Asset Pricing:
Utility Theory and Pricing Kernel Models
Andrea Macrina
Department of Mathematics
King’s College London
African Institute for Mathematical Sciences
16-18 February 2012
Cape Town
Asset Pricing: Utility Theory and Pricing Kernel Models
-2-
AIMS, Cape Town
Contents
1. Dynamic asset pricing; pricing kernels; stochastic interest rate and market
price of risk; numeraire and change-of-measure martingales; asset pricing
formulae.
2. Utility functions; marginal utility; risk aversion and psycological discount
factor; utility maximization; optimal consumption.
3. Marginal utility of consumption and the pricing kernel; equilibrium asset
pricing; Sidrauski utility, money supply and liquidity benefit; inflation.
4. Pricing kernels driven by time–inhomogeneous Markov processes; explicit
“A–Z example” of asset pricing including stochastic interest rates and market
risk premium, stochastic volatility, and option pricing for various asset classes.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
-3-
AIMS, Cape Town
Dynamic asset pricing
The financial market is modelled by a probability space (Ω, F, P), and the
market information is modelled by the filtration {Ft}0≤t.
We may assume that the market filtration is generated by a multi–dimensional
Brownian motion {Wt}0≤t.
Asset prices. Each asset is characterised by (a) its price process and (b) the
dividend or cash–flow stream that it generates. In the case of an asset
generating a continuous cash flow, the asset is defined by a pair of
{Ft}–adapted processes {St} and {Dt} which we call the price process and the
dividend flow–rate process, respectively.
Martingale. A process Mt is a martingale if it is {Ft}–adapted and satisfies
E [|Mt|] < ∞ and E [Mt | Fs] = Ms for any 0 ≤ s ≤ t.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
-4-
Pricing kernel. A process {πt} is called a pricing kernel for the asset pair
{St, Dt} if the process {Mt} defined by
Mt = πt St +
Z
t
πu Du du
(1)
0
is a martingale.
The existence of a (universal) pricing kernel ensures the absence of arbitrage in
the market.
The pricing kernel establishes the inter–temporal relationship between asset cash
flows.
Money market account. We assume the existence of a money market
account {Bt}0≤t that is a non–dividend–paying asset with a price process of the
form
Z t
rs ds ,
Bt = B0 exp
(2)
0
where {rt}0≤t is the {Ft}–adapted short rate of interest.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
-5-
Market price of risk. The pricing kernel {πt} satisfies the dynamical equation
dπt
= −rt dt − λt dWt,
πt
(3)
where {λt}0≤t is the (vector–valued) market price of risk process.
Proof: Let {ρt}0≤t be defined by ρt = πtBt. Then:
ρt
= ρt d(Bt−1) + Bt−1dρt.
Bt
(4)
d(Bt−1) = −Bt−2dBt = −rtBt−1dt.
(5)
dπt = d
Also,
By the martingale representation theorem, we may write
dρt = −ρt λt dWt,
(6)
and therefore
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
-6-
ρt
ρt
λt dWt,
dπt = − rt dt −
Bt
Bt
= −πt rt dt − πt λt dWt.
(7)
Asset pricing formula. Recall that,
Mt = πt St +
Z
t
πuDu du,
(8)
0
where {Mt}0≤t is an ({Ft}, P)–martingale.
It follows that:
16-18 February 2012
Z T
1 P
St = E πt St +
πuDu du Ft .
πt
t
(9)
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
-7-
Market numeraire. The process {ξt}0≤t defined by
ξt = πt−1,
(10)
is called the market numeraire (or natural numeraire).
By an application of Ito’s formula, we obtain
dξt = rt +
λ2t
ξt dt + λt ξt dWt.
(11)
Asset price dynamics. Let {St} denote the asset price process, and let {Dt}
denote the dividend rate process of the asset. The asset price process {St}
satisfies the following SDE:
dSt
= (rt − δt + σtλt) dt + σt dWt,
St
(12)
where δt := Dt/St, and where {σt}0≤t is some volatility process.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
-8-
Proof: We recall that
Mt = πt St +
Then, we have that
Z
Mt
1
St =
−
πt
πt
t
πuDu du.
(13)
0
Z
= ξ t Mt − ξ t
t
πsDs ds,
0
Z
t
0
Ds
ds.
ξs
(14)
An application of Ito’s formula gives
St
dSt = dξt + ξt dMt + dξt dMt − Dt dt.
ξt
Since {Mt} is a martingale, we may write
dMt = θt dWt,
(16)
where {θt}0≤t is a vector–valued {Ft}–adapted volatility process.
