The Martingale Approach to Operational and Financial Hedgingpages.stern.nyu.edu/~rcaldent/papers/Presentation-Duke.pdf · The Martingale Approach to Operational and Financial Hedging

The Martingale Approach to Operationaland Financial Hedging

Rene Caldentey Martin Haugh

Stern School of Business IE & OR

NYU Columbia University

Fuqua School of BusinessDuke University

October 2005.

Motivation

ProcurementSuppliers' Management

Inventory/ProductionCapacity/Processes

Product Design

DistributionPricing Policies

Demand Forecast

Corporation

The Martingale Approach to Operational and Financial Hedging 2

Motivation

ProcurementSuppliers' Management

Inventory/ProductionCapacity/Processes

Product Design

DistributionPricing Policies

Demand Forecast

Corporation

Problem:

Select an operational strategy and a financial strategy that optimize the value of thecorporation.


Motivation

Operations and services models are now beginning to consider issues related to risk aversion and

hedging

There are many good reasons for hedging

– costs associated with financial distress

– taxes

– costs associated with raising capital

– principal-agent issues

– smoothing earnings for the Wall Street analysts!

And in practice corporations often do hedge

– not inconsistent with Modigliani - Miller theory of corporate finance

Is there a modelling framework that might be useful for operational and financial decision-making?


Motivation

A Methodology that effectively integrates Operational and Financial activities by

– Maximizing the Economic Value of the corporation.

– Modeling the risk preferences of decision makers.

– Using financial markets for hedging purposes.

– Recognizing the “incompleteness”of the problem.

– Taking advantage of all available information.


Motivation







max Economic Valuesubject to problem dynamics

and subject to risk management constraint(s).


Motivation







max Economic Valuesubject to problem dynamics

and subject to risk management constraint(s).

Operational risk in customers’ taste, machine breakdowns, unreliable suppliers, etc...

Financial risk in exchange rates, commodity prices, interest rates, etc...


Outline of Talk

Part I: The Modelling Framework

- Motivation for maximizing economic value of operating profits.

- Financial market & informational structure

Part II: Supply Contracts with Financial Hedging

- The Wholesale Price Contract with Budget Constraint

Part III: Models with Tail Constraints

- A Procurement Example

Conclusions & Further Research


Part I

Modelling Framework

Financial Market & Information


Motivating the Modelling Framework

Consider two series cash-flows arising from two operating policies I1 and I2:

C(1)

(I1) = (c(1)1 (I1), c

(1)2 (I1), . . . , c

(1)n (I1))

C(2)

(I2) = (c(2)1 (I2), c

(2)2 (I2), . . . , c

(2)n (I2))

If probability distribution of C(1) and C(2) identical then most operations models are indifferent

between the two.

However, if C(1) positively correlated with financial market for example, and C(2) negatively

correlated with financial market then economic value of C(2) greater than economic value of C(1).

– models should take this into account as they get ever more complex and include additional

sources of uncertainty

Do so by using an equivalent martingale measure, Q, or stochastic discount factor

– consistent with only a hedging motivation for financial trading


The Modelling Framework

H(I) the time T cumulative payoff from some operating policy, I

– The operating policy may be static or dynamic.





G(θ) the time T gain from a self-financing trading strategy, θ






Operational and financial hedging problem is

minI,G(θ)

EQ [H(I)]

subject to problem dynamics

and subject to risk management constraint(s) ≡R(H(I), G(θ)) ≤ 0 a.s.

EP [R(H(I), G(θ))] ≤ 0.






Operational and financial hedging problem is

minI,G(θ)

EQ [H(I)]

subject to problem dynamics

and subject to risk management constraint(s) ≡R(H(I), G(θ)) ≤ 0 a.s.

EP [R(H(I), G(θ))] ≤ 0.

consistent with only a hedging motivation for financial trading

– G(θ) a martingale under Q so EQ[G(θ)] = 0

– assuming initial capital assigned to hedging strategy is zero


Financial Market Model

Probability space (Ω,F, P).

(W1t , W2t) a standard Brownian motion.

