Spot and forward dynamic utilities and their associated pricing systems Thaleia Zariphopoulou UT, Austin 1
Joint work with Marek Musiela (BNP Paribas, London) References A valuation algorithm for indifference prices in incomplete markets, 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) 2
Incomplete markets Derivative Valuation Stochastic volatility Non-traded assets Real Options, R & D projects Optimal Portfolio Decisions Stochastic Sharpe ratio Correlated predictors Stochastic labor income 3
Pricing elements for contingent claims Extreme cases Arbitrage free valuation theory Actuarial principles for insurance General case Pricing via equilibrium arguments Investment optimality 4
Arbitrage free valuation theory Hedgeable risks Riskless bond B (zero interest rate): B 0 = B T =1 Probability space: Ω {ω 1,ω 2 }, P(ω 1 )=p, P(ω 2 )=1 p Traded asset S: Claim C : C T = C(S T ) Arbitrage free measure Q : S = S d,s u with 0 <S d <S u Pricing by arbitrage Q {ω 1 } = q = S 0 S d S u S d, Q {ω 2} =1 q Arbitragefreepriceν(C T ) ν(c T )=E Q (C(S T )) 5
Incomplete Models Probability space Ω={ω 1,ω 2,ω 3,ω 4 }, P {ω i } = p i, i =1,..., 4 Two risks S 0 S u Y u Y 0 S d Y d Random variables S T and Y T S T (ω 1 )=S u,y T (ω 1 )=Y u S T (ω 2 )=S u,y T (ω 2 )=Y d S T (ω 3 )=S d,y T (ω 3 )=Y u S T (ω 4 )=S d,y T (ω 4 )=Y d 6
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 7
Arbitrary risks: C T = C(Y T,S T ) 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 8
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 9
Investment opportunities We invest the amount β in bond and the amount α in stock Wealth variable X 0 = x, X T = β + αs T = x + α(s T S 0 ) Indifference price For a general claim C T, we define the value function V C T (x) =max α E( e γ(x T C T ) ) The indifference price is the amount ν(c T ) for which, V 0 (x) =V C T (x + ν(c T )) 10
Structural result Duality techniques yield ν =sup(e Q (C T ) ϑ(c T )), Q ϑ(c T )= 1 H(Q/P ) inf γ Q ( ) 1 γ H(Q/P ) Rouge and El Karoui (2000) Frittelli (2000) Kabanov and Stricker (2002) Delbaen et al. (2000) Considerations Price respresented via a non-intuitive optimization problem Pricing measure depends on the payoff Certain pricing elements are lost 11
A new probabilistic algorithm for indifference prices Arbitragefreeprices ν(c T )=E Q (C T ) E(.) :linear pricing functional Q : the (unique) risk neutral martingale measure Indifference prices ν(c T )=E Q (C T ) E : pricing functional (possibly) nonlinear payoff independent wealth independent preference dependent Q : pricing measure payoff independent preference independent 12
Choice of pricing measure Q Q needs to be a martingale measure Q needs to preserve the conditional distribution of the unhedgeable risks, given the hedgeable ones, from their historical values Q(Y T S T )=P(Y T S T ) Indifference price components C T = 1 γ log ( E P (e γc(s T,Y T ) S T ) ) = 1 γ log ( E Q (e γc(s T,Y T ) S T ) ) ν(c T )=E Q ( C T ) 13
The indifference price ν(c T )=E Q ( 1 γ log(e Q(e γc(s T,Y T ) S T )) ) = E Q (C T ) Q(Y T S T )=P(Y T S T ) 14
Valuation Procedure ( ν(c T )=E 1γ Q log(e Q (e γc(s T,Y T ) S T ) ) = E Q (C T ) Q(Y T S T )=P(Y T S T ) Step 1: Specification, isolation and pricing of unhedgeable risks The original payoff C T is altered to the preference adjusted payoff C T = 1 γ log ( E Q (e γc(s T,Y T ) S T ) ) Observe that C T 1 γ log(e Q (eγc(s T,Y T ) )) 1 γ log(e P(e γc(s T,Y T ) )) but C T = 1 γ log(e Q(e γc(s T,Y T ) S T )) = 1 γ log(e P(e γc(s T,Y T ) S T )) Step 2: Pricing by arbitrage of the remaining hedgeable risks 15
Conclusions ν(c T )=E Q (C T ) Pricing functional E Q (.) Nonlinear Payoff independent Preference dependent in the first (unhedgeable risk) step only Pricing measure Q Preserves conditional distribution of unhedgeable risks, given the hedgeable ones Preference independent Payoff independent 16