16-18 February 2012
(15)
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
-9-
Recalling the dynamics of the natural numeraire {ξt}0≤t, we obtain
dSt = (rtSt − Dt + ψtλt) dt + ψt dWt,
(17)
where ψt := Stλt + ξtθt.
It follows that
dSt
= (rt − δt + σtλt) dt + σt dWt,
St
(18)
where δt = Dt/St and σt = ψt/St.
Example: GBM. Suppose the asset price process {St} is given by
St = S0 exp (r − δ + σλ)t −
1 2
2σ t
+ σWt ,
(19)
(20)
and the pricing kernel {πt} is given by
πt = exp −rt −
16-18 February 2012
1 2
2λ t
− λWt .
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 10 -
Then the pricing formula
St =
1 P
E πT ST +
πt
Z
is satisfied. In other words, the process
Mt = πt St +
Z
T
t
πuδ du Ft
(21)
t
πuδSu du
(22)
0
is an ({Ft}, P)–martingale.
Pricing of a discount bond. Let {PtT }0≤t≤T denote the price process of a
T –maturity zero–coupon bond (non–dividend–paying asset) with payoff function
PT T = 1. Then,
PtT =
16-18 February 2012
1 P
E [πT | Ft] .
πt
(23)
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 11 -
The dynamical equation of the discount bond price process is given by
dPtT
= (rt + λtΩtT )dt + ΩtT dWt,
PtT
(24)
where {ΩtT }0≤t≤T is the bond volatility process.
Risk-Neutral Valuation
The price St at time t ≥ 0 of a limited–liability non–dividend–paying asset is
1 P
St = E [πT ST | Ft] ,
πt
(25)
where ST can be viewed as the random cash flow that will occur at (the future)
fixed time T ≥ t.
Recalling that πt := ρt/Bt, we have that
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 12 -
Bt P
E
St =
ρt
ρT
ST Ft .
BT
(26)
Assuming that {ρt}0≤t≤T is a change–of–measure density martingale from
P → Q, we obtain
St = B t E Q
ST Ft .
BT
(27)
This is the so–called risk–neutral valuation formula, and Q is the risk–neutral
measure.
Bond price process under Q. Under the risk–neutral measure Q, the bond
price process {PtT } takes the form
PtT
1 = Bt EQ
Ft ,
BT
Z T
ru du Ft .
= EQ exp −
(28)
t
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 13 -
AIMS, Cape Town
Often this is the starting point in interest–rate modelling.
However, what about the model for the market price of risk {λt}, which is
necessary for asset pricing under the real probability measure P?
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 14 -
AIMS, Cape Town
Utility functions
Utility is an abstract concept for the notion of happiness, satisfaction, or simply
for the pleasure derived from consumption of a specified amount of goods and
services.
Some “goods and services” can provide immediate utility, e.g. an ice cream, or a
shower; other “goods”, like education, provide continuous utility.
Money provides immediate utility since the notion of the “value of money” is
universally recognised.
However, money provides also a continuous flow of utility in the form of
“status”, security, flexibility, etc.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 15 -
AIMS, Cape Town
Utility function
A function U : R+ → R that satisfies
(i) U (x) ∈ C 2
(ii) U ′(x) > 0
(iii) U ′′(x) < 0
(iv) limx→0 U ′(x) = ∞, limx→∞ U ′(x) = 0
is called a utility function.
The derivative U ′(x) is called the marginal utility.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 16 -
Example: Logarithmic utility. U (x) = a ln(x), a > 0.
Example: Power utility. U (x) =
a
p
xp, a > 0, p ∈ (−∞, 1)\{0}.
The parameter p in the power utility function can be interpreted as a measure
for how risk–averse an investor is. The parameter p controls the steepness of the
utility function.
The concavity condition on the utility function, U ′′(x) < 0, implies risk aversion.
Consider a gambling game in which a player receives a random amount X in
exchange for a wager equal to E[X]. Jensen’s inequality implies that
E [U (X)] ≤ U (E[X]) .
(29)
This says that a risk–averse player requires an incentive to play a risky game,
and requests to receive a positive “risk premium” as an additional reward for
accepting the risk involved in the game.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 17 -
Inverse marginal utility
A function I : R+ → R characterised by
I [U ′(x)] = x,
U ′ [I(y)] + y
(30)
is called the inverse function of the marginal utility U ′(x).
The inverse function I(y) always exists for a standard utility function since
U ′(x) > 0 and U ′′(x) < 0.
The inverse marginal utility function is monotonically decreasing.
