 0 ≤ t ≤ T (2.10)
dP Ft t
dPe(cid:12)(cid:12)
1(cid:12) =eξ , 0≤ t ≤ T (2.11)
dP Ft 1,t
dPe (cid:12)(cid:12)
2(cid:12) = ξe , 0≤ t ≤ T . (2.12)
dP Ft 2,t
e (cid:12)
(cid:12)
Then, by Girsanov-Maruyama theore(cid:12)m (seee, for example, [28]), one can show that
t
W := W + θ du (2.13)
t t u
Z0
t
f
B := B + α du (2.14)
t t u
Z0
e
are the standard (P,F) as well as (P ,F)-Brownian motions, and that
1
e At :e= At−t (2.15)
t
D := D − 1 ds (2.16)
et t {Qs−>0}
Z0
e
are (P,F) as well as (P ,F)-martingales. The following lemma tells us that the filtration
2
G can be generated by these simple martingales, too. This is crucial for the filtering
techneique we shall useebelow.
Lemma 1 The filtration G = FS,Y,A,D is the augmented filtration generated by
(W,B,A,D).
Pfroofe: Seincee σ and ρ are assumed to be always invertible, we can write
t
W = σ−1(u,S ,Y )dS (2.17)
t u u u
Z0
t
Bf = ρ−1(u,S ,Y ) dY −σ¯(u,S ,Y )σ−1(u,S ,Y )dS . (2.18)
t u u u u u u u u
Z0 (cid:16) (cid:17)
e
7
In addition,
A = A −t (2.19)
t t
t
D = D − 1 ds (2.20)
et t {Q0+As−−Ds−>0}
Z0
e
and Q ∈ G . Hence it is clear that FW,B,A,D ⊂ G. On the other hand, we have
0 0 f e e e
t
S = S + σ(u,S ,Y )dW
t 0 u u u
Z0
t t
f
Y = Y + σ¯(u,S ,Y )dW + ρ(u,S ,Y )dB
t 0 u u u u u u
Z0 Z0
At = At+t f e
t
D = D + 1 du (2.21)
t et {Q0+Au−−Du−>0}
Z0
e
and hence G ⊂ FW,B,A,D. (cid:3)
f e e e
3 Filtering equations
In order to obtain tractable filtering equations for the unobservable processes (θ,α,X),
we want to use the method of the “reference” measure where every increment of the
stochastic factors becomes independent from the past filtration. The following lemmas
are modifications of Proposition 3.15 in [1] to our setup.
Lemma 2 Let Ψ be an integrable F -measurable (t ∈ [0,T]) random variable. Then,
t t
EPe Ψt|GT = EPe Ψt|Gt . (3.1)
(cid:2) (cid:3) (cid:2) (cid:3)
Proof: Let us put
G = σ W −W ,B −B ,A −A ,D −D ;u ∈ [t,T] , (3.2)
t,T u t u t u t u t
(cid:16) (cid:17)
and then f f e e e e e e
G = G ∨G := σ(G ∪G ). (3.3)
T t t,T t t,T
If G is independent of F under the measure P, it is clear that (3.1) holds as explained
t,T t
in [1]. Unfortunately, this is not the case in our setup due to the information carried by
the jump intensity of D, which is 1 . Heowever, in measure P, (A,D,Q) consists
{Q−>0}
of a completely decoupled Queueing system with a single server, where the entrance of
new queue is given byethe Poisson process with unit intensity andethe service (or exit)
intensity is also 1 unless the queue is empty. Thus, all the information dependent on F
t
contained in G is restricted to the Queueing system {(A ,D ,Q ),t < s ≤ T}. Since it
t,T s s s
is irrelevant for Ψ , (3.1) holds true. (cid:3)
t
8
Let D (C) be the set of all E-valued c`adla`g (Rn-valued continuous) functions in the time
interval [0,T], respectively.
