In this talk, we discuss two kinds of dynamic financial decision - dynamic hedging of contingent claim and dynamic asset allocation - in the presence of concerns on financial risk and the implementation of financial risk management. For the dynamic hedging decision, we focus on the asset pricing implications of financial risks, such as liquidity risk and model risk. We also discuss some challenges of deep dynamic hedging of contingent claim. Then we discuss the dynamic asset allocation under several important financial risk management measures and formulate the equilibrium (game) under financial risks.

1. Dynamic Decisions under Financial Risks
Weidong Tian
University of North Carolina at Charlotte
GDRR-SAMSI Workshop, August, 2019
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 1 / 27

2. Introduction
Financial risks are pervasive
Market risk (due to moves in market factors)
Credit risk and counterparty risk (due to the credit or default to its
counterparty)
Liquidity risk (due to the illiquidity on both macro and micro-level
environment)
Operational risk (the loss rusting from inadequate or failed process,
people, system or external events)
Model risk (the adverse consequence from decisions based on incorrect
or misused model outputs and reports)
Risk category including data reporting, fair lending, fintech, financial
crime, etc.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 2 / 27

3. Introduction
Financial risks are pervasive
Market risk (due to moves in market factors)
Credit risk and counterparty risk (due to the credit or default to its
counterparty)
Liquidity risk (due to the illiquidity on both macro and micro-level
environment)
Operational risk (the loss rusting from inadequate or failed process,
people, system or external events)
Model risk (the adverse consequence from decisions based on incorrect
or misused model outputs and reports)
Risk category including data reporting, fair lending, fintech, financial
crime, etc.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 2 / 27

4. Introduction
Financial risks are pervasive
Market risk (due to moves in market factors)
Credit risk and counterparty risk (due to the credit or default to its
counterparty)
Liquidity risk (due to the illiquidity on both macro and micro-level
environment)
Operational risk (the loss rusting from inadequate or failed process,
people, system or external events)
Model risk (the adverse consequence from decisions based on incorrect
or misused model outputs and reports)
Risk category including data reporting, fair lending, fintech, financial
crime, etc.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 2 / 27

5. Introduction
Financial risks are pervasive
Market risk (due to moves in market factors)
Credit risk and counterparty risk (due to the credit or default to its
counterparty)
Liquidity risk (due to the illiquidity on both macro and micro-level
environment)
Operational risk (the loss rusting from inadequate or failed process,
people, system or external events)
Model risk (the adverse consequence from decisions based on incorrect
or misused model outputs and reports)
Risk category including data reporting, fair lending, fintech, financial
crime, etc.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 2 / 27

6. Introduction
Financial risks are pervasive
Market risk (due to moves in market factors)
Credit risk and counterparty risk (due to the credit or default to its
counterparty)
Liquidity risk (due to the illiquidity on both macro and micro-level
environment)
Operational risk (the loss rusting from inadequate or failed process,
people, system or external events)
Model risk (the adverse consequence from decisions based on incorrect
or misused model outputs and reports)
Risk category including data reporting, fair lending, fintech, financial
crime, etc.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 2 / 27

7. Introduction
Financial risks are pervasive
Market risk (due to moves in market factors)
Credit risk and counterparty risk (due to the credit or default to its
counterparty)
Liquidity risk (due to the illiquidity on both macro and micro-level
environment)
Operational risk (the loss rusting from inadequate or failed process,
people, system or external events)
Model risk (the adverse consequence from decisions based on incorrect
or misused model outputs and reports)
Risk category including data reporting, fair lending, fintech, financial
crime, etc.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 2 / 27

8. Introduction
Market risk measure from Basel
Minimal capital requirement
Value ar risk requirement
Var(p) = min {m : P{V(0) − V(∆t) ≥ m} ≤ 1 − p} , p = 0.99
Expected shortfall
ES(p) = EP
[V(0) − V(∆t)|V(0) − V(∆t) ≥ Var(p)]
Risk management constraint (Stressed VaR, stress testing), and capital
charge
BCBS, “Minimal Capital Requirements for Market Risks" (standards),
January 2016.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 3 / 27

9. Introduction
Market risk measure from Basel
Minimal capital requirement
Value ar risk requirement
Var(p) = min {m : P{V(0) − V(∆t) ≥ m} ≤ 1 − p} , p = 0.99
Expected shortfall
ES(p) = EP
[V(0) − V(∆t)|V(0) − V(∆t) ≥ Var(p)]
Risk management constraint (Stressed VaR, stress testing), and capital
charge
BCBS, “Minimal Capital Requirements for Market Risks" (standards),
January 2016.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 3 / 27

10. Introduction
Market risk measure from Basel
Minimal capital requirement
Value ar risk requirement
Var(p) = min {m : P{V(0) − V(∆t) ≥ m} ≤ 1 − p} , p = 0.99
Expected shortfall
ES(p) = EP
[V(0) − V(∆t)|V(0) − V(∆t) ≥ Var(p)]
Risk management constraint (Stressed VaR, stress testing), and capital
charge
BCBS, “Minimal Capital Requirements for Market Risks" (standards),
January 2016.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 3 / 27

11. Introduction
Market risk measure from Basel
Minimal capital requirement
Value ar risk requirement
Var(p) = min {m : P{V(0) − V(∆t) ≥ m} ≤ 1 − p} , p = 0.99
Expected shortfall
ES(p) = EP
[V(0) − V(∆t)|V(0) − V(∆t) ≥ Var(p)]
Risk management constraint (Stressed VaR, stress testing), and capital
charge
BCBS, “Minimal Capital Requirements for Market Risks" (standards),
January 2016.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 3 / 27

12. Introduction
Market risk measure from Basel
Minimal capital requirement
Value ar risk requirement
Var(p) = min {m : P{V(0) − V(∆t) ≥ m} ≤ 1 − p} , p = 0.99
Expected shortfall
ES(p) = EP
[V(0) − V(∆t)|V(0) − V(∆t) ≥ Var(p)]
Risk management constraint (Stressed VaR, stress testing), and capital
charge
BCBS, “Minimal Capital Requirements for Market Risks" (standards),
January 2016.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 3 / 27