Examples: U (x) = ln(x) ⇒ I(y) = y1 , and
U (x) = p1 xp ⇒ I(y) = y q−1, where p1 + 1q = 1.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 18 -
Convex dual
A standard utility function has so–called convex dual given by
Ũ (y) = max {U (z) − zy} ,
(31)
z>0
for y > 0.
The convex dual of a standard utility function is guaranteed to exist, and is
given by
Ũ (y) = U [I(y)] − yI(y).
(32)
This is shown by observing that for fixed y, the maximum of U (z) − zy over z is
obtained when
d
[U (z) − zy] = U ′(z) − y = 0.
(33)
dz
This is a genuine maximum because U ′′(z) < 0.
Therefore we have y = U ′(z) and z = I(y) at the maximum.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 19 -
AIMS, Cape Town
Maximised expected utility
Let Ht be a random cash flow at the fixed time T > 0. The expected utility
E [U (HT )] is maximised subject to the budget constraint H0 = E[πT HT ] if
HT = I(βπT ), and β is chosen such that the budget constraint is satisfied.
Proof:
We consider an alternative investment target ZT that also satisfies the budget
constraint. We next show that the expected utility derived from ZT cannot
exceed that of HT
E [U (HT )] − E [U (ZT )] =
=
=
=
E [U (HT ) − βπT HT ] − E [U (ZT ) − βπT ZT ]
E [U (I(βπT )) − βπT I(βπT )] − E [U (ZT ) − βπT ZT ]
E Ũ (βπT ) − E [U (ZT ) − βπT ZT ]
E Ũ (βπT ) − (U (ZT ) − βπT ZT )
(34)
However, by virtue of the definition of Ũ (x), we have that
Ũ (βπT ) > (U (ZT ) − βπT ZT ) ,
and therefore E [U (HT )] − E [U (ZT )] ≥ 0.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 20 -
Example: Optimal investment with log–utility. Consider U (x) = a ln(x),
so U ′(x) = a/x. The first order condition is then is given by
a
a
′
= βπT ⇒ HT =
.
(35)
U (HT ) =
HT
βπT
From the budget constraint, we obtain
H0 = E [πT HT ] =
and therefore
a
,
β
(36)
H0
= H0 ξ T .
HT =
πT
(37)
Optimal consumption
The expected utility of consumption over the period [0, T ] given by
E
Z
T
U (ct)dt
0
(38)
is maximised by the random consumption plan ct = I(β πt), where β is chosen
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 21 -
such that the budget constraint
E
Z
T
πtctdt
0
(39)
is satisfied.
Example: log–utility of consumption. Let U (ct) = ln(ct). Then the
optimal consumption strategy is given by
H0
ct =
ξt
T
where {ξt} is the price process of the natural numeraire asset.
(40)
Optimal consumption with impatience
Let the utility of consumption be discounted by the exponential impatience
factor exp(−γt). Then subject to the budget constraint
Z T
πtctdt ,
(41)
H0 = E
0
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 22 -
the expected total discounted utility
Z T
eγtU (ct)dt
E
AIMS, Cape Town
(42)
0
is maximised by
eγtU ′(ct) = βπt.
(43)
Example: log–utility with impatience. We consider U (x) = ln(x). It
follows that U ′(ct) = 1/ct, and that
eγ t
.
(44)
ct =
β πt
The budget constraint simplifies to
1
−γ T
H0 =
.
(45)
1−e
βγ
Therefore, we have:
H0 γ
e−γ t
−γt
=
e
πt ,
(46)
ct =
−γT
βπt
1−e
or
1 − e−γT
ct .
(47)
πt =
H0γ e−γt
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 23 -
Monetary Models for Interest Rates and Inflation
1. Modelling framework for inflation
2. Real and nominal pricing kernels
3. Index–linked bonds
4. Real interest rates
5. Dynamics of the consumer price index
6. Macroeconomic models for inflation
We now have a look at past inflation levels in the UK, the USA, and Japan.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 24 -
Interest rate models and asset pricing
To begin, let’s say what we mean by an interest rate model.
An interest rate model consists of the following ingredients:
(i) A probability space (Ω, F, P) with a market filtration {Ft}0≤t<∞ to which
asset price processes are adapted.
(ii) A process {Bt}0≤t<∞ which we call the “nominal money market account”,
Z t
Bt = exp
rsds ,
(48)
0
where {rt} is called the “short rate”.
(iii) A system of nominal discount bond price processes {PtT }0≤t≤T <∞.
(iv) A pricing kernel {πt}0≤t<∞ with the property that {πtBt} and {πtPtT } are
martingales.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 25 -
More generally, for any asset with price process {St}0≤t≤∞, and continuous
dividend process {Dt} we require that the process {Mt} defined by
Z t
πsDsds.
