Stochastic Optimization

and

Indifference Valuation

Thaleia Zariphopoulou

UT, Austin

1

Contents

• Historical perspective

• Beyond derivatives

• Investments and stochastic optimization

• Optimal behavior and valuation

• Integrated pricing systems

2

Replication

3

Concept of value

• The option price corresponds to the cost of perfect replication

of the option payoff.

• Perfect replication requires strong assumptions about the model.

• In 1973, Black, Scholes and Merton developed such a model.

• Another example was proposed in 1979 by Cox, Ross and Rubinstein.

4

Does it work in practice?

• In reality no risk can be perfectly replicated.

• The discrepancies depend a lot on the type of risk we look at and

on the variables that affect it.

• So why is this idea so fundamental?

• It laid conceptual foundations for the development of a multibillion

derivatives industry.

5

Milestones

6

Stochastic Calculus and Finance

• In 1979, Harrison and Kreps created a link between pricing by perfect repli-

cation and the general theory of martingales.

• The price was shown to correspond to the expected value, calculated under

the risk neutral probability, of the discounted option payoff.

• In 1981, Harrison and Pliska presented this stochastic calculus based ap-

proach to a large community of researchers from that field.

7

Term structure models

• Black-Scholes model assumes constant interest rates.

• In reality, the interest rate is not constant and the entire curve of interest

rates of different maturities is traded.

• In 1992, Heath, Jarrow and Morton developed a framework for the derivatives

pricing under stochastic interest rates.

• At this stage, infinite dimensional analysis and the stochastic PDEs started

to play a role in Mathematical Finance.

8

Volatility models

• Black-Scholes model assumes constant volatility.

• In reality the volatility is not constant and is traded according to a market

specific convention.

• It depends on the option strike and maturity.

• This dependence is known as the volatility smile or skew.

• In 1993, Dupire constructed a martingale diffusion that is consistent with

the market smile.

9

Models consistent with smile

• In general, asset evolution is represented by a martingale and the option

price may not be defined uniquely by replication.

• The marginals are given by the market smile/skew — there are many ways

to construct such a martingale.

• Solutions differ by the nature of the forward volatilities they generate.

• The notion of value drifts away from the concept of payoff replication. The

price is defined as the expected value under a martingale measure of the

discounted option payoff.

10

Credit derivatives

• The Black-Scholes model did not have a major impact on the development

of this market.

• Is there anything quantitative that the credit markets embraced on their

side to the same extent as the other markets accepted the Black-Scholes

formula?

• A credit default swap provides information about the distribution of the

default time for the underlying name.

• The pricing of tranches of synthetic indices requires the knowledge of the

joint distribution of all default times.

11

Credit derivatives

• The price is defined by an expectation of the payoff, calculated with respect

to the joint distribution of the default times.

• The marginals are given by the relevant CDSs.

• The joint distribution is constructed by choosing a copula function.

• The notion of price has nothing to do with the concept of payoff replication.

12

Beyond derivatives

13

Funds management

• Funds management industry uses relatively simple mathematics when com-

pared with the derivatives side.

• Mathematical models of investment and consumption are very complex, to

an extent more complex than the models used in the derivatives pricing.

• The challenge is to bridge this gap.

• One needs to connect the concept of replication used in the derivatives

pricing with the investment optimization.

14

From replication towards

optimal investment

• In a sense one needs to move from: ‘I can do it for you’ – replication,

towards: ‘I will do my best’ – optimal investment.

• First steps in the academic literature are being undertaken now.

• Pricing of derivatives has been linked with optimal investment.

15

Market uncertainty, preferences

and investments

16

Investments and stochastic optimization

• Maximal expected utility models

• Methods

Primal problem (HJB eqn)

Dual problem (BS eqn)

• Optimal policies: consumption and portfolios

17

Maximal expected utility models

• Market uncertainty

(Ω,F , P), W = (W 1, . . . , Wd)∗ d-dim Brownian motion

Trading horizon: [0, T ], (0, +∞)

Asset returns: dRt = µt dt + σt dWt

µ ∈ L1(Rm), σ ∈ L2(R

d×m)

riskless asset

Wealth process: dXt = πt dRt − Ct dt

Control processes: consumption rate Ct, asset allocation πt

18

Maximal expected utility models

• Preferences: U : R → R

increasing, concave, asymptotically elastic....