Lemma 3 Let Ψ be a map Ψ :[0,T]×Ω×D → R in such a way that {Ψ (x),0 ≤ t ≤ T}
t
is an integrable G-predictable process for any given step function x ∈ D. Then, using the
hidden Markov-chain X in (2.5), we have
EPe2 Ψt({Xs,0 ≤ s ≤ t}) GT = EPe2 Ψt({Xs,0 ≤ s ≤ t}) Gt . (3.4)
h (cid:12) i h (cid:12) i
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof: (A,D,Q) consists of a completely decoupled Queueing system with unit entrance
andserviceintensities alsoinmeasureP . Although(W,B)carriesnontrivialinformation
2
θ
through its drift z = , it does not affect the dynamics of X by the model setup. (cid:3)
α e f e
(cid:18) (cid:19)
Similarly, we also need the following lemma.
Lemma 4 Let Ψ be a map Ψ :[0,T]×Ω×C → R in such a way that {Ψ (x),0 ≤ t ≤ T}
t
is an integrable G-predictable process for any given continuous function x ∈ C. Then,
using the hidden process z in (2.2), we have
EPe1 Ψt({zs,0 ≤ s ≤ t}) GT = EPe1 Ψt({zs,0 ≤ s ≤ t}) Gt . (3.5)
h (cid:12) i h (cid:12) i
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof: In measure P , (W,B) becomes a n-dimensional standard Brownian motion and
1
hence the information generated by its increments is independent of F . On the other
t
hand, the observatioen offA aend D provides non-trivial information through their intensi-
ties, (λA(s,X ),λD(s,X )). However, by Assumption (A2) (i), any available informa-
s− s−
tion on diffusions can only appear in the form generated by (W,B) and X is irrelevant
for z. (cid:3)
f e
We would like to obtain the filtering equations for
θˆ := E θ |G , αˆ := E α |G (3.6)
t t t t t t
(cid:2) (cid:3) (cid:2) (cid:3)
and
Xˆ := E X |G . (3.7)
t t t
(cid:2) (cid:3)
Since X is valued in E = {~e ,··· ,~e }, we have
t 1 N
λˆA := E λA(t,X )|G = E λA(t,X )|G
t t− t t− t−
= λ(cid:2)A(t,~e)·Xˆt− (cid:3), (cid:2) (cid:3) (3.8)
(cid:0) (cid:1)
9
and similarly for λˆD. Here, we have used the inner product defined by
t
N
λA(t,~e)·Xˆ := λA(t,~e )Xˆi (3.9)
t− i t−
i=1
(cid:0) (cid:1) X
where Xˆi is the i-th element of Xˆ.
For notational simplicity, let us put
E[θ |G ]
zˆ := E[z |G ]= t t . (3.10)
t t t E[α |G ]
t t
(cid:18) (cid:19)
Using Kallianpur-Striebel formula, we have
EP1 ξ z |G
e 1,t t t
zˆ = (3.11)
t
EPe1(cid:2) ξ1,t|Gt (cid:3)
and (cid:2) (cid:3)
EP2 ξ X |G
Xˆ = e 2,t t t (3.12)
t
EPe(cid:2)2 ξ2,t|Gt (cid:3)
(cid:2) (cid:3)
where ξ := 1/ξ and ξ := 1/ξ . Note that {ξ ,0 ≤ t ≤ T} and {ξ ,0 ≤ t ≤ T}
1,t 1,t 2,t 2,t 1,t 2,t
are (P ,F) and (P ,F) martingales, respectively. This fact can be easily proved by Bayes
1 2
formula and Asseumption (A3). Theey define the inverse measure-change by:
e e
dP dP
= ξ , = ξ . (3.13)
1,t 2,t
dP1 Ft dP2 Ft
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
e e
Remark 2: Of course, (zˆt,Xˆt) can also be given by the Bayes formula with EPe[·|Gt]
and a (P,F)-martingale ξ := 1/ξ which defines
t t
dP
e e
= ξ , (3.14)
t
dP Ft
(cid:12)
(cid:12)
oranyotherequivalent probabilitymeasur(cid:12)eswiththecorrespondingRadon-Nikodymden-
e
sities. However, other choices do not lead to a tractable filtering equation since z and X
appear together in a single equation, or the properties proved in Lemma 3 and 4 do not
hold which then mixes the filter and the smoother of the unobservables. (cid:4)
Applying Itˆo formula, one can easily find
t
ξ = 1+ ξ z⊤dw
1,t 1,s s s
Z0
t 1 t
= exp z⊤dwe− ||z ||2ds (3.15)
s s 2 s
(cid:18)Z0 Z0 (cid:19)
e
10