13. Introduction
Counterparty credit risk management from Basel
Credit risk value at risk to capture the credit downgrade or default
Regulator capital (advanced internal credit model) for counterparty risk:
the effective expected exposure
Additional cost or adjustments for credit risk (XVA)
BCBS, “Margin requirements for non-centrally cleared derivatives",
September 2013; “Review of the credit valuation adjustment risk
framework", July 2015.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 4 / 27

14. Introduction
Counterparty credit risk management from Basel
Credit risk value at risk to capture the credit downgrade or default
Regulator capital (advanced internal credit model) for counterparty risk:
the effective expected exposure
Additional cost or adjustments for credit risk (XVA)
BCBS, “Margin requirements for non-centrally cleared derivatives",
September 2013; “Review of the credit valuation adjustment risk
framework", July 2015.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 4 / 27

15. Introduction
Counterparty credit risk management from Basel
Credit risk value at risk to capture the credit downgrade or default
Regulator capital (advanced internal credit model) for counterparty risk:
the effective expected exposure
Additional cost or adjustments for credit risk (XVA)
BCBS, “Margin requirements for non-centrally cleared derivatives",
September 2013; “Review of the credit valuation adjustment risk
framework", July 2015.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 4 / 27

16. Introduction
Counterparty credit risk management from Basel
Credit risk value at risk to capture the credit downgrade or default
Regulator capital (advanced internal credit model) for counterparty risk:
the effective expected exposure
Additional cost or adjustments for credit risk (XVA)
BCBS, “Margin requirements for non-centrally cleared derivatives",
September 2013; “Review of the credit valuation adjustment risk
framework", July 2015.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 4 / 27

17. Introduction
Liquidity and Operational risk management from Basel
Liquidity coverage ratio
Operational risk capital y
Prob (Loss portfolio <= y) = 0.001
BCBS, “Liquidity Coverage Ratio and liquidity risk monitoring tools",
January 2013
BCBS, “Standardised Measurement Approach for operational risk",
March 2016
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 5 / 27

18. Introduction
Liquidity and Operational risk management from Basel
Liquidity coverage ratio
Operational risk capital y
Prob (Loss portfolio <= y) = 0.001
BCBS, “Liquidity Coverage Ratio and liquidity risk monitoring tools",
January 2013
BCBS, “Standardised Measurement Approach for operational risk",
March 2016
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 5 / 27

19. Introduction
Liquidity and Operational risk management from Basel
Liquidity coverage ratio
Operational risk capital y
Prob (Loss portfolio <= y) = 0.001
BCBS, “Liquidity Coverage Ratio and liquidity risk monitoring tools",
January 2013
BCBS, “Standardised Measurement Approach for operational risk",
March 2016
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 5 / 27

20. Introduction
Liquidity and Operational risk management from Basel
Liquidity coverage ratio
Operational risk capital y
Prob (Loss portfolio <= y) = 0.001
BCBS, “Liquidity Coverage Ratio and liquidity risk monitoring tools",
January 2013
BCBS, “Standardised Measurement Approach for operational risk",
March 2016
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 5 / 27

21. Introduction
Model risk management
Black-Scholes model and 1987 Black Monday
Gaussian copula model and 2007-2008 financial crisis
Model risk capital (inherent risk, residual risk, aggregate risk)
Bayesian model average approach
The worst-case scenario approach
Federal Reserve Supervisory Bulletin 2011-7; Office of the Comptroller
of the Currency (OCC) 2011-12
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 6 / 27

22. Introduction
Model risk management
Black-Scholes model and 1987 Black Monday
Gaussian copula model and 2007-2008 financial crisis
Model risk capital (inherent risk, residual risk, aggregate risk)
Bayesian model average approach
The worst-case scenario approach
Federal Reserve Supervisory Bulletin 2011-7; Office of the Comptroller
of the Currency (OCC) 2011-12
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 6 / 27

23. Introduction
Model risk management
Black-Scholes model and 1987 Black Monday
Gaussian copula model and 2007-2008 financial crisis
Model risk capital (inherent risk, residual risk, aggregate risk)
Bayesian model average approach
The worst-case scenario approach
Federal Reserve Supervisory Bulletin 2011-7; Office of the Comptroller
of the Currency (OCC) 2011-12
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 6 / 27

24. Introduction
Model risk management
Black-Scholes model and 1987 Black Monday
Gaussian copula model and 2007-2008 financial crisis
Model risk capital (inherent risk, residual risk, aggregate risk)
Bayesian model average approach
The worst-case scenario approach
Federal Reserve Supervisory Bulletin 2011-7; Office of the Comptroller
of the Currency (OCC) 2011-12
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 6 / 27

25. Introduction
Model risk management
Black-Scholes model and 1987 Black Monday
Gaussian copula model and 2007-2008 financial crisis
Model risk capital (inherent risk, residual risk, aggregate risk)
Bayesian model average approach
The worst-case scenario approach
Federal Reserve Supervisory Bulletin 2011-7; Office of the Comptroller
of the Currency (OCC) 2011-12
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 6 / 27

26. Introduction
Model risk management
Black-Scholes model and 1987 Black Monday
Gaussian copula model and 2007-2008 financial crisis
Model risk capital (inherent risk, residual risk, aggregate risk)
Bayesian model average approach
The worst-case scenario approach
Federal Reserve Supervisory Bulletin 2011-7; Office of the Comptroller
of the Currency (OCC) 2011-12
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 6 / 27

27. Introduction
Outlines
We discuss six examples from the following topics:
Dynamic asset allocation under risk measures or capital requirement.
Dynamic asset allocation under model risk.
Asset pricing under model risk.
Asset pricing under VaR and other risk measures.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 7 / 27

28. Introduction
Outlines
We discuss six examples from the following topics:
Dynamic asset allocation under risk measures or capital requirement.
Dynamic asset allocation under model risk.
Asset pricing under model risk.
Asset pricing under VaR and other risk measures.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 7 / 27

29. Introduction
Outlines
We discuss six examples from the following topics:
Dynamic asset allocation under risk measures or capital requirement.
Dynamic asset allocation under model risk.
Asset pricing under model risk.
Asset pricing under VaR and other risk measures.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 7 / 27