U (x) =1

γxγ, log x, −e−γx

• Objective: maximize intermediate utility of consumption and

utility of terminal wealth

V (x, t) = sup(C,π)

EP

(∫ T

tU1(Cs) ds + U2(XT )/Xt = x

)

• Generalizations: infinite horizon, long-term average, ergodic criteria...

19

Primal maximal expected utility problem

• V solves the Hamilton-Jacobi-Bellman eqn⎧⎪⎨⎪⎩Vt + F (x, Vx, Vxx; U1) = 0

V (x, T ) = U2(x)

• Viscosity theory (Crandall-Lions)

Z., Soner, Touzi, Duffie-Z., Elliott, Davis-Z., Bouchard

• Optimal policies in feedback form

C∗s = C((V −1

x )′(X∗s , s)) , π∗s = π(Vx(X∗

s , s), Vxx(X∗s , s))

• Degeneracies, discontinuities, state and control constraints

20

Dual maximal expected utility problem

• Dual utility functional

U∗(y) = maxx

(U (x) − xy)

• Dual problem becomes linear – direct consequence of market completeness

and representation, via risk neutrality, of replicable contingent claims

• Problem reduces to an optimal choice of measure – intuitive connection with

the so-called state prices

Cox-Huang, Karatzas, Shreve, Cvitanic, Schachermayer, Zitkovic,

Kramkov, Delbaen et al, Kabanov, Kallsen, ...

21

Extensions

• Recursive utilities and Backward Stochasticc Differential Equations (BSDEs)

Kreps-Porteus, Duffie-Epstein, Duffie-Skiadas, Schroder-Skiadas, Skiadas,

El Karoui-Peng-Quenez, Lazrak and Quenez, Hamadene, Ma-Yong,

Kobylanski

• Ambiguity and robust optimization

Ellsberg, Chen-Epstein, Epstein-Schneider, Anderson et al.,

Hansen et al, Maenhout, Uppal-Wang, Skiadas

22

• Mental accounting and prospect theory

Discontinuous risk curvatureHuang-Barberis, Barberis et al., Thaler et al., Gneezy et al.

• Large trader models

Feedback effectsKyle, Platen-Schweizer, Bank-Baum, Frey-Stremme, Back, Cuoco-Cvitanic

• Social interactions

Continuous of agents – Propagation of frontsMalinvaud, Schelling, Glaesser-Scheinkman, Horst-Scheinkman, Foellmer

• Fund management

Non-zero sum stochastic differential gamesHuggonier-Kaniel

23

Bridging the gap between

optimal behavior and valuation

24

Pricing elements for contingent claims

“Extreme cases”

• Arbitrage free valuation theory: ν(CT ) = EQ(CT )

• Actuarial principles for insurance: ν(CT ) = U−1(EQ(U (CT )))

P historical measure, Q risk neutral measure,

U utility function

“General case”

• Pricing via equilibrium arguments and investment optimality

25

Arbitrary risks: CT = C(YT, ST )

Pricing from a perspective of optimal investment

• Use the market to assess both types of risk

• Formulate optimality criteria in terms of individual preferences

• Derive the concept of value from indifference to the various

investment opportunities

26

Issues

• There are hedgeable and unhedgeable risks to specify

• The hedgeable risks could be priced by arbitrage, and the hedgeable ones by

certainty equivalent

• Two probability measures, the nested risk neutral measure and the historical

measure, are then involved

27

Indifference valuation

• Optimal behavior with and without the claim

V 0(Xt, t) = supπ

EP (U (XT )/Ft)

V C(Xt, t) = supπ

EP (U (XT − CT )/Ft)

• Partial equilibrium

V 0(Xt, t) = V C(Xt + νt, t)

νt : indifference price process

Hodges-Neuberger, Constantinides-Z., Davis, Rouge-El Karoui, Hugonnieret al, Delbaen et all, Musiela-Z., Kramkov-Sirbu, Henderson,...

28

Structural result

Duality techniques yield

ν = supQ

(EQ(CT ) − ϑ(CT )),

ϑ(CT ) =1

γH(Q/P ) − inf

Q

(1

γH(Q/P )

)

Rouge–El Karoui, Frittelli, Kabanov and Stricker, Delbaen et al.

