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Equilibrium Pricing in Incomplete Markets
Ulrich Horst1
Humboldt-Universitat zu BerlinDepartment of Mathematics
Energy and Finance Conference, University Duisburg-EssenOctober 7th, 2010
1This talk is based two projects with P. Cheridito, M. Kupper, T Pirvu andM. Kupper, C. Mainberger, respectively.
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Outline
• Mathematical finance vs. economics
• Equilibrium pricing in incomplete markets
• The model• Characterization, existence and uniqueness of equilibrium• A generalized one fund theorem
• Computing equilibria: backward stochastic differenceequations
• Investor characteristics and volatility smiles
• Explicit solutions in continuous time: forward dynamics.
• Conclusion
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Mathematical Financevs.
Economic Theory
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Some Mathematical Finance
In mathematical finance, one usually assumes that asset pricesfollow a diffusion (St) on some probability space
(Ω,F ,P).
In a complete market, under the assumption of no arbitrage, everyclaim f (ST ) can be replicated and hence uniquely priced:
πt = EP∗ [f (ST ) | Ft ] for a unique P∗ ≈ P.
Pricing uses replication and no arbitrage arguments; no referenceto preferences of the market participants.
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However ...
Challenge 1: Most real world markets are incomplete:
• a perfect hedge might not be possible
• risk needs to be properly accessed
Challenge 2: Non-tradable underlyings:
• weather derivatives written on temperature processes;
• CAT bonds
Challenge 3: Liquidity risk:
• trading the underlying may move the market
• sometimes liquidity is unavailable
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Some Economic Theory
In general equilibrium theory one assumes that agents maximizetheir utility subject to their budget constraints:
maxx∈C a
ua(x) s.t. p(x) ≤ p(ea).
Here
• C a ⊂ C is the consumption set of the agent a ∈ A• ea is her initial endowment
• p ∈ C ∗ is a linear pricing functional
Prices are derived endogenously by market clearing condition∑a
(xa(p∗)− ea) = 0
and so prices depend on preferences and endowments.
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Some Economic Theory
The equilibrium approach might be more suitable for pricingweather derivatives, carbon emission credits, however while
the case of complete markets is very well understood:
• equilibria are efficient (no waste of resources)
• existence via a representative agent approach
for incomplete markets no general theory is yet available:
• equilibria are not necessarily efficient
• representative agent approach breaks down
Often very abstract- equilibria are difficult to compute -
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Our Approach- The General Structure -
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The Model
We consider a dynamic equilibrium model in discrete time whereuncertainty is represented by a general filtered probability space
(Ω,F , (Ft)Tt=0,P).
Each agent a ∈ A is endowed with an uncertain payoff
Ha ∈ L0(FT ) bounded from below.
The agents can lend and borrow from a money market account atan exogenous interest rate. Lending and borrowing is unrestricted;all prices will be expressed in discounted terms.
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Preferences
At any time t ∈ 1, 2, ...,T the agent a ∈ A maximizes apreference functional
Uat : L(FT ) → L(Ft)
which is normalized, monotone, Ft-translation invariant
Uat (X + Z ) = Ua
t (X ) + Z for all Z ∈ Ft
Ft-concave and (strongly) time consistent, i.e.,
Uat+1(X ) ≥ Ua
t+1(Y ) ⇒ Uat (X ) ≥ Ua
t (Y ).
Agents maximize terminal utility from trading in a financial market.
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Examples of Preferences
• Expected exponential utility:
Uat (X ) = − 1
γaln E[e−γaX | Ft ].
• Robust expectations:
Uat (X ) = min
P∈PEP [X | Ft ]
• Mean-risk preferences:
uat (X ) = λE[X | Ft ]− (1− λ)%t(X ).
• Optimized certainty equivalents, g-expectations, ...
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The Financial Market
The agents can trade a finite number of stocks and securities. Therespective holdings over the time period [t, t + 1) are denoted
ηat+1 and ϑa
t+1.