30. Introduction
Outlines
We discuss six examples from the following topics:
Dynamic asset allocation under risk measures or capital requirement.
Dynamic asset allocation under model risk.
Asset pricing under model risk.
Asset pricing under VaR and other risk measures.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 7 / 27

31. Dynamic asset allocation under risk measures or capital requirement
The economy
A financial market with asset prices S1, · · · , SN
An investor’s trading (percentage of the wealth) strategy (process) is
π1, · · · , πN; and consumption rate c
The wealth process W satisfies
dW = π1W
dS1
S1
+ · · · + πNW
dSN
SN
− cdt
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 8 / 27

32. Dynamic asset allocation under risk measures or capital requirement
Objective function
Markowitz’s mean-variance setting:
max E[WT] −
A
2
Var[WT]
Merton’s dynamic portfolio choice setting:
max E
T
0
e−ρt
u(ct)dt + e−ρT
V(WT)
Roy’s safey-first setting:
max Prob {WT ≥ LT}
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 9 / 27

33. Dynamic asset allocation under risk measures or capital requirement
Constraints from the risk measure requirement
Minimal wealth requirement
WT ≥ KT
Minimal capital requirement or VaR requirement
Var(p) ≤ LT
or Expected shortfall constraint
ES(p) ≤ MT
Ratio constraints (leverage ratio, liquidity ratio etc): The position on the
risk-free asset or liquid asset is higher enough.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 10 / 27

34. Dynamic asset allocation under risk measures or capital requirement
Example 1. Mean-variance under VaR measure
In a complete financial market with unique state price density process
(ζt).
Pre-commitment optimal strategy
max
E[ζT WT ]≤W0,P(WT ≥K)≥α
E[WT] −
A
2
Var(WT)
Consider a sequence of the optimal variance problem for each x
min
E[ζT WT ]≤W0,P(WT ≥K)≥α,E[WT ]=x
E[W2
T]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 11 / 27

35. Dynamic asset allocation under risk measures or capital requirement
Example 1. Mean-variance under VaR measure
In a complete financial market with unique state price density process
(ζt).
Pre-commitment optimal strategy
max
E[ζT WT ]≤W0,P(WT ≥K)≥α
E[WT] −
A
2
Var(WT)
Consider a sequence of the optimal variance problem for each x
min
E[ζT WT ]≤W0,P(WT ≥K)≥α,E[WT ]=x
E[W2
T]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 11 / 27

36. Dynamic asset allocation under risk measures or capital requirement
Example 1. Mean-variance under VaR measure
In a complete financial market with unique state price density process
(ζt).
Pre-commitment optimal strategy
max
E[ζT WT ]≤W0,P(WT ≥K)≥α
E[WT] −
A
2
Var(WT)
Consider a sequence of the optimal variance problem for each x
min
E[ζT WT ]≤W0,P(WT ≥K)≥α,E[WT ]=x
E[W2
T]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 11 / 27

37. Dynamic asset allocation under risk measures or capital requirement
Example 1 (continued)
The corresponding unconstrained problem is to maximize
E[−W2
T] − λ1(E[ζTWT] − W0) − λ2 (α − E [1WT ≥K]) − λ3 (E[WT] − x)
The static optimization problem at each scenario is WT(λ1, λ2, λ3)
satisfying three budget constraint equations.
WT(λ1, λ2, λ3) is the optimal wealth under the VaR constraint.
Reference. Basak and Shapiro (RFS, 2001); Basak, Shapiro and Tepla
(MS, 2006); Boyle and Tian (MF, 2007)
In general, time-consistent strategy for the dynamic mean-variance
preference. See Basak and Chabakauri (RFS, 2010). A Nash equilibrium
for all shelves.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 12 / 27

38. Dynamic asset allocation under risk measures or capital requirement
Example 1 (continued)
The corresponding unconstrained problem is to maximize
E[−W2
T] − λ1(E[ζTWT] − W0) − λ2 (α − E [1WT ≥K]) − λ3 (E[WT] − x)
The static optimization problem at each scenario is WT(λ1, λ2, λ3)
satisfying three budget constraint equations.
WT(λ1, λ2, λ3) is the optimal wealth under the VaR constraint.
Reference. Basak and Shapiro (RFS, 2001); Basak, Shapiro and Tepla
(MS, 2006); Boyle and Tian (MF, 2007)
In general, time-consistent strategy for the dynamic mean-variance
preference. See Basak and Chabakauri (RFS, 2010). A Nash equilibrium
for all shelves.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 12 / 27

39. Dynamic asset allocation under risk measures or capital requirement
Example 1 (continued)
The corresponding unconstrained problem is to maximize
E[−W2
T] − λ1(E[ζTWT] − W0) − λ2 (α − E [1WT ≥K]) − λ3 (E[WT] − x)
The static optimization problem at each scenario is WT(λ1, λ2, λ3)
satisfying three budget constraint equations.
WT(λ1, λ2, λ3) is the optimal wealth under the VaR constraint.
Reference. Basak and Shapiro (RFS, 2001); Basak, Shapiro and Tepla
(MS, 2006); Boyle and Tian (MF, 2007)
In general, time-consistent strategy for the dynamic mean-variance
preference. See Basak and Chabakauri (RFS, 2010). A Nash equilibrium
for all shelves.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 12 / 27

40. Dynamic asset allocation under risk measures or capital requirement
Example 1 (continued)
The corresponding unconstrained problem is to maximize
E[−W2
T] − λ1(E[ζTWT] − W0) − λ2 (α − E [1WT ≥K]) − λ3 (E[WT] − x)
The static optimization problem at each scenario is WT(λ1, λ2, λ3)
satisfying three budget constraint equations.
WT(λ1, λ2, λ3) is the optimal wealth under the VaR constraint.
Reference. Basak and Shapiro (RFS, 2001); Basak, Shapiro and Tepla
(MS, 2006); Boyle and Tian (MF, 2007)
In general, time-consistent strategy for the dynamic mean-variance
preference. See Basak and Chabakauri (RFS, 2010). A Nash equilibrium
for all shelves.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 12 / 27