(49)
Mt = πt St +
0
is a martingale.
We assume that the market filtration is generated by a multi–dimensional
Brownian motion.
With these ingredients in place, we say that we have a “model for the nominal
interest rate system”.
The pricing of contingent claim with payoff HT is then given by
Et[πT HT ]
Ht =
.
πt
16-18 February 2012
(50)
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 26 -
AIMS, Cape Town
Models for inflation
The modelling of inflation has a number of subtle features.
We need to model nominal interest rates, real interest rates, and the consumer
price index in a consistent way. The basic setup is as follows.
For the nominal discount bond system we have
Et[πT ]
PtT =
.
(51)
πt
The associated dynamics are given by:
dPtT
= (rt + λtΩtT ) dt + ΩtT dWt.
(52)
PtT
Here rt (nominal short rate) and λt (nominal market price of risk) are defined by
dπt
= −rtdt − λtdWt,
(53)
πt
and ΩtT is the nominal discount–bond vector volatility. The dynamics of the
discount bond system gives rise to the interpretation of rt and λt.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 27 -
AIMS, Cape Town
Next we consider the consumer price index {Ct}.
The rate of inflation over a given period is defined to be the percentage increase
in the price index over the given period, expressed on an annualised basis.
The question is: how do we model {Ct}?
One is tempted to write down the usual arbitrage-free dynamics for the price
process of a risky asset.
But since {Ct} is a price index (including the prices of non-durable items such
as a haircut, a pint of beer, a meal in a restaurant, or a night at the cinema) one
cannot immediately conclude that the usual arbitrage–free dynamics apply.
Instead, we consider a related family of prices: the prices of index–linked bonds.
These are financial assets, and hence are subject to the laws of no arbitrage.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 28 -
Index-linked bonds
An index-linked discount bond pays out the amount CT at time T .
I
The value PtT
of an index-linked bond at time t is therefore
I
PtT
=
Et[πT CT ]
.
πt
(54)
I
If we divide PtT
by Ct we get the value (in real terms) of a bond that pays one
unit of goods and services at time T :
Et[πT CT ]
R
.
(55)
PtT
=
πt Ct
Now suppose we define the real pricing kernel πtR by writing
πtR = πtCt.
(56)
Then clearly for the price index we have
πtR
Ct =
.
πt
16-18 February 2012
(57)
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 29 -
In terms of πtR , the real discount bond prices take the form
Et[πTR ]
.
=
R
πt
The dynamics of the real discount bond system are given by:
R
dPtT
R
R R
R
dt
+
Ω
Ω
+
λ
=
r
tT dWt .
tT
t
t
R
PtT
R
PtT
(58)
(59)
Here rtR and λR
t are defined by
dπtR
R
R
dt
−
λ
=
−r
t dWt ,
t
πtR
(60)
and ΩR
tT is the real discount bond volatility.
The dynamics of the real discount bond show that rtR and λR
t have the
interpretation of being the real short rate, and the real market price of risk,
respectively.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 30 -
The CPI process
Now we can deduce the dynamics of the consumer price index. In particular, by
an application of Ito calculus to the relation Ct = πtR /πt we obtain:
dCt R
R
R
(61)
= rt − rt + λt λt − λt dt + λt − λt dWt.
Ct
Thus we can write
dCt R
(62)
= rt − rt + λtνt dt + νtdWt.
Ct
Here the CPI volatility vector is defined by: νt = λt − λR
t .
We see that {Ct} has the arbitrage-free dynamics of a risky asset that pays a
continuous dividend at the rate rtR .
Absence of statistical arbitrage opportunities implies νt2 > λtνt > 0.
With these relations at hand, we are now in a position to build models for
pricing inflation-linked derivatives.
In particular, it is apparent that the two required ingredients are πt and πtR .
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 31 -
Monetary systems
Inflation is a monetary phenomenon. To model a monetary economy we need
three more ingredients:
1. The rate of real consumption {kt}.
2. The money supply {Mt}.
3. The convenience yield or benefit rate {βt} associated with the money supply.
The process {βt} represents the intangible rate of benefit delivered, in dollars
per unit of time and per unit of money supply.
If the money supply at time t is Mt it confers a benefit at the rate of βtMt
dollars per year.
Thus one thinks of the benefit as a kind of “consumable” commodity.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 32 -
AIMS, Cape Town
What really “counts” is the real rate of benefit derivable from the presence of
the money supply.
This is given by lt = βtMt/Ct.
The consumer price index is thus used as a kind of “exchange rate” to convert
the nominal benefit into a real benefit.