Stochastic risk preferences γ T = γ (S T ) F S T -measurable random variable (in reciprocal to wealth units) Risk tolerance (in units of wealth) δ T = 1 γ T The indifference price of C T is given by ( ( 1 ν (C T ; γ T )=E Q log E γ Q e γ T C )) T 1+r S T T 17
Value functions and optimal behavior Value function without the claim V 0 (x; γ T )= exp ( x E Q (δ T ) H (Q P) ) Value function with the claim ( V C T (x; γ T )= exp ( x ν (CT ; γ T ) E Q (δ T ) ) H (Q P) ) 18
Two minimal entropy measures dq dq (ω) = δ T (ω) E Q (δ T ). Note: E Q (S T (1 + r) S 0 )=0 E Q (γ T (S T (1 + r) S 0 )) = 0 Structural constraints between the market environment and the risk preferences 19
Optimal policies for stochastic risk preferences in the presence of the claim α C T, = α 0, + α 1, + α 2,, Optimal demand due to incompleteness: α 0, α 0, = H (Q P) S 0 E Q (δ T ) 20
Optimal demand due to changes in risk tolerance: α 1, α 1, = log E Q (δ T ) x S 0 Optimal demand due to liability: α 2, α 2, = E Q (δ T ) S 0 ( ) ν (CT ; γ T ) E Q (δ T ) 21
Indifference Pricing Systems and the Term Struture of Risk Preferences Consistent Valuation for different Maturities Consistent Valuation across Numeraires Consistent Risk Transfering across Investors 22
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 that captures market incompleteness This leads to the new notions of Forward and Spot Dynamic Utilities (MZ 2005) 23
The multiperiod binomial model Traded asset: S t,t=0, 1,..., T (S t > 0, t) ξ t+1 = S t+1 S t,ξ t+1 = ξ d t+1,ξu t+1 with 0 <ξ d t+1 < 1 <ξu t+1 Second traded asset is riskless yielding zero interest rate Non-traded asset: Y t,t=0, 1,..., T η t+1 = Y t+1 Y t,η t+1 = η d t+1,ηu t+1 with η d t <η u t {S t,y t : t =0, 1,..., T } : a two-dimensional stochastic process Probability space (Ω, (F t ), P) Filtrations F S t and F Y t : generated by the random variables S i (ξ i ) and Y i (η i ),fori =0, 1,...,t. 24
Traditional indifference pricing mechanism State wealth process: X s,s= t +1,..., T α s,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] X T = x + T s=t+1 α s S s Claim C T (Path dependence/early exercise are allowed) Value function: V C T (X t,t; T )= Indifference price: ν t (C T ) ( sup E P e γ(x T C T ) ) F t α t+1,...,α T V 0 (X t,t; T )=V C T (X t + ν t (C T ),t; T ) 25
Auxiliary quanities quantifying the model incompleteness Local entropy terms: h i with h i = q i log q i P (A i F i 1 ) +(1 q i)log 1 q i 1 P (A i F i 1 ) A i = {ξ i = ξ u i } and q i = Q (A i F i 1 ) for i =0, 1,.., T. 26
Aggregate entropy: H T (Q ( F t ) P ( F t )) H T (Q ( F t ) P ( F t )) = E Q T i=t+1 h i F t h i = H i (Q ( F i 1 ) P ( F i 1 )). Traditional value function: V (x, t) V (x, t) = sup α t+1,...,α T E P ( e γx T /X t = x ) = e γx H T 27
An example Assume T =2 and U (x;2)= e γx Value function: V (x, t;2), for t =0, 1, 2. V (x, 0; 2) = e γx E Q(h 1 +h 2 F 0 ) V (x, 1; 2) = e γx E Q(h 2 F 1 ) V (x, 2; 2) = e γx V (x, 0; 2) V (x, 0; 1) and V (x, 0; 2) >V(x, 0; 1), if P (A 2 F 1 ) q 2 28
Specification of trading horizon? Time invariant maximal expected utilities: the forward case Forward dynamic utility, normalized at T =2 U f (x, t;2)= e γx E Q(h 1 +h 2 F 0 ) t =0 e γx E Q(h 2 F 1 ) t =1 e γx t =2 29
Forward dynamic value function V f (x, 0,t;2) V f (x, 0,t;2)= sup α 1,α t E P ( U f (X t,t;2) F t ) for t =1, 2 with X t = x + t i=1 α i S i. This choice of dynamic risk preferences makes the associated dynamic value function invariant with respect to the trading horizon V f (x, 0, 0; 2) = V f (x, 0, 1; 2) = V f (x, 0, 2; 2) = e γx E Q(h 1 +h 2 F 0 ) for x R 30
Time invariant maximal expected utilities: the spot case Spot dynamic utility, normalized at current time 0, U s (x, t;0)= Spot dynamic value function V s (x, 0,t;0) e γx t =0 e γx+h 1 t =1 e γx+σ2 i=1 h i t =2 V s (x, 0,t;0)= sup α 1,α t E P (U s (X t,t;0) F 0 ) for t =1, 2 with X 0 = x and X t = x + t i=1 α i S i 31
This choice of spot dynamic risk preferences makes the associated spot dynamic value function invariant with respect to the trading horizon V s (x, 0, 0; 0) = V s (x, 0, 1; 0) = V s (x, 0, 2; 0) = e γx 32
Spot and forward dynamic utilities Forward dynamic utility For T =1, 2.., the process { U f (x, t; T ):t =0, 1,..., T } defined, for x R, by U f (x, t; T )= e γx if t = T e γx H T (Q( F t ) P( F t )) if 0 t T 1 is called the forward to time T, or, normalized at time T, dynamic exponential utility 33
Spot dynamic utility For s =0, 1,.., the process {U s (x, t; s) :t = s, s +1,...} defined, for x R, by U s (x, t; s) = e γx if t = s e γx+ t u=s+1 h u if t s +1 is called the spot, normalized at time s, dynamic exponential utility. Spot and forward dynamic utilities U f (x, t; T ) F t, 0 t T U s (x, t; s) F t 1, 0 s t T 34
Spot and forward dynamic value functions Forward dynamic value function Let U f 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 )= ( sup E P U f ) (X t,t; T ) F s α s+1,..,α t with X t = X s + t i=s+1 α i S i and X s = x. 35
Spot dynamic value function Let U s be the spot dynamic utility process, normalized at s, for s =0, 1,... The associated spot dynamic value function V s is defined, for x R, T = s +1,... and s t T as V s (x, t, T ; s) = sup E P (U s (X T,T; s) F t ) α t+1,..,α T with X T = X t + T i=t+1 α i S i and X t = x 36
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, t 1 ; T )=V f (x, s, t 2 ; T ) for t 1 t 2 and s min (t 1,t 2 ) V f (x, s, t; T )=V f (x, s, s; T )= e γx E Q (Σ T i=s+1 h i F s ) The forward dynamic value function V f coincides with the forward dynamic utility V f (x, s, t; T )=U f (x, s; T ) for x R and s t T 37
Fundamental properties of the spot dynamic value function The spot dynamic value function V f has the following properties: V s (x, s, s; s) = e γx for x R V s (x, t, T 1 ; s) =V s (x, t, T 2 ; s) for T 1 T 2 and s t min (T 1,T 2 ) V s (x, t, T ; s) =V s (x, t, t; s) = e γx+σt i=s+1 h i The spot dynamic value function V s coincides with the spot dynamic utility V s (x, t, T ; s) =U s (x, T ; s) for x R and s t T 38
Forward indifference price Forward dynamic utility U f, normalized at time T Let T T and consider a claim, written at time t 0 0, yielding payoff C T at time T,withC T F T Let, also, V f,c T and V f,0 be, respectively, the forward dynamic value functions with and without the claim For t 0 t T,theforward indifference price, associated with the forward, to time T, utility U f, is defined as the amount ν f (C T,t; T ) for which V f,0 ( x, t, T ; T ) = V f,c T ( x + ν f (C T,t; T ),t, T ; T ) for x R 39
Spot indifference price Spot dynamic utility U s, normalized at time s Consider a claim, written at time t 0 s, yielding payoff C T at time T,with C T F T Let, also, V s,c T and V s,0 be, respectively, the spot dynamic value functions with and without the claim For t 0 t T, the spot indifference price, associated with the spot, normalized at time s, utility U s, is defined as the amount ν s (C T,t; s) for which V s,0 (x, t, T ; s) =V f,c T (x + ν s (C T,t; s),t,t; s) for x R 40
Important observation The spot and forward indifference prices do not in general coincide, i.e., for s t 0 t T T ν s (C T,t; s) ν f (C T,t; T ) 41
Forward Indifference Prices under Forward Risk Preferences Forward Dynamic Utility U f (x, t; T )= Inverse Forward Dynamic Utility e γx if t = T e γx H T (Q( F t ) P( F t )) if 0 t T 1 ( U f ) 1 1 (x, t; T )= γ log ( x) 1 γ E Q Term structure forward utility notation T i=t+1 h i F t U f (x, t; T )=U f t,t (x) and ( U f ) 1 (x, t; T )= ( U f ) 1 t,t (x) 42
The forward valuation algorithm Pricing blocks Z is a random variable on (Ω, F, P) and F t and F S t be the market filtrations The pricing measure Q to be a martingale measure with the minimal relative to P entropy, that is, satisfying, for t =0, 1,... Q ( η t+1 Ft F S t+1) = P ( ηt+1 Ft F S t+1), The single step forward price functional E f,(t,t+1) ( Q (Z; T )=E Q ( U f ) ( 1 (E t+1,t Q U f t+1,t ( Z) Ft Ft+1 S The iterative forward price functional E f,(t,t ) Q (Z; T )=E f,(t,t+1) Q (E f,(t+1,t+2) Q (...(E f,(t 1,t ) Q (Z; T )); T ); T ) 43 )) ) F t
The forward valuation algorithm (cont d) The forward indifference price ν f (C T,t; T ) is given, for t 0 t T,bythe algorithm ν f (C T,t; T )=E f,(t,t+1) Q (ν f (C T,t+1;T ); T ), ν f (C T,T; T )=C T, The forward indifference price process ν f (C T,t; T ) F t and satisfies ν f (C T,t; T )=E f,(t,t ) Q (C T ; T ), t 0 t T The forward pricing algorithm is consistent across time in that, for 0 t 0 t t T, the semigroup property ν f (C T,t; T )=E f,(t,t ) Q (E f,(t,t ) Q (C T ; T );T ) holds. = E f,(t,t ) Q ((ν f (C T,t ; T ); T )=ν f (E f,(t,t ) Q (C T ; T ),t; T ) 44
Spot Indifference Prices under Spot Risk Preferences Spot Dynamic Utility U s (x, t; s) = e γx if t = s e γx+ t u=s+1 h u if t s +1 Inverse Spot Dynamic Utility (U s ) 1 (x, t; s) = 1 γ log ( x) 1 γ Term structure spot utility notation t h i i=s+1 U s (x, t; s) =U s s,t (x) and (U s ) 1 (x, t; s) =(U s ) 1 s,t (x) for t s 45
The spot valuation algorithm Pricing blocks Z is a random variable on (Ω, F, P) and F t and F S t be the market filtrations. The pricing measure Q to be a martingale measure with the minimal relative to P entropy, that is, satisfying, for t =0, 1,... Q ( η t+1 Ft F S t+1) = P ( ηt+1 Ft F S t+1), The single step spot price functional E s,(t,t+1) ( Q (Z; s) =E Q (U s ) 1 ( ( s,t+1 EQ U s s,t+1 ( Z) Ft Ft+1 S )) ) Ft The iterative spot price functional E s,(t,t ) Q (Z; s) =E s,(t,t+1) Q (E s,(t+1,t+2) Q (...(E s,(t 1,t ) Q (Z; s)));s) 46
The spot valuation algorithm (cont d) The spot indifference price ν s (C T,t; s) is given, for s t T,bythealgorithm ν s (C T,t; s) =E s,(t,t+1) Q (ν s (C T,t+1;s); s), ν s (C T,T; s) =C T The spot indifference price process ν s (C T,t; s) F t and satisfies ν s (C T,t; s) =E s,(t,t ) Q (C T ; s), s t 0 t T The spot pricing algorithm is consistent across time in that, for 0 s t 0 t t T, the semigroup property ν s (C T,t; s) =E s,(t,t ) Q (E s,(t,t ) Q (C T ; s);s) holds. = E s,(t,t ) Q (ν s (C T,t ; s); s) =ν s (E s,(t,t ) Q (C T ; s),t; s) 47
When do the forward and spot prices coincide? Reduced binomial model Transition probabilities of the traded asset are not affected by the non-traded one P ( ξ t+1 = ξ u t+1 F t) = P ( ξt+1 = ξ u t+1 F S t ) The local entropy terms satisfy h t Ft 1 S,for0 t T and thus they create replicable claims Then, for all C T and t [s, T ], the two prices coincide, ν f (C T,t; s) =ν s (C T,t; T ) 48
Forward and spot prices They both capture information about market incompleteness in a dynamic and consistent way The forward pice captures information in a coarse, aggregate way. The local entropies are priced as additional non-traded claims. The forward price functional depends functionally on the evolution of the market The spot price captures information in a more dynamic and finer way, along the path. The spot price functional is static in preferences and offers a more intuitive pricing perspective 49
Payoff decomposition for forward and spot dynamic preferences (MZ (2005a) and MZ (2005b)) Optimal residual wealth Optimal residual risk L t = X C T, t X t R t = ν (C T ; t) L t The residual wealth process L t represents the replicable part and it is, naturally, a martingale under all martingale measures Q Q e The residual risk process R t represents the non-replicable part and it is a supermartingale under the pricng measure Q. It has the important property of zero indifference value ν (R T ; t) =0 (Not trivial under forward preferences!!) 50
Payoff decomposition Doob-Meyer decomposition of the Residual Risk supermartingale R t = R m t + R d t Doob-Meyer decomposition of the Indifference price supermartingale with ν (C T ; t) =L t + R t = M t + R d t M t = L t + R m t The payoff C T can be decomposed as C T = L T + R T 51
Extensions Forward and spot prices in diffusion models Connection with non-linear expectations Forward and spot prices for stochastic risk tolerance The term structure of utilities Indifference pricing systems and contract theory Zero- and Non zero-sum stochastic differential games Infinite dimensional models 52