41. Dynamic asset allocation under risk measures or capital requirement
Example 1 (continued)
The corresponding unconstrained problem is to maximize
E[−W2
T] − λ1(E[ζTWT] − W0) − λ2 (α − E [1WT ≥K]) − λ3 (E[WT] − x)
The static optimization problem at each scenario is WT(λ1, λ2, λ3)
satisfying three budget constraint equations.
WT(λ1, λ2, λ3) is the optimal wealth under the VaR constraint.
Reference. Basak and Shapiro (RFS, 2001); Basak, Shapiro and Tepla
(MS, 2006); Boyle and Tian (MF, 2007)
In general, time-consistent strategy for the dynamic mean-variance
preference. See Basak and Chabakauri (RFS, 2010). A Nash equilibrium
for all shelves.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 12 / 27

42. Dynamic asset allocation under risk measures or capital requirement
Example 2. Safety-first under VaR measure
Consider the problem
max
E[ζT WT ]≤W0,P(WT ≥L)≥α
P(WT ≥ K)
The static unconstrained problem for scenario ω is
max 1WT (ω)≥K + λ1 (W0 − ζTWT) + λ2 1WT (ω)≥L − α
The optimal one is written as WT(λ1, λ2) under budget constraints.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 13 / 27

43. Dynamic asset allocation under risk measures or capital requirement
Example 2. Safety-first under VaR measure
Consider the problem
max
E[ζT WT ]≤W0,P(WT ≥L)≥α
P(WT ≥ K)
The static unconstrained problem for scenario ω is
max 1WT (ω)≥K + λ1 (W0 − ζTWT) + λ2 1WT (ω)≥L − α
The optimal one is written as WT(λ1, λ2) under budget constraints.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 13 / 27

44. Dynamic asset allocation under risk measures or capital requirement
Example 2. Safety-first under VaR measure
Consider the problem
max
E[ζT WT ]≤W0,P(WT ≥L)≥α
P(WT ≥ K)
The static unconstrained problem for scenario ω is
max 1WT (ω)≥K + λ1 (W0 − ζTWT) + λ2 1WT (ω)≥L − α
The optimal one is written as WT(λ1, λ2) under budget constraints.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 13 / 27

45. Dynamic asset allocation under risk measures or capital requirement
Example 2 (continued)
Given another feasible wealth WT,
1WT (λ1,λ2;ω)≥K + λ1(W0 − ζTWT(λ1, λ2; ω)) + λ2(1WT (λ1,λ2;ω)≥L − α
≥ 1WT (ω)≥K + λ1 (W0 − ζTWT) + λ2 1WT (ω)≥L − α
Taking expectation on both sides, we have
P(WT(λ1, λ2) ≥ K) ≥ P(WT ≥ K) + λ1 (W0 − E[ζTWT])
+λ2 (P(WT ≥ L) − α)
≥ P(WT ≥ K).
References: Browne (MS, 2000); Cvitanic and Karatzas (FS, 1992);
Follmer and Leukert (FS, 1999); Spivak and Cvitanic (AAP, 1999).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 14 / 27

46. Dynamic asset allocation under risk measures or capital requirement
Example 2 (continued)
Given another feasible wealth WT,
1WT (λ1,λ2;ω)≥K + λ1(W0 − ζTWT(λ1, λ2; ω)) + λ2(1WT (λ1,λ2;ω)≥L − α
≥ 1WT (ω)≥K + λ1 (W0 − ζTWT) + λ2 1WT (ω)≥L − α
Taking expectation on both sides, we have
P(WT(λ1, λ2) ≥ K) ≥ P(WT ≥ K) + λ1 (W0 − E[ζTWT])
+λ2 (P(WT ≥ L) − α)
≥ P(WT ≥ K).
References: Browne (MS, 2000); Cvitanic and Karatzas (FS, 1992);
Follmer and Leukert (FS, 1999); Spivak and Cvitanic (AAP, 1999).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 14 / 27

47. Dynamic asset allocation under risk measures or capital requirement
Example 2 (continued)
Given another feasible wealth WT,
1WT (λ1,λ2;ω)≥K + λ1(W0 − ζTWT(λ1, λ2; ω)) + λ2(1WT (λ1,λ2;ω)≥L − α
≥ 1WT (ω)≥K + λ1 (W0 − ζTWT) + λ2 1WT (ω)≥L − α
Taking expectation on both sides, we have
P(WT(λ1, λ2) ≥ K) ≥ P(WT ≥ K) + λ1 (W0 − E[ζTWT])
+λ2 (P(WT ≥ L) − α)
≥ P(WT ≥ K).
References: Browne (MS, 2000); Cvitanic and Karatzas (FS, 1992);
Follmer and Leukert (FS, 1999); Spivak and Cvitanic (AAP, 1999).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 14 / 27

48. Dynamic asset allocation under risk measures or capital requirement
Equilibrium under (dynamic) measures
Equilibrium under minimal capital wealth constraint. Grossman and
Zhou (JF, 1996); El Karoui, Jeanbalc-Piques and Lacoste (JEDC 2005);
Equilibrium under Value at risk measure. Jiang and Tian (JFE, 2016).
Equilibrium under liquidity constraint, leverage constraint. Detemple
and Murthy (RFS, 1996), Detemple and Serrat (RFS, 2003).
Equilibrium under margin constraint, Chabakauri (JME 2016), Rytchkov
(JF, 2014); capital requirement, Chabakauri and Han (2016); operational
risk constraint, Basak and Buffa (2016).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 15 / 27

49. Dynamic asset allocation under risk measures or capital requirement
Equilibrium under (dynamic) measures
Equilibrium under minimal capital wealth constraint. Grossman and
Zhou (JF, 1996); El Karoui, Jeanbalc-Piques and Lacoste (JEDC 2005);
Equilibrium under Value at risk measure. Jiang and Tian (JFE, 2016).
Equilibrium under liquidity constraint, leverage constraint. Detemple
and Murthy (RFS, 1996), Detemple and Serrat (RFS, 2003).
Equilibrium under margin constraint, Chabakauri (JME 2016), Rytchkov
(JF, 2014); capital requirement, Chabakauri and Han (2016); operational
risk constraint, Basak and Buffa (2016).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 15 / 27