We consider a partial (full) equilibrium model where:
• stock prices follow an exogenous stochastic process StTt=1
• securities are priced to match supply and demand
• security prices follow an endogenous process RtTt=1
• securities pay a bounded dividend R at maturity hence
RT = R
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Equilibrium
An equilibrium consists of a trading strategy (ηat , ϑ
at ) for every
agent a ∈ A and a price process Rt with RT = R s.t.:
• Individual optimality:
Uat
(Ha +
T−1∑s=t
ηas ·∆Ss+1 + ϑa
s ·∆Rs+1
)
≥ Uat
(Ha +
T−1∑s=t
ηas ·∆Ss+1 + ϑa
s ·∆Rs+1
)
for all a ∈ A, t = 1, ...,T and (ηat+1, ϑ
at+1), ..., (η
aT , ϑ
aT ).
• Market clearing in the securities market:∑a∈A
ϑat = n for all t = 1, ...,T − 1.
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Equilibrium
The key observation is that the evolution of utilities follows abackward dynamics. In fact, we associate with
(ηat , ϑ
at ) and Rt
the process Hat defined by Ha
T = Ha and
Hat = Ua
t
(Ha +
T−1∑s=t
ηas ·∆Ss+1 + ϑa
s ·∆Rs+1
).
By translation invariance we obtain that
Hat = Ua
t
(Ha
t+1 + ηat ·∆St+1 + ϑa
t ·∆Rt+1
).
This suggests that the dynamic problem of equilibrium pricing canbe reduced to a recursive sequence of static one-period models.
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Theorem (Characterization of Equilibrium)
A family of stochastic processes Rt , (ηat , ϑ
at )a∈AT
t=1 is anequilibrium if and only if
RT = R
and for all t ∈ 1, 2, ...,T the following holds:
• Pareto optimality:
u(n) := supηa,
∑a ϑ
a=n
∑a
Ua(Ha
t+1 + ηat ·∆St+1 + ϑa · Rt+1
)=
∑a
Uat (Ha
t+1 + ηat ·∆St+1 + ϑa
t ·∆Rt+1).
• Prices equal marginal utility: Rt ∈ ∂ut(n).
• Security markets clear:∑
a ϑat = n.
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Theorem (Existence and Uniqueness of Equilibrium)
An equilibrium exists if the agents are sensitive to large losses, i.e.,
limλ→∞
Ua1 (λX ) = −∞ if P[X < 0] > 0.
The equilibrium is unique if the preferences are differentiable. Inthis case
St = EQa [ST | Ft ], Rt = EQa [RT | Ft ]
where
dQa
dP=
T∏t=1
∇Uat
(Ha +
T∑t=1
ηat ·∆St + ϑa
t ·∆Rt
).
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Theorem (One Fund Theorem)
Assume that (as in the classical CAPM)
Uat (X ) = (γa)−1Ut(γ
aX )
for some γa > 0 and that Ha is “attainable”, i.e.,
Ha = ca +T∑
t=1
ηat ·∆St + ϑa
t ·∆Rt .
Then in equilibrium the market participants share the gains fromtrading according to
(γa)−1(ηt ·∆St + γ(n + ϑ) ·∆Rt
)i.e., according to their risk aversion.
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Theorem (One Fund Theorem Continued)
If preferences are differentiable, then the equilibrium pricing densityis unique and given by
dQt
dP= ∇Ut
(T∑
u=1
ηu ·∆Su + γ(n + ηR) ·∆Ru
).
Corollary
Consider a full equilibrium model where an asset in unit net supplyis priced in equilibrium. Then introducing another security in zeronet supply does not alter the asset price process.
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Example
Assume that all agents have exponential utility functions:
Uat (X ) = − 1
γaE[e−γaX | Ft ].
There is one stock in unit net supply that pays a dividend D atT > 0 and call options with different strikes on the stock in zeronet supply.
By the one fund theorem the unique equilibrium pricing kernel is
dQdP
=e−γD
E[e−γD ]where γ−1 =
∑a
(γa)−1.
In particular, option prices for different strikes can be computedindependently of each other.
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Computing Equilibria- Discrete BSDEs -
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Event TreesLet the uncertainty be generated by independent random walks
bit =
t∑s=1
∆bis (i = 1, ..., d).
Then some form of predictable representation property holds: anyrandom variable X ∈ L(Ft+1) can be represented as
X = E[X |Ft ] +M∑i=1
Z i∆bit+1
where the random coefficients Z i are given by
Z i = E[X∆bit+1|Ft ].