A single risky stock with price dynamics

dXt = µXt dt + σXt dW1t.

Cash account available with r ≡ 0.

Ft is the filtration generated by (W1t , W2t).

FXt is the filtration generated by Xt.

Operating profits, H(I) ∈ FT , a function of (W1t , W2t).

The gain process, G(θ)t of a self-financing trading strategy θt is

G(θ)t =

∫ t

0

θs dXs.

An equivalent martingale measure, Q, under which Xt and G(θ)t are martingales.


Information

In general, operating profits are function of (Xt ,W2t), that is, H(I) ∈ FT .


Information


Incomplete Information:

– Demand, product quality, customer tastes not always observable.– Only information related to Xt is available.

⇒ Trading strategies must satisfy: θt ∈ FXt , t ∈ [0, T ].


Information


Incomplete Information:

– Demand, product quality, customer tastes not always observable.– Only information related to Xt is available.

⇒ Trading strategies must satisfy: θt ∈ FXt , t ∈ [0, T ].

Complete Information:

– The evolution of Xt and B2t are both available.

⇒ Trading strategies satisfy: θt ∈ Ft, t ∈ [0, T ].


Part II

Supply Contracts with Financial Hedging

The Wholesale Price Contract with Budget Constaint


The Wholesale Price Contract

ClearancePrice

! "

# ! $% ! &$

Non-Cooperative Operation:

Solution Concept: Stackelberg game.

1. At time t = 0, Manufacturer offers a wholesale price w to the Retailer.2. At time t = 0, Retailer orders a quantity q.3. At time t = T , a random clearance price P (q) is realized: P (q) = A− q.

The random price A is correlated to the financial market Xt.

Retailer operates under a budget constraint: wq ≤ B.


Types of Contracts

timet = 0 t = T

!"Production takes place


Types of Contracts

timet = 0 t = T


timet = 0 t = T

ττττ ττττ

t = τ

!ττττ

"#

Production takes place

$ % % & %'(( )


Types of Contracts

timet = 0 t = T


timet = 0 t = T

ττττ ττττ

t = τ

!ττττ

"#


$ % % & %'(( )

timet = 0 t = T

ττττ ττττ

t = τ

!" #$% &ττττ ' ( )$ * *

ττττ ' (

Production takes placeFinancial hedging takes place

+% , , %- ,.// 0


Types of Contracts

timet = 0 t = T


timet = 0 t = T

ττττ ττττ

t = τ

!ττττ

"#


$ % % & %'(( )

timet = 0 t = T

ττττ ττττ

t = τ

!" #$% &ττττ ' ( )$ * *

ττττ ' (


+% , , %- ,.// 0

Remark: The time τ can be a decision variable: a deterministic time or a F Xτ -stopping time.


Some Notation

∗) (Ω,F,Q) probability space.

∗) Xt: Time t value of a Q-martingale tradable security (t ∈ [0, T ]).

∗) Ft = σ(Xs ; 0 ≤ s ≤ t) filtration generated by Xt with FT ⊆ F .

∗) EQτ [E] := EQ[E|Fτ ] for all E ∈ F .

∗) P (q) = A− q: retail price at time T with A ∈ F .

∗) Aτ = EQ[A|Fτ ] for any Ft-stopping time τ ≤ T .

∗) cτ : per unit manufacturing cost at time τ ∈ [0, T ].

∗) wτ ∈ Fτ : manufacturer’s wholesale price menu.

∗) qτ ∈ Fτ : retailer’s ordering quantity menu.

∗) Gτ ∈ Fτ : retailer’s trading gains (or losses) with EQ[Gτ ] = 0.

∗) Budget constraint: wτ qτ ≤ Bτ := B + Gτ .

Assumption. For all τ ∈ [0, T ], Aτ ≥ cτ .


Flexible Contract

Consider a fixed τ ∈ [0, T ].

Step 1: At t = 0, the manufacturer offers a wholesale menu wτ ∈ Fτ .

Step 2: A retailer, with no access to the financial market, decides the ordering level qτ ∈ Fτ solving

ΠFR(wτ) = EQ0

[maxqτ ≥