50. Dynamic asset allocation under risk measures or capital requirement
Equilibrium under (dynamic) measures
Equilibrium under minimal capital wealth constraint. Grossman and
Zhou (JF, 1996); El Karoui, Jeanbalc-Piques and Lacoste (JEDC 2005);
Equilibrium under Value at risk measure. Jiang and Tian (JFE, 2016).
Equilibrium under liquidity constraint, leverage constraint. Detemple
and Murthy (RFS, 1996), Detemple and Serrat (RFS, 2003).
Equilibrium under margin constraint, Chabakauri (JME 2016), Rytchkov
(JF, 2014); capital requirement, Chabakauri and Han (2016); operational
risk constraint, Basak and Buffa (2016).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 15 / 27

51. Dynamic asset allocation under risk measures or capital requirement
Equilibrium under (dynamic) measures
Equilibrium under minimal capital wealth constraint. Grossman and
Zhou (JF, 1996); El Karoui, Jeanbalc-Piques and Lacoste (JEDC 2005);
Equilibrium under Value at risk measure. Jiang and Tian (JFE, 2016).
Equilibrium under liquidity constraint, leverage constraint. Detemple
and Murthy (RFS, 1996), Detemple and Serrat (RFS, 2003).
Equilibrium under margin constraint, Chabakauri (JME 2016), Rytchkov
(JF, 2014); capital requirement, Chabakauri and Han (2016); operational
risk constraint, Basak and Buffa (2016).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 15 / 27

52. Dynamic asset allocation under model risk
Example 3. Robust approach for mean-variance investor
Mean-variance objective in terms of Sharpe ratio
f(x, µ, Σ) =
x µ
√
x Σx
µ is the expected return vector, Σ is the covariance matrix.
The model risk is that we do not know (µ, Σ) precisely. We have a
confidence level that (µ, Σ) belongs to convex, compact region A.
To address the model risk, the robust approach is to maximize
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 16 / 27

53. Dynamic asset allocation under model risk
Example 3. Robust approach for mean-variance investor
Mean-variance objective in terms of Sharpe ratio
f(x, µ, Σ) =
x µ
√
x Σx
µ is the expected return vector, Σ is the covariance matrix.
The model risk is that we do not know (µ, Σ) precisely. We have a
confidence level that (µ, Σ) belongs to convex, compact region A.
To address the model risk, the robust approach is to maximize
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 16 / 27

54. Dynamic asset allocation under model risk
Example 3. Robust approach for mean-variance investor
Mean-variance objective in terms of Sharpe ratio
f(x, µ, Σ) =
x µ
√
x Σx
µ is the expected return vector, Σ is the covariance matrix.
The model risk is that we do not know (µ, Σ) precisely. We have a
confidence level that (µ, Σ) belongs to convex, compact region A.
To address the model risk, the robust approach is to maximize
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 16 / 27

55. Dynamic asset allocation under model risk
Example 3 (continued)
We apply the Sion’s theorem,
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ) = inf
(µ,Σ)∈A
max f(x, µ, Σ)
which is equivalents to
inf
(µ,Σ)∈A
µ Σµ
1/2
A zero-sum game interpretation; Parameter uncertainty
The maximin expected (Gilboa and Schmeidler) utility for ambiguity
averse agent
Reference: Kim and Boyd (SIAM J Optim, 2008); Garlappi, Uppal and
Wang (RFS, 2007); Grinblatt and Linnainman (RFS, 2011).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 17 / 27

56. Dynamic asset allocation under model risk
Example 3 (continued)
We apply the Sion’s theorem,
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ) = inf
(µ,Σ)∈A
max f(x, µ, Σ)
which is equivalents to
inf
(µ,Σ)∈A
µ Σµ
1/2
A zero-sum game interpretation; Parameter uncertainty
The maximin expected (Gilboa and Schmeidler) utility for ambiguity
averse agent
Reference: Kim and Boyd (SIAM J Optim, 2008); Garlappi, Uppal and
Wang (RFS, 2007); Grinblatt and Linnainman (RFS, 2011).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 17 / 27

57. Dynamic asset allocation under model risk
Example 3 (continued)
We apply the Sion’s theorem,
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ) = inf
(µ,Σ)∈A
max f(x, µ, Σ)
which is equivalents to
inf
(µ,Σ)∈A
µ Σµ
1/2
A zero-sum game interpretation; Parameter uncertainty
The maximin expected (Gilboa and Schmeidler) utility for ambiguity
averse agent
Reference: Kim and Boyd (SIAM J Optim, 2008); Garlappi, Uppal and
Wang (RFS, 2007); Grinblatt and Linnainman (RFS, 2011).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 17 / 27

58. Dynamic asset allocation under model risk
Example 3 (continued)
We apply the Sion’s theorem,
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ) = inf
(µ,Σ)∈A
max f(x, µ, Σ)
which is equivalents to
inf
(µ,Σ)∈A
µ Σµ
1/2
A zero-sum game interpretation; Parameter uncertainty
The maximin expected (Gilboa and Schmeidler) utility for ambiguity
averse agent
Reference: Kim and Boyd (SIAM J Optim, 2008); Garlappi, Uppal and
Wang (RFS, 2007); Grinblatt and Linnainman (RFS, 2011).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 17 / 27

59. Dynamic asset allocation under model risk
Example 4. Hansen-Sargent robust approach under model
uncertainty
A standard Black-Scholes economy: constant risk-free interest rate r and
a risky asset S with lognormal return.
In Merton’s model
E
T
0
e−δt c1−A
t
1 − A
dt
The wealth process
dWt = [Wt(r + πt(µ − r)) − ct]dt + σtπtWtdZt
= µ(Wt)dt + σ(Wt)dZt
Two control variables πt, ct. The value function V(W, t) satisfies
0 = supπ,c
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 18 / 27