Considerations

• Price respresented via a non-intuitive optimization problem

• Pricing measure depends on the payoff

• Certain pricing elements are lost

29

Towards a constitutive analogue of

the Black and Scholes theory

30

Fundamental elements of an indifference pricing system

• Monotonicity, scaling and concavity with respect to payoffs

• Monotonicity, robustness and regularity with respect to risk aversion

• Consistency with the no-arbitrage principle

• Translation invariance with respect to replicable risks

• Risk quantification and monitoring

• Numeraire independence

• Additivity with respect to incremental risks

• Risk transfering across parties

31

Price representation and payoff decomposition

(Musiela-Z.)

• Complete markets

ν(CT , t) = EQ(CT/Ft)

CT = ν(CT , t) +∫ T

t

“∂ν(CT , s)

∂S

” dSs

Ss

• Incomplete markets

ν(CT , t) = EQ(CT/Ft)

CT = ν(CT , t) +∫ T

t

“∂ν(CT , s)

∂S

” dSs

Ss+ Rt,T

Rt,T : residual risk

32

Joint work with Marek Musiela (BNP Paribas, London)

References

• “A valuation algorithm for indifference prices in incomplete

Finance and Stochastics (2004)

• “An example of indifference prices under exponential preferences”,

Finance and Stochastics (2004)

• “Spot and forward dynamic utilities and their associated pricing systems:

Case study of the binomial model”

Indifference Pricing, PUP (2005)

33

A new probabilistic representation for indifference prices

The static case

• Arbitrage free prices ν(CT ) = EQ∗(CT )

E(.) : linear pricing functionalQ∗ : the (unique) risk neutral martingale measure

• Indifference prices ν(CT ) = EQ(CT )

E : pricing functional Q : pricing measure

nonlinear payoff independent

payoff independent preference independent

wealth independent

preference dependent

34

The indifference price

ν(CT ) = EQ

(1

γlog(EQ(eγC(ST ,YT ) | ST ))

)= EQ(CT )

Q(YT | ST ) = P(YT | ST )

35

Choice of pricing measure Q

• Q needs to be a martingale measure

• Q needs to preserve the conditional distribution of the unhedgeablerisks, given the hedgeable ones, from their historical values

Q(YT | ST ) = P(YT | ST )

Indifference price components

CT =1

γlog

(EP(eγC(ST ,YT ) |ST )

)

=1

γlog

(EQ(eγC(ST ,YT ) |ST )

)ν(CT ) = EQ(CT )

36

Valuation Procedure

ν(CT ) = EQ

(1γ log(EQ(eγC(ST ,YT ) | ST )

)= EQ(CT )

Q(YT | ST ) = P(YT | ST )

• Step 1: Specification, isolation and pricing of unhedgeable risks

The original payoff CT is altered to the preference adjusted payoff

CT =1

γlog

(EQ(eγC(ST ,YT ) |ST )

)Observe that CT = 1

γlog(EQ∗(eγC(ST ,YT ))) = 1

γlog(EP(eγC(ST ,YT )))

but CT =1

γlog(EQ(eγC(ST ,YT ) | ST )) =

1

γlog(EP(eγC(ST ,YT ) | ST ))

• Step 2: Pricing by arbitrage of the remaining hedgeable risks

37

Payoff decomposition

The static case

Risk quantification and monitoring for unhedgeable risks

• Optimal investments in the presence of the claim

αCT ,∗ = α0,∗ +∂ν(CT )

∂S0,

where

α0,∗ = −1

γ

∂H (Q |P)

∂S0

• Optimal wealths with and without the claim

XCT,∗T = x + ν(CT ) + αCT ,∗(ST − S0) and X

CT,∗0 = x + ν(CT )

X0,∗T = x + α0,∗(ST − S0) and X

0,∗T = x

38

• The optimal residual wealth

Lt = XCT,∗t − X

0,∗t for t = 0, T

Indifference hedge and replicability

L0 = ν(CT )

and

LT = ν(CT ) +∂ν(CT )

∂S0(ST − S0)

Conditional certainty equivalent and extraction of unhedgeable risks

LT = CT

Martingale property

EQ(LT ) = L0 = ν(CT ) for Q ∈ Qe

39

• The residual risk Rt

Rt = Ct − Lt for t = 0, T

Nonreplicable components of the claim and of the residual risk

R0 = 0 and RT = CT − CT ; RT = 0

The indifference price of the residual risk

ν(RT ) = 0

Supermartingale property under the pricing measure Q

EQ(RT ) ≤ Rt = 0

Actuarial certainty equivalent

1

γlog EP(eγRT ) = 0

40

Residual risk decomposition

The supermartingale Rt, for t = 0, T , admits the decomposition

Rt = Rmt + Rd

t

where

Rm0 = 0 and Rm

T = RT − EQ(RT ),

and

Rd0 = 0 and Rd

T = EQ(RT ).