Optimisation
Our goal now is to determine a set of relations between {kt}, {Mt}, {βt}, and
the consumer price index {Ct}.
We assume the existence of a standard bivariate utility function U (x, y) such
that the expected total utility
Z T
e−γtU (kt, lt)dt
(63)
E
0
is maximised over some time horizon.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 33 -
This is subject to the budget constraint
Z T
Z
πtCtktdt +
W0 = E
0
T
πtβtMtdt .
0
(64)
Here W0 denotes the total wealth of the economy (including, e.g., the present
value of any forthcoming income) available for the given period. A standard
argument shows that the maximum is achieved if
Ux(kt, lt) = µeγtπtCt,
(65)
and
Uy (kt, lt) = µeγtπtCt
for some constant µ which is fixed by the budget constraint.
(66)
These relations can then be solved to give πt, πtR , and Ct as functions of kt, Mt,
and βt.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 34 -
A simple example
A simple choice for U (x, y) is given by
U (x, y) = A ln(x) + B ln(y),
(67)
where A, and B are constants.
It follows that
A
Ux(kt, lt) = ,
kt
Equating these we get
B
Uy (kt, lt) = .
lt
A βtMt
A
.
kt = l t =
B
B Ct
Hence, rearranging, we have
Ct =
A βtMt
.
B kt
(68)
(69)
(70)
Thus the consumer price index is determined completely by the nominal money
supply benefit and the rate of real consumption.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 35 -
But we can also determine the pricing kernels. A calculation shows that
Be−γt
πt =
,
µβtMt
πtR
Ae−γt
.
=
µkt
(71)
Thus both the real and the nominal interest rate systems are determined by the
model.
Now suppose that HT is the payoff of a contingent claim.
Then in our monetary policy model it follows that
H0 = β0 M0 e−γT E
HT
.
βT MT
(72)
This example shows how expectations concerning monetary policy can have a
major effect on the valuation of long-dated derivatives.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 36 -
AIMS, Cape Town
Information-sensitive pricing kernels with
Time-Inhomogeneous Markov Processes
Stochastic discount bond systems can be constructed by modelling the pricing
kernel (stochastic discount factor) that we denote {πt}0≤t.
The price PtT at time t of a discount bond with unit payoff at maturity T is
EP [πT | Ft]
PtT =
,
(73)
πt
where P is the real probability measure.
Next we model the pricing kernel {πt} and the filtration {Ft} following the
scheme of information-based asset pricing developed in
D.C. Brody, L.P. Hughston, A. Macrina (2008) Information-Based Asset Pricing.
International Journal of Theoretical and Applied Finance Vol. 11, 107-142.
We fix a time U > T and introduce a random variable XU with real probability
density p(x).
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 37 -
The random variable XU may represent a macroeconomic factor (e.g. the GDP
level of a country at time U ) revealed at time U .
Suppose that at time t < U , market investors have access only to incomplete
information about the macroeconomic factor XU .
We model this incomplete information by an information process {LtU } with the
property that LU U = G(XU ), where G(x) is an invertible function.
Next, we assume that the market filtration is given by Ft = σ ({LsU }0≤s≤U ).
Let the bond price process {PtT }0≤t≤T <U be adapted to {Ft}.
We consider a pricing kernel {πt} that is modelled by a function of the value
LtU at time t, and possibly time t:
πt := π(t, LtU ).
(74)
The function π(t, x) shall be chosen such that the pricing kernel is guaranteed
to be a positive supermartingale.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 38 -
AIMS, Cape Town
Armed with the models for the pricing kernel and the market filtration, the price
PtT of the discount bond is
PtT
EP [π(T, LT U ) | LtU ]
.
=
π(t, LtU )
(75)
Here we have recalled that the market filtration {Ft} is generated by {LtU }
which is taken to be a time-inhomogeneous Markov process.
To obtain explicit models for the bond price PtT , we need to explicitly construct
(i) pricing kernel models and (ii) information processes {LtU }.
One method to construct information-based pricing kernels is presented in:
L. P. Hughston & A. Macrina (2012) Pricing Fixed-Income Assets in an
Information-Based Framework. Applied Mathematical Finance.
arXiv.org: No. 0911.1610.
However, we consider another method, today.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 39 -
AIMS, Cape Town
Weighted heat kernel approach with time-inhomogeneous
Markov processes
The details of the interest-rate framework, which is presented in this lecture, can
be found in:
J. Akahori & A. Macrina (2012) Heat Kernel Interest Rate Models with
Time-Inhomogeneous Markov Processes. International Journal of Theoretical
and Applied Finance.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 40 -
We consider a time-inhomogeneous Markov process {LtU }0≤t≤U , and introduce
a real-valued measurable function p(u, t, x).