60. Dynamic asset allocation under model risk
Example 4. Hansen-Sargent robust approach under model
uncertainty
A standard Black-Scholes economy: constant risk-free interest rate r and
a risky asset S with lognormal return.
In Merton’s model
E
T
0
e−δt c1−A
t
1 − A
dt
The wealth process
dWt = [Wt(r + πt(µ − r)) − ct]dt + σtπtWtdZt
= µ(Wt)dt + σ(Wt)dZt
Two control variables πt, ct. The value function V(W, t) satisfies
0 = supπ,c
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 18 / 27

61. Dynamic asset allocation under model risk
Example 4. Hansen-Sargent robust approach under model
uncertainty
A standard Black-Scholes economy: constant risk-free interest rate r and
a risky asset S with lognormal return.
In Merton’s model
E
T
0
e−δt c1−A
t
1 − A
dt
The wealth process
dWt = [Wt(r + πt(µ − r)) − ct]dt + σtπtWtdZt
= µ(Wt)dt + σ(Wt)dZt
Two control variables πt, ct. The value function V(W, t) satisfies
0 = supπ,c
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 18 / 27

62. Dynamic asset allocation under model risk
Example 4. Hansen-Sargent robust approach under model
uncertainty
A standard Black-Scholes economy: constant risk-free interest rate r and
a risky asset S with lognormal return.
In Merton’s model
E
T
0
e−δt c1−A
t
1 − A
dt
The wealth process
dWt = [Wt(r + πt(µ − r)) − ct]dt + σtπtWtdZt
= µ(Wt)dt + σ(Wt)dZt
Two control variables πt, ct. The value function V(W, t) satisfies
0 = supπ,c
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 18 / 27

63. Dynamic asset allocation under model risk
Example 4 (continued)
D(π,c)
V(W, t) =
dE[V]
dt
= Vw[W(r + π(µ − r)) − c] + Vt +
1
2
Vwwπ2
σ2
W2
What if the investor has concerns about the model of the wealth dWt?
The agent accepts it as a “reference model" but it might be “model
mispecification". Then the agent considers alternative models and the
agent guard against an adverse alternative model because of model
uncertainty concern.
Alternative models
dWt = µ(Wt)dt + σ(Wt)[σ(Wt)u(Wt)dt + dZt]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 19 / 27

64. Dynamic asset allocation under model risk
Example 4 (continued)
D(π,c)
V(W, t) =
dE[V]
dt
= Vw[W(r + π(µ − r)) − c] + Vt +
1
2
Vwwπ2
σ2
W2
What if the investor has concerns about the model of the wealth dWt?
The agent accepts it as a “reference model" but it might be “model
mispecification". Then the agent considers alternative models and the
agent guard against an adverse alternative model because of model
uncertainty concern.
Alternative models
dWt = µ(Wt)dt + σ(Wt)[σ(Wt)u(Wt)dt + dZt]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 19 / 27

65. Dynamic asset allocation under model risk
Example 4 (continued)
D(π,c)
V(W, t) =
dE[V]
dt
= Vw[W(r + π(µ − r)) − c] + Vt +
1
2
Vwwπ2
σ2
W2
What if the investor has concerns about the model of the wealth dWt?
The agent accepts it as a “reference model" but it might be “model
mispecification". Then the agent considers alternative models and the
agent guard against an adverse alternative model because of model
uncertainty concern.
Alternative models
dWt = µ(Wt)dt + σ(Wt)[σ(Wt)u(Wt)dt + dZt]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 19 / 27

66. Dynamic asset allocation under model risk
Example 4 (continued)
D(π,c)
V(W, t) =
dE[V]
dt
= Vw[W(r + π(µ − r)) − c] + Vt +
1
2
Vwwπ2
σ2
W2
What if the investor has concerns about the model of the wealth dWt?
The agent accepts it as a “reference model" but it might be “model
mispecification". Then the agent considers alternative models and the
agent guard against an adverse alternative model because of model
uncertainty concern.
Alternative models
dWt = µ(Wt)dt + σ(Wt)[σ(Wt)u(Wt)dt + dZt]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 19 / 27

67. Dynamic asset allocation under model risk
Example 4 (continued)
The agent chooses the adjustment u(Wt) to minimize the expected payoff
but adjusted to reflect an entropy penalty (penalty control term on model
mispecification)
inf
u
DV + u(Wt)σ(Wt)2
Vw +
1
2θ
u(Wt)2
σ(Wt)2
The parameter θ ≥ 0 measures the strength of the reference model for
robustness (θ = 0 corresponds to Merton’s model). θ = θ(Wt, t).
The equation is
0 = supπ,c inf
u
[
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
+Vwπ2
σ2
W2
u +
1
2θ(W, t)
π2
σ2
W2
u2
].
Reference: Hansen and Sargent (AER 2001); Maenhout (RFS, 2004);
Uppal and Wang (JF, 2003)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 20 / 27

68. Dynamic asset allocation under model risk
Example 4 (continued)
The agent chooses the adjustment u(Wt) to minimize the expected payoff
but adjusted to reflect an entropy penalty (penalty control term on model
mispecification)
inf
u
DV + u(Wt)σ(Wt)2
Vw +
1
2θ
u(Wt)2
σ(Wt)2
The parameter θ ≥ 0 measures the strength of the reference model for
robustness (θ = 0 corresponds to Merton’s model). θ = θ(Wt, t).
The equation is
0 = supπ,c inf
u
[
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
+Vwπ2
σ2
W2
u +
1
2θ(W, t)
π2
σ2
W2
u2
].
Reference: Hansen and Sargent (AER 2001); Maenhout (RFS, 2004);
Uppal and Wang (JF, 2003)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 20 / 27

69. Dynamic asset allocation under model risk
Example 4 (continued)
The agent chooses the adjustment u(Wt) to minimize the expected payoff
but adjusted to reflect an entropy penalty (penalty control term on model
mispecification)
inf
u
DV + u(Wt)σ(Wt)2
Vw +
1
2θ
u(Wt)2
σ(Wt)2
The parameter θ ≥ 0 measures the strength of the reference model for
robustness (θ = 0 corresponds to Merton’s model). θ = θ(Wt, t).
The equation is
0 = supπ,c inf
u
[
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
+Vwπ2
σ2
W2
u +
1
2θ(W, t)
π2
σ2
W2
u2
].
Reference: Hansen and Sargent (AER 2001); Maenhout (RFS, 2004);
Uppal and Wang (JF, 2003)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 20 / 27