The component Rmt is an F (S,Y )

T -martingale under Q while Rdt is decreasing

and adapted to the trivial filtration F (S,Y )0 .

41

Payoff decomposition

Let CT and RT be, respectively, the conditional certainty equivalent and the

residual risk associated with the claim CT . Let also Rmt and Rd

t be the Doob

decomposition components of the residual risk supermartingale.

Define the process MCt , for t = 0, T , by

MC0 = ν (CT ) and MC

T = ν (CT ) +∂ν (CT )

∂S0(ST − S0)

The claim CT admits the unique, under Q, payoff decomposition

CT = CT + RT

= ν(CT ) +∂ν(CT )

∂S0(ST − S0) + RT = MC

T + RmT + Rd

T

42

Convex risk measures ρ : FT → R

Positions: C1T , C2

T ∈ FT

Properties

Convexity: ρ(αC1T + (1 − α)C2

T ) ≤ αρ(C1T ) + (1 − α)ρ(C2

T )

Monotonicity: if C1T ≤ C2

T then ρ(C1T ) ≥ ρ(C2

T )

Translation invariance w.r.t. replicable risks: ρ(C1T + m) ≤ ρ(C1

T ) − m

Artzner et al., Foellmer and Shied, Frittelli and Gianini, Delbaen, ...

Convex measures are represented as expectations with penalty terms

(entropic terms-special case)

43

Claim: CT ∈ FT

ρ(CT ) = ν (−CT ) = EQ

(1γ log EQ

(e−γCT |ST

))ρ(CT ) : Capital requirement in order to accept the position CT

ν (CT ) : Indifference value of the claim CT

The concept of indifference price, deduced from the desire to behaveoptimally as an investor, is consistent with an axiomatic approachused to determine the capital amount, for a position to be acceptedby a supervising body.

• Convex risk measures yield the capital requirement but not the relevant riskmonitoring implementation strategy

• Convex risk measures are traslation invariant w.r.t. to constant risks

• Convex risk measures are static

44

Dynamic models and the term

structure of risk preferences

45

The dynamic case

• Specification of the non linear price functional that yields the indifference

price representation

νt(CT ) = E(t,T )Q (CT/Ft)

• Semigroup property

νt(CT ) = E(t,t′)Q (νt′(CT )/Ft) = E(t,t′)

Q (E(t′,T )Q (CT )/Ft)

• Payoff decomposition

• Residual risk and error quantification

• Proper specification of preferences

46

A reduced binomial model

47

The multiperiod model

• Traded asset: St, t = 0, 1, ..., T (St > 0, ∀t)

ξt+1 =St+1

St, ξt+1 = ξd

t+1, ξut+1 with 0 < ξd

t+1 < 1 < ξut+1

Second traded asset is riskless yielding zero interest rate

• Non-traded asset: Yt, t = 0, 1, ..., T

ηt+1 =Yt+1

Yt, ηt+1 = ηd

t+1, ηut+1 with ηd

t < ηut

St, Yt : t = 0, 1, ..., T : a two-dimensional stochastic process

• Probability space (Ω, (Ft) , P)

Filtrations FSt and FY

t : generated by the random variables Ss (ξs)

and Ys (ηs), for s = 0, 1, . . . , t.