A so-called propagator {p(u, t, LtU )} associated with the process {LtU } has the
property that
E [p (u, t, LtU ) | LsU ] = p (u + t − s, s, LsU ) ,
(76)
for s < t, 0 < u, and 0 < u + t < U .
An example of a propagator is
p(u, t, LtU ) = E [F (u + t, Lu+t,U ) | LtU ] ,
(77)
where F (t, x) is taken to be a measurable positive function.
Next we introduce a so-called weight function w(t, u) that is positive and
measurable, and has the property that
w(t, u − s) ≤ w(t − s, u),
(78)
where t, u ∈ [0, U ) and s ≤ t ∧ u.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 41 -
A weighted heat kernel is then defined by
g(t, LtU ) =
Z
U −t
p(u, t, LtU ) w(t, u) du,
(79)
0
for 0 ≤ t < U < ∞. In the case of the propagator (77), we have
g(t, LtU ) =
Z
U −t
0
E [F (u + t, Lu+t,U ) | LtU ] w(t, u) du.
(80)
It can be proved that {g(t, LtU )} is a positive supermartingale by showing that
Z U −s
p(s, u, LsU ) w(t, u − t + s) du
(81)
E [g(t, LtU ) | LsU ] =
Zt−s
U −s
p(s, u, LsU ) w(s, u) du,
(82)
≤
0
= g(s, LsU ),
(83)
for s ≤ t.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 42 -
Applying the pricing kernels constructed by weighted heat kernels, the bond
price PtT can be expressed by
PtT =
R U −t
T −t
p(u, t, LtU ) w(T, u − T + t) du
.
R U −t
p(u, t, LtU ) w(t, u) du
0
(84)
In the case that the propagator is given by (77), we have
PtT =
R U −t
T −t
E [F (u + t, Lu+t,U ) | LtU ] w(T, u − T + t) du
R U −t
E [F (u + t, Lu+t,U ) | LtU ] w(t, u) du
0
(85)
Explicit formulae for the bond price are obtained by specifying the functions
F (t, x) and w(t, u), and the information process {LtU }.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 43 -
Bond pricing with Brownian bridge information
We consider the case where {LtU }0≤t≤U is a so-called Brownian bridge
information process:
LtU = σXU t + βtU ,
(86)
where XU is taken to be independent of the Brownian bridge process {βtU }.
We assume that the market filtration is defined by
Ft = σ {LsU }0≤s≤t .
(87)
Remarks:
(i) {LtU } is a time-inhomogeneous {Ft}-Markov process.
(ii) XU is FU -measurable.
(iii) The information flow rate σ in (86) is constant.
(iv) Var[βtU ] = t(U − t)/U .
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 44 -
We recall that the bond price can be expressed by
PtT
E [π(T, LT U )| LtU ]
.
=
π(t, LtU )
(88)
In order to work out the conditional expectation, we assume with no loss of
generality, that
π(t, LtU ) = Mt g(t, LtU ),
(89)
where the (P, {Ft})-martingale {Mt} is defined by
σU
dMt
=−
E [XU | LtU ] dWt,
Mt
U −t
for 0 ≤ t < U and where
Wt = LtU +
Z
t
0
LsU
ds − σU
U −s
Z
t
0
(90)
1
E [XU | LsU ] ds.
U −s
(91)
The martingale {Mt} induces a change of measure from P to the so-called
bridge measure B under which {LtU } has the distribution of a Brownian bridge.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 45 -
The bond price can thus be expressed as follows:
PtT
EP [π (T, LT U ) | LtU ]
=
,
π(t, LtU )
(92)
EP [MT g(T, LT U )| LtU ]
=
,
Mt g(t, LtU )
(93)
EB [g(T, LT U )| LtU ]
.
=
g(t, LtU )
(94)
We emphasize that the pricing kernel {π(t, LtU )} is a P-supermartingale if
{g(t, LtU )} is a supermartingale under B (and vice versa).
We now may make use of
Z
g(t, LtU ) =
0
16-18 February 2012
U −t
EB [F (u + t, Lu+t,U ) | LtU ] w(t, u) du,
(95)
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 46 -
and hence of
PtT =
R U −t
T −t
EB [F (Lt+u,U ) | LtU ] w(T, u − T + t) du
.
R U −t
B
E [F (Lu+t,U ) | LtU ] w(t, u) du
0
(96)
The information process {LtU }0≤t<U has the distribution of a Brownian bridge
under B so that the conditional expectation can be worked out explicitly.