70. Dynamic asset allocation under model risk
Example 4 (continued)
The agent chooses the adjustment u(Wt) to minimize the expected payoff
but adjusted to reflect an entropy penalty (penalty control term on model
mispecification)
inf
u
DV + u(Wt)σ(Wt)2
Vw +
1
2θ
u(Wt)2
σ(Wt)2
The parameter θ ≥ 0 measures the strength of the reference model for
robustness (θ = 0 corresponds to Merton’s model). θ = θ(Wt, t).
The equation is
0 = supπ,c inf
u
[
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
+Vwπ2
σ2
W2
u +
1
2θ(W, t)
π2
σ2
W2
u2
].
Reference: Hansen and Sargent (AER 2001); Maenhout (RFS, 2004);
Uppal and Wang (JF, 2003)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 20 / 27

71. Asset pricing under model risk
Asset Pricing under No-arbitrage approach
(Fundamental theorem of asset pricing) A financial market is absence of
arbitrage if and only if there exists one equivalent martingale measure.
Traded assets S1, · · · , SN, one numeaire asset B (which is always
positive). Q is an equivalent martingale measure if {Si
B } is a martingale
under Q for each i = 1, · · · , N.
Reference: Delbaen and Schachermayer, “The Mathematics of
Arbitrage, Springer Finance, 2006.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 21 / 27

72. Asset pricing under model risk
Asset Pricing under model uncertainty
Consider the economy with N traded assets i = 1, · · · , N and one
risk-free asset (as a numeaire) B,.
The agent has several models about the risky asset, say Sα
i (t)
representing the asset i’s price at time t in model α ∈ A.
How to compute the “right" price of a derivative X under this model
uncertainty?
What is arbitrage under model uncertainty? A trading strategy is
arbitrage if this strategy yields “arbitrage" in each model since the agent
is not certain which model is a right model.
Replication principle: It holds in all feasible models at the same time.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 22 / 27

73. Asset pricing under model risk
Example 5. Asset Pricing under model uncertainty
An equivalent martingale measure in this setting is one average of
measures under each feasible model (model average principle).
The market is free of arbitrage under model uncertainty if there exists
one such equivalent martingale measure
All available price of X are bounded by infQ α EQα [Xα/B] and
supQ α EQα [Xα/B].
The model risk measure can be measured by the difference of these
bounds.
Reference: Jiang and Tian (IRF, 2017); Cont (MF, 2006); Beissner (ET,
2017); Beissner and Riedel (Econometrica, 2019).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 23 / 27

74. Asset pricing under model risk
Example 5. Asset Pricing under model uncertainty
An equivalent martingale measure in this setting is one average of
measures under each feasible model (model average principle).
The market is free of arbitrage under model uncertainty if there exists
one such equivalent martingale measure
All available price of X are bounded by infQ α EQα [Xα/B] and
supQ α EQα [Xα/B].
The model risk measure can be measured by the difference of these
bounds.
Reference: Jiang and Tian (IRF, 2017); Cont (MF, 2006); Beissner (ET,
2017); Beissner and Riedel (Econometrica, 2019).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 23 / 27

75. Asset pricing under model risk
Example 5. Asset Pricing under model uncertainty
An equivalent martingale measure in this setting is one average of
measures under each feasible model (model average principle).
The market is free of arbitrage under model uncertainty if there exists
one such equivalent martingale measure
All available price of X are bounded by infQ α EQα [Xα/B] and
supQ α EQα [Xα/B].
The model risk measure can be measured by the difference of these
bounds.
Reference: Jiang and Tian (IRF, 2017); Cont (MF, 2006); Beissner (ET,
2017); Beissner and Riedel (Econometrica, 2019).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 23 / 27

76. Asset pricing under model risk
Example 5. Asset Pricing under model uncertainty
An equivalent martingale measure in this setting is one average of
measures under each feasible model (model average principle).
The market is free of arbitrage under model uncertainty if there exists
one such equivalent martingale measure
All available price of X are bounded by infQ α EQα [Xα/B] and
supQ α EQα [Xα/B].
The model risk measure can be measured by the difference of these
bounds.
Reference: Jiang and Tian (IRF, 2017); Cont (MF, 2006); Beissner (ET,
2017); Beissner and Riedel (Econometrica, 2019).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 23 / 27

77. Asset pricing under model risk
Example 5. Asset Pricing under model uncertainty
An equivalent martingale measure in this setting is one average of
measures under each feasible model (model average principle).
The market is free of arbitrage under model uncertainty if there exists
one such equivalent martingale measure
All available price of X are bounded by infQ α EQα [Xα/B] and
supQ α EQα [Xα/B].
The model risk measure can be measured by the difference of these
bounds.
Reference: Jiang and Tian (IRF, 2017); Cont (MF, 2006); Beissner (ET,
2017); Beissner and Riedel (Econometrica, 2019).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 23 / 27

78. Asset pricing under constraint and VaR measures
Asset Pricing under “Convex-type" constraint
One-period economy with finite nature of states Ω = {ω1, · · · , ωK} and
a subjective probability P
N assets S1(ω), · · · , SN(ω), ω ∈ Ω, with time zero price
S1(0), · · · , SN(0). One asset is always positive (for instance, the first
asset).
One investor’s trading strategy H1, · · · , HN.
By a convex-type constraint we mean the range of the trading strategy
belongs to a “convex" subset of RN.
Fundamental theorem of asset pricing under convex-type constraint.
No-arbitrage price of a general contingent claim X by all “feasible
trading strategies" (short-sell, capital requirement, leverage, margin,
transaction-cost, etc).
References: Jouini and Kallal (MF, 1995, JET 1996); Garleanu and
Pedersen (RFS, 2011)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 24 / 27

79. Asset pricing under constraint and VaR measures
Example 6. A no-arbitrage problem under VaR measure
The investor’s loss portfolio is
V0 − V1(ω) =
N
i=1
Hi (Si(0) − Si(ω))
The investor’s admissible strategy satisfies
Prob {ω : V1(ω) − V0(ω) ≥ K} ≥ 95%
H is an arbitrage if V1 − V0 ≥ 0, and E[V1 − V0] > 0.
What is the fundamental theorem in the presence of VaR constraint?
Characterization of no-arbitrage and feasible trading strategy in this
market.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 25 / 27