48

Indifference pricing mechanism

• State wealth process: Xs, s = t + 1, ..., T

αs, s = t + 1, t + 2, ..., T : the number of shares of the traded asset held

in this portfolio over the time period [s − 1, s]

XT = x +T∑

s=t+1

αs Ss

• Claim CT (Path dependence/early exercise are allowed)

• Value function: V CT (Xt, t; T ) = supαt+1,...,αT

EP

(−e−γ(XT−CT ) | Ft

)• Indifference price: νt(CT )

V 0(Xt, t; T ) = V CT (Xt + νt(CT ), t; T )

49

Fundamental multiperiod pricing blocks

Let Zs, 0 ≤ s ≤ t be an Fs-adapted process and Q a martingale measure

E(t,s)Q (Zs) E(t,s−1)

Q (E(s−1,s)Q (Zs))

where

E(s−1,s)Q (Zs) EQ

(1γ log EQ(eγZs|Fs−1 ∨ FS

s )|Fs−1

)and

E(s,s)Q (Zs) = Zs

• Note that for t < s − 1

E(t,s)Q (Zs) = EQ

(1

γlog EQ(Zs|Ft ∨ FS

s )|Ft

)

50

The valuation algorithm

Let Q be a martingale measure satisfying, for t = 0, 1, . . . , T

Q(ηt+1 | Ft ∨ FSt+1) = P(ηt+1 | Ft ∨ FS

t+1)

• The indifference price νt(CT ) satisfies

νt(CT ) = E(t,t+1)Q (νt+1(CT )) νT (CT ) = CT

• The indifference price process is given by

νt(CT ) = E(t,T )Q (CT )

• The pricing algorithm is consistent across time

νt(CT ) = E(t,s)Q (E(s,T )

Q (CT )) = E(t,s)Q ((νs(CT )) = νt(E(s,T )

Q (CT ))

51

Interpretation of the pricing algorithm

Valuation is done via an iterative pricing schemeapplied backwards in time.

νt(CT ) = E (t,t+1)Q (E (t+1,t+2)

Q (...(E (T−1,T )Q (CT )))

νt(CT ) = E (t,t+1)Q (νt+1(CT ))

– Specification, isolation and pricing of unhedgeable risks

νt+1(CT ) : the end of the period payoff is altered to thepreference adjusted payoff

νt+1(CT ) =1

γlog EQ

(eγνt+1(CT )

∣∣∣Ft ∨ FSt+1

)– Pricing of remaining hedgeable risks by arbitrage

νt(CT ) = EQ(νt+1(CT ))

52

Indifference prices and quasilinear pdes

53

Continuous time reduced models

The diffusion case

• Probabilistic setting

– Traded asset: St, dSs = µ(Ss, s)Ssds + σ(Ss, s)SsdW 1s

– Riskless bond Bt

– Nontraded asset: Yt, dYs = b(Ys, s)ds + a(Ys, s)dWs

– Probability space: (Ω,F , (Fs), P)

Fs = σ(W 1u, Wu; 0 ≤ u ≤ s) d(W 1,W )(s) = ρds

– Claim: CT = C(ST, YT )

54

Indifference price representation

The indifference price process is given by

νs(CT ) = H(Ss, Ys, s)

where H(S, y, t) is the unique viscosity solution of the quasilinear price equation

Ht + L(H(S, y, t)) +1

2γ(1 − ρ2)a2(y, t)H2

y = 0

H(S, y, T ) = C(S, y)

The operator L(.) is given by

L(.) =1

2σ2(S, t)

ϑ2

ϑS2 + ρa(y, t)σ(S, t)ϑ2

ϑSϑy+

1

2a2(S, t)

ϑ2

ϑy2

+

(b(y, t) − ρ

µ(S, t)

σ(S, t)a(y, t)

)ϑ

ϑy

55

• Observation 1

i) H may be directly related to a quadratic stochastic control problem

ii) H may be related to a BSDE

But, there is no direct Feynman-Kac type representation of H available up

to now!

• Observation 2

Solutions of the above equation are directly structurally related

to optimal investment policies in models with stochastic Sharpe

ratio, recursive utilities, and stochastic labor income

Stoikov-Z.

56

Special case

(Musiela-Z., Henderson)

• Lognormal dynamics for the traded asset St :

Ss = µSsds + σSsdW 1s

• Payoff depending exclusively on the nontraded asset Yt

C(ST , YT ) = C(YT )

H(y, t) =1

γ(1 − ρ2)ln(EQ(eγ(1−ρ2)C(YT ) | Yt = y)

)

The above formula is not a naive extension of the classical certainty

equivalent price!