Example: quadratic family
Let F (x) = x2, and w(t, u) = U − t − u.
A calculation shows that
EB
h
i u(U − t − u)
(Lu+t,U )2 LtU =
+
U −t
U −t−u
U −t
2
L2tU .
(97)
With this intermediate result at hand, we can write the weighted heat kernel
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 47 -
process as follows:
Z U −t
EB [F (Lu+t,U ) | LtU ] w(t, u) du,
(98)
g(t, LtU ) =
0
#
2
Z U −t "
u(U − t − u)
U −t−u
=
+
L2tU (U − t − u)du.
U −t
U −t
0
(99)
The integral in the expression for the weighted heat kernel can be calculated in
closed form, so that we obtain the supermartingale
1
1
3
g(t, LtU ) = (U − t) + (U − t)2 L2tU .
12
4
The bond price PtT at time t, derived in this example, is thus given by
PtT =
1
12
(U
1 (T −t)(U −T )3
1 (U −T )4
− T ) + 4 (U −t) + 4 (U −t)2
1
1
3
2 2
12 (U − t) + 4 (U − t) LtU
3
L2tU
.
(100)
(101)
The simulation of the bond price is straightforward since the process {LtU },
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 48 -
LtU = σ XU t + βtU ,
(102)
is Gaussian conditional on the outcome of the underlying economic factor XU .
The short rate process {rt} can be worked out by calculating the instantaneous
forward rate associated with the bond price {PtT }0≤t≤T <U .
The result is:
L2tU
,
1
r(t, LtU ) = 1
2
4 (U − t) 3 (U − t) + LtU
(103)
for 0 ≤ t < U . We emphasize that this is a positive interest rate model.
The market price of risk {λt} associated with the quadratic family is
σU P
λ(t, LtU ) =
E [XU | LtU ] −
U −t
16-18 February 2012
1
12
(U
1
2
(U
−
t)
LtU
2
− t)3 + 41 (U −
t)2 L2tU
.
(104)
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 49 -
Example: exponential quadratic family
1
2
Let F (x) = exp 2 γt+u x , where γt+u is deterministic.
In this case the propagator takes the form
B
2
1
E exp 2 γt+u Lt+u,U | LtU
2
γt+u at+u
1
exp
L2tU , (105)
=√
2 (1 − u γt+u at+u)
1 − u γt+u at+u
where at+u = (U − t − u)/(U − t).
By setting γt+u = (U − t − u)−1, and by choosing the weight function to be
1
w(t, u) = (U − t − u)η− 2
(η > 21 ),
(106)
we obtain an analytical expression for the supermartingale {g(t, LtU )}:
L2tU
1
η
(U
−
t)
exp
g(t, LtU ) =
.
2(U − t)
η − 12
16-18 February 2012
(107)
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 50 -
The supermartingale (107) leads to a deterministic bond price, even though the
related pricing kernel is stochastic.
However we can modify slightly the supermartingale {g(t, LtU )}:
Let f0(t) and f1(t) be positive, decreasing and differentiable functions.
Consider the supermartingale
g̃(t, LtU ) = f0(t) + f1(t)(U − t)γ exp
L2tU
2(U − t)
.
Then the associated bond price system has stochastic dynamics:
2 L
f0(T ) + f1(T )(U − T )γ−1/2(U − t)1/2 exp 2(UtU−t)
2 PtT =
,
L
f0(t) + f1(t)(U − t)γ exp 2(UtU−t)
(108)
(109)
for t ∈ [0, U ) and u ∈ [0, U − t].
We note that further examples can be constructed: a semi-analytic formula is
obtained for the exponential linear family defined by F (x) = exp(−µ x).
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 51 -
AIMS, Cape Town
Credit-risky discount bonds with stochastic discounting
In the last part of this talk, we give an idea how the information-based asset
pricing framework can be extended so as to incorporate stochastic discounting.
In particular we aim at generalizing the information-based credit-risk models
presented in
D. C. Brody, L. P. Hughston & A. Macrina (2007) Beyond hazard rates: a new
framework for credit-risk modelling. In Advances in Mathematical Finance,
Festschrift Volume in Honour of Dilip Madan, edited by R. Elliott, M. Fu,
R. Jarrow & J.-Y. Yen. Birkhäuser, Basel and Springer, Berlin.
We proceed as follows:
We fix two dates T and U , where T < U , and attach two independent factors
XT and XU to these dates.
The payoff of the credit-risky bond is modelled by making use of the random
variable XT .
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 52 -
We assume that XT is a discrete random variable that takes values in
{x0, x1, . . . , xn} with a priori probabilities {p0, p1, . . . , pn}, where
0 ≤ x0 < x1 < . . . < xn−1 < xn ≤ 1.