After quite some math ...
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Theorem (Equilibrium Dynamics)
There exist random functions gRt and ga
t such that the equilibriumprice and equilibrium processes Rt and Ha
t satisfy the coupledsystem of discrete BSDEs
Rt = Rt+1 − gR((Z a
t+1)a∈A,ZRt+1
)+ ZR
t+1 ·∆bt+1
Hat = Ha
t+1 − ga(Z at+1,Z
Rt+1) + Z a
t+1 ·∆bt+1
with terminal conditions
RT = R and HaT = Ha.
Here Z at and ZR
t are derived from the representations of Hat and Rt
in terms of the increments ∆b1t+1, ...,∆bM
t+1.
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A Numerical Example
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A Numerical Example
Consider a CAPM in discrete time with a stock and a put optionon the stock.
The stocks pays a dividend DT at time T > 0 where
∆Dt = µh +√
vt∆b1t
∆vt = (κ− λvt)h + σ√
vt∆b2t
The agents have exponential utility functions and (negative)endowments in the put options:
Ha = ca,SST + ca,P(K − ST )+
The equilibrium dynamics (partial or full) can be computed;implied volatilities can be calculated.
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Volatility Smiles: The Supply Effect
10 11 12 13 14 15 16 17 18 19 200.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Strike Price K
Im
pli
ed
Vola
til
ity
Net Supply = -0.2
Net Supply = 0
Net Supply = -0.5
Figure: Volatility smiles in the full equilibrium model.
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Volatility Smiles: The VolVol Effect
10 11 12 13 14 15 16 17 18 19 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Strike Price K
Im
pli
ed
Vola
til
ity
Volvol = 0.3
Volvol = 0.4
Volvol = 0.5
Figure: Volatility smiles in the full equilibrium model.
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Stock Positions in a Partial Equilibrium Model
0 5 10 15 20 25 30 35 40 45 50−25
−20
−15
−10
−5
0
5
Absolute Value Net Supply Put Option
Avera
ge
Nu
mb
er
of
Stock
Held
Start Trading Period
Midth Trading Period
End Trading Period
Figure: Average stock holding in a partial equilibrium model.
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Outlook:The CAPM in Continuous Time
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The CAPM in Continuous Time
Consider a CAPM in continuous time with a stock in unit netsupply and an option on zero net supply.
The stocks pays a dividend DT at time T > 0 where
dDt = µdt +√
vtdW 1t
dvt = (κ− λvt) + σ√
vtdW 2t
Due to the affine structure of Y = (D, v)
E[eu·Y | Ft ] = eφ(τ,u)+ψ(τ,u)·Yt
for all u ∈ C2 with τ = T − t. The functions (φ, ψ) solve a systemof Riccati equations and can be given in closed form.
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The CAPM in Continuous Time
Assume all agents have exponential utility functions. Then
dQdP
=e−γST
E[e−γST ].
Theorem (Equilibrium Stock Prices)
The equilibrium stock price process is given by
St = EQ[ST | Ft ] = F (t,Dt , vt) |u=(−γ,0)
where
F (t,D, v) = ∂u1φ(τ, u) + ∂u1ψ1(τ, u)D + ∂u1ψ2(τ, u)v .
In particular, the equilibrium price process is given in terms of aforward dynamics.
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The CAPM in Continuous Time
Equilibrium prices for options in zero net supply with differentstrikes are all computed under the same measure Q:
Ct = EQ[(ST − K )+ | Ft ].
We have an integral representation for Ct that involves only thefunctions φ and ψ and the Fourier-transform of e−γx(x − K )+.
One goal is to calibrate the local volatility
2∂τCt(τ,K )
K 2∂KKCt(τ,K )
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Conclusion
• Dynamic GEI model with translation invariant preferences
• Dynamic equilibrium problem can be reduced to a sequence ofone-period problems
• Standard properties from the theory of complete markets carryover to incomplete markets:
• Equilibria are (constrained) Pareto optimal• Equilibria exist iff a representative agent exists• ...
• One fund theorem with testable hypothesis
• CAPM in continuous time with explicit solution: possibility ofmodel calibration
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Thank You!