57

Valuation algorithm

• Pricing measure Q: a martingale measure satisfying

Q(Yt+dt

∣∣∣Ft ∨ FSt,t+dt) = P(Yt+dt

∣∣∣Ft ∨ FSt,t+dt)

The indifference price νt(Ss, Ys, s) is given by the iterative algorithm

νt(Ss, Ys, s) =

= EQ

(1γ log EQ(eγν(Ss+ds,Ys+ds,s+ds) | Fs ∨ FS

s,s+ds) | Fs

)νT (ST , YT , T ) = C(ST , YT )

58

A novel probabilistic representation of solutionsto quasilinear pdes arises

The solution H(S, y, t) of the quasilinear price equation

Ht + L(H(S, y, t)) +1

2γ(1 − ρ2)a2(y, t)H2

y = 0

with

L(.) = 12σ

2(S, t) ϑ2

ϑS2 + ρa(y, t)σ(S, t) ϑ2

ϑSϑy + 12a

2(S, t) ϑ2

ϑy2

+(b(y, t) − ρ

µ(S,t)σ(S,t)

a(y, t))

ϑϑy

is given byH(S, y, t) = limds→0,s→t νt(Ss+ds, Ys+ds, s + ds)

Proof: Based on convergence results of supermartingales, behavior of nonadditivemeasures and robustness properties of viscosity solutions.

59

Extensions

• Early exercise claims: Cτ = C(Sτ , Yτ ) ; τ ∈ F[0,T ]

νat (Cτ ) = sup

τ∈F[0,T ]

νt(Cτ ) = supτ∈F[0,T ]

(E(t,τ )Q (Cτ ))

(Musiela-Z., Oberman-Z., Sokolova)

• Path dependent claims: C(t,T ) = C(∫ T

tc(Ss, Ys, s)ds; ω)

• Stochastic interest rates (infinite dimensional problems/PSDEs)

(Tehranchi and Zariphopoulou)

• Defaultable claims (Bielecki et al.)

60

Stochastic modelling and indifference prices

• Quasilinear PDEs

• BSDEs

• Stochastic PDEs

• Non-linear expectations

61

Utility-based pricing systems

for general market models

62

Important issues

• Traditional utilities are “fixed” w.r.t. a given horizon but we need to be able

to price claims across all maturities.

• Typically, risk aversion coefficient is chosen in isolation from the market

environment.

• Numeraire independence and semigroup property of prices imposes

endogeneous restrictions on preferences.

63

Towards a utility-based pricing system

Ingredients for a term structure of risk preferences that ultimately

yields consistent prices across units, numeraires, investment horizons

and maturities

• Stochastic risk tolerance

• Time evolution of preferences that captures market incompleteness

This leads to the new notions of

Forward and Spot Dynamic Utilities (Musiela-Z., 2005)

64

The diffusion case

• Market dynamics

dSs = µ(Ys, s)Ss ds + σ(Ys, s)Ss dW 1s

dYs = b(Ys, s) ds + a(Ys, s) dWs

ρ = cor(W 1,W ), (Ω,F , P), λs(Ys, s) =µ(Ys, s)

σ(Ys, s)

• Minimal relative entropy measure: Q

dQ

dP= exp

(−∫ T

0λs dW 1

s −∫ T

0λ1

s dW 1,⊥s − 1

2

∫ T

0(λ2

s + (λ⊥s )2) ds

)

λ⊥s = λ⊥s (Ys, s); λ⊥(y, t) ∼ gradient to the sln of a quasilinear pde

(Hobson, Rheinlander, Stoikov-Z., Benth-Karlsen)

65

Spot and forward dynamic utilities

• Forward dynamic utility

For T = 1, 2.., the processUf (x, t; T ) : t = 0, 1, ..., T

defined,

for x ∈ R, by

Uf (x, t; T ) =

⎧⎪⎨⎪⎩−e−γx if t = T

−e−γx−HT (Q(·|Ft)| P(·|Ft)) if 0 ≤ t ≤ T − 1

with

HT (Q(·|Ft)/P(·|Ft)) = EQ

(∫ T

t

1

2(λ2

s + (λ⊥s )2) ds/Ft

)

is called the forward to time T , or, normalized at time T ,dynamic exponential utility

66

• Spot dynamic utility

For s = 0, 1, .., the process Us (x, t) : t ≥ s, ... defined,for x ∈ R, by

Us (x, t; s) =

⎧⎪⎪⎨⎪⎪⎩−e−γx if t = s

−e−γx+∫ ts

12λ

2u du if t > s

is called the spot, normalized at time s, dynamic exponential utility.

Spot and forward dynamic utilities

• Uf aggregates information

• Us follows information along the path

67

Spot and forward dynamic value functions

Forward dynamic value function

Let Uf be the forward dynamic utility process, normalized at T . The associated

forward dynamic value function V f is defined, for x ∈ R, s = 0, 1, .. and

s ≤ t ≤ T , as

V f (x, s, t; T ) = supπ

EP

(Uf (Xt, t; T ) |Ft

)with

dXt = µtπt dt + σtπt dW 1t , t ≥ s

and Xs = x.

68

Spot dynamic value function

Let Us be the spot dynamic utility process, normalized at s ≥ 0.

The associated spot dynamic value function V s is defined, for x ∈ R,

and s ≤ t ≤ T as

V s (x, t, T ) = supπ

EP (Us (XT, T ) |Ft)

with

dXu = µuπu du + σuπu dW 1u

and Xt = x

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Fundamental properties of the forward dynamic value function

The forward dynamic value function V f has the following properties:

V f (x, T, T ; T ) = −e−γx for x ∈ R

V f (x, s, t1; T ) = V f (x, s, t2; T ) for t1 = t2 and s ≤ min (t1, t2)

V f (x, s, t; T ) = V f (x, s, s; T ) = −e−γx−HT (Q(·|Ft)/P(·|Ft))

The forward dynamic value function V f coincides with the forwarddynamic utility

V f (x, s, t; T ) = Uf (x, s; T ) for x ∈ R and s ≤ t ≤ T

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Fundamental properties of the spot dynamic value function

The spot dynamic value function V f has the following properties:

V s (x, s, s) = −e−γx for x ∈ R

V s (x, t, T1) = V s (x, t, T2) for T1 = T2 and s ≤ t ≤ min (T1, T2)

V s (x, t, T ) = V s (x, t, t) = −e−γx+∫ ts

12λ

2u du

The spot dynamic value function V s coincides

with the spot dynamic utility

V s (x, t, T ) = Us (x, T ) for x ∈ R and s ≤ t ≤ T

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Forward indifference price

• Forward dynamic utility Uf , normalized at time T

• Let T ≤ T and consider a claim, written at time t0 ≥ 0, yielding payoff CTat time T , with CT ∈ FT

• Let, also, V f,CT and V f,0 be, respectively, the forward dynamic value func-

tions with and without the claim

For t0 ≤ t ≤ T , the forward indifference price, associated with the forward,

to time T, utility Uf, is defined as the amount νf (CT , t; T ) for which

V f,0(x, t, T ; T

)= V f,CT

(x + νf (CT , t; T ), t, T ; T

)for x ∈ R

72

Spot indifference price

• Spot dynamic utility Us, normalized at time s

• Consider a claim, written at time t0 ≥ s, yielding payoff CT at time T , with

CT ∈ FT

• Let, also, V s,CT and V s,0 be, respectively, the spot dynamic value functions

with and without the claim

For t0 ≤ t ≤ T , the spot indifference price, associated with the spot,

normalized at time s, utility Us, is defined as the amount νs(CT , t; s) for which

V s,0 (x, t, T ) = V s,CT (x + νs(CT , t), t, T ) for x ∈ R

73

Important observation

The spot and forward indifference prices do not in general coincide,

i.e., for

s ≤ t0 ≤ t ≤ T ≤ T

νs(CT , t; s) = νf (CT , t; T )

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Consistency across maturities

Both V f and V s are independent of the investment horizon

Spot and forward parity

−νs

(∫ T

t

1

2

λ2u

γdu, t

)= νf

(∫ T

t−1

2

λ2u

γdu; T

)

νs

(∫ T

tCT , t

)+ νs

(∫ T

t−1

2

λ2u

γdu, t

)= νf

(CT −

∫ T

t

1

2

λ2u

γdu, t; T

)

Spot and foward prices coincide if Sharpe ratios are replicable.

75

Extensions

• Forward and spot prices in Markovian models

Connection with non-linear expectations

• Forward and spot prices for stochastic risk tolerance

The term structure of utilities, stochastic pdes

• Indifference pricing systems and contract theory

Zero- and Non zero-sum stochastic differential games

• Infinite dimensional models

76