(110)
We assume that the economic factor XU is a continuous random variable.
With XT and XU we associate the independent information processes
{LtT }0≤t≤T and {LtU }0≤t≤U defined by
LtT = σ1 t XT + βtT ,
LtU = σ2 t XU + βtU .
(111)
The market filtration {Ft} is generated by {LtT } and {LtU }:
Ft = σ ({LsT }0≤s≤t, {LsU }0≤s≤t)
(112)
The price BtT at t ≤ T of a defaultable discount bond with payoff HT at
T < U is given by
BtT
16-18 February 2012
EP[πT HT | Ft]
.
=
πt
(113)
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 53 -
Let the pricing kernel {πt} be defined by
πt = Mt g(t, LtU )
AIMS, Cape Town
(114)
and let the payoff of the credit-risky bond be given by
HT = H (XT , LT U ) .
(115)
The formula for the price BtT of the credit-risky bond is then worked out by
applying the weighted heat kernel approach for the pricing kernel, and eventually
by specifying the payoff function H(XT , LT U ).
However, we leave this task for another time...
Meantime these results can be found in:
A. Macrina & P. A. Parbhoo (2009) Security Pricing with Information-Sensitive
Discounting. In: Recent Advances in Financial Engineering 2009. Proceedings of
the KIER-TMU International Workshop on Financial Engineering 2009 (World
Scientific).
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 54 -
AIMS, Cape Town
Lévy Random Bridges
We fix a probability space (Ω, F, Q).
We say that {LtT }0≤t≤T has the law LRB C ([0, T ], {ft}, ν) if the following hold:
LT T has marginal law ν.
There exists a Lévy process {Lt} ∈ C[0, T ] such that Lt has density ft(x) for all
t ∈ (0, T ].
ν concentrates mass where fT (z) is positive and finite, i.e. 0 < fT (z) < ∞ for
ν-a.e. z.
For every n ∈ N+, every 0 < t1 < · · · < tn < T , every (x1, . . . , xn) ∈ Rn, and
ν-a.e. z, we have
Q [Lt1,T ≤ x1, . . . , Ltn,T ≤ xn | LT T = z ]
= Q [Lt1 ≤ x1, . . . , Ltn ≤ xn | LT = z ] .
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
- 55 -
AIMS, Cape Town
LRBs have the following properties:
We can interpret an LRB to be a Lévy process conditioned to have a specified
marginal law at the terminal time T .
LRBs are time-inhomogeneous Markov processes.
LRBs have stationary, dependent increments.
Lévy processes and Lévy bridges are special cases of LRBs.
Credit risk simulation.
Next we show some simulations of bond price processes driven by various
different LRBs
The associated information processes (LRBs) are plotted in red—the bond price
processes in blue.
In the following simulations, we assume that the bond ends at maturity in a
state of default.
16-18 February 2012
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
AIMS, Cape Town
- 56 -
Brownian information: ft(x) =
√1
2πt
h
exp − 12
x2
t
i
.
XtT
ΞtT
1.0
0.4
0.8
0.2
0.6
1
2
3
4
5
Time
0.4
-0.2
0.2
-0.4
-0.6
16-18 February 2012
0
1
2
3
4
5
Time
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
VG information: ft(x) =
AIMS, Cape Town
- 57 -
mt 1
m 2 +4
√
π Γ(mt)
2 mt2 − 14
x
4
Kmt− 1
2
√
mx2 ,
m > 0.
XtT
ΞtT
1.0
0.6
0.8
0.5
0.6
0.4
0.3
0.4
0.2
0.2
0.1
0.2
16-18 February 2012
0.4
0.6
0.8
1.0
Time
0.0
0.2
0.4
0.6
0.8
1.0
Time
Andrea Macrina, King’s College London
Asset Pricing: Utility Theory and Pricing Kernel Models
Cauchy information: ft(x) =
AIMS, Cape Town
- 58 -
1
ct
2
π x +c2 t2 ,
c > 0.
XtT
ΞtT
1.0
0.7
0.8
0.6
0.5
0.6
0.4
0.4
0.3
0.2
0.2
0.1
0.2
0.4
0.6
0.8
1.0
Time
0.0
0.2
0.4
0.6
0.8
1.0
Time
Further details in:
E. Hoyle, L.P. Hughston, and A. Macrina (2011) Levy Random Bridges and the
Modelling of Financial Information. Stochastic Processes and Their Applications
121, 856-884.
16-18 February 2012
Andrea Macrina, King’s College London