80. Asset pricing under constraint and VaR measures
Example 6. A no-arbitrage problem under VaR measure
The investor’s loss portfolio is
V0 − V1(ω) =
N
i=1
Hi (Si(0) − Si(ω))
The investor’s admissible strategy satisfies
Prob {ω : V1(ω) − V0(ω) ≥ K} ≥ 95%
H is an arbitrage if V1 − V0 ≥ 0, and E[V1 − V0] > 0.
What is the fundamental theorem in the presence of VaR constraint?
Characterization of no-arbitrage and feasible trading strategy in this
market.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 25 / 27

81. Asset pricing under constraint and VaR measures
Example 6. A no-arbitrage problem under VaR measure
The investor’s loss portfolio is
V0 − V1(ω) =
N
i=1
Hi (Si(0) − Si(ω))
The investor’s admissible strategy satisfies
Prob {ω : V1(ω) − V0(ω) ≥ K} ≥ 95%
H is an arbitrage if V1 − V0 ≥ 0, and E[V1 − V0] > 0.
What is the fundamental theorem in the presence of VaR constraint?
Characterization of no-arbitrage and feasible trading strategy in this
market.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 25 / 27

82. Asset pricing under constraint and VaR measures
Example 6. A no-arbitrage problem under VaR measure
The investor’s loss portfolio is
V0 − V1(ω) =
N
i=1
Hi (Si(0) − Si(ω))
The investor’s admissible strategy satisfies
Prob {ω : V1(ω) − V0(ω) ≥ K} ≥ 95%
H is an arbitrage if V1 − V0 ≥ 0, and E[V1 − V0] > 0.
What is the fundamental theorem in the presence of VaR constraint?
Characterization of no-arbitrage and feasible trading strategy in this
market.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 25 / 27

83. Asset pricing under constraint and VaR measures
Example 6 (continued)
In a dynamic model with assets Si(t), t = 1, · · · , T. For simplicity we
assume that one risk-free asset B (money market account)
In the absence of VaR constraint, there is no arbitrage if and only if there
exists an equivalent martingale measure Q.
The feasible trading strategy satisfies, at each time t, the conditional
probability that Vt − Vt+1 that across a VaR limit is smaller than 5%.
The characterization of no-arbitrage feasible trading strategy.
The feasible trading strategy in terms of other risk measures, ratio
requirement, or capital requirement.
Challenge: Non-convex issue in the optimization problem
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 26 / 27

84. Asset pricing under constraint and VaR measures
Example 6 (continued)
In a dynamic model with assets Si(t), t = 1, · · · , T. For simplicity we
assume that one risk-free asset B (money market account)
In the absence of VaR constraint, there is no arbitrage if and only if there
exists an equivalent martingale measure Q.
The feasible trading strategy satisfies, at each time t, the conditional
probability that Vt − Vt+1 that across a VaR limit is smaller than 5%.
The characterization of no-arbitrage feasible trading strategy.
The feasible trading strategy in terms of other risk measures, ratio
requirement, or capital requirement.
Challenge: Non-convex issue in the optimization problem
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 26 / 27

85. Asset pricing under constraint and VaR measures
Example 6 (continued)
In a dynamic model with assets Si(t), t = 1, · · · , T. For simplicity we
assume that one risk-free asset B (money market account)
In the absence of VaR constraint, there is no arbitrage if and only if there
exists an equivalent martingale measure Q.
The feasible trading strategy satisfies, at each time t, the conditional
probability that Vt − Vt+1 that across a VaR limit is smaller than 5%.
The characterization of no-arbitrage feasible trading strategy.
The feasible trading strategy in terms of other risk measures, ratio
requirement, or capital requirement.
Challenge: Non-convex issue in the optimization problem
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 26 / 27

86. Asset pricing under constraint and VaR measures
Example 6 (continued)
In a dynamic model with assets Si(t), t = 1, · · · , T. For simplicity we
assume that one risk-free asset B (money market account)
In the absence of VaR constraint, there is no arbitrage if and only if there
exists an equivalent martingale measure Q.
The feasible trading strategy satisfies, at each time t, the conditional
probability that Vt − Vt+1 that across a VaR limit is smaller than 5%.
The characterization of no-arbitrage feasible trading strategy.
The feasible trading strategy in terms of other risk measures, ratio
requirement, or capital requirement.
Challenge: Non-convex issue in the optimization problem
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 26 / 27

87. Asset pricing under constraint and VaR measures
Example 6 (continued)
In a dynamic model with assets Si(t), t = 1, · · · , T. For simplicity we
assume that one risk-free asset B (money market account)
In the absence of VaR constraint, there is no arbitrage if and only if there
exists an equivalent martingale measure Q.
The feasible trading strategy satisfies, at each time t, the conditional
probability that Vt − Vt+1 that across a VaR limit is smaller than 5%.
The characterization of no-arbitrage feasible trading strategy.
The feasible trading strategy in terms of other risk measures, ratio
requirement, or capital requirement.
Challenge: Non-convex issue in the optimization problem
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 26 / 27

88. Asset pricing under constraint and VaR measures
Example 6 (continued)
In a dynamic model with assets Si(t), t = 1, · · · , T. For simplicity we
assume that one risk-free asset B (money market account)
In the absence of VaR constraint, there is no arbitrage if and only if there
exists an equivalent martingale measure Q.
The feasible trading strategy satisfies, at each time t, the conditional
probability that Vt − Vt+1 that across a VaR limit is smaller than 5%.
The characterization of no-arbitrage feasible trading strategy.
The feasible trading strategy in terms of other risk measures, ratio
requirement, or capital requirement.
Challenge: Non-convex issue in the optimization problem
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 26 / 27

89. Conclude
Conclusion
Financial risks are modelled in a stochastic setting.
Financial risk management measures for risks
Dynamic asset allocations under risk control
No-arbitrage asset pricing under risk control
There are more challenge than what we know in both theory and practice.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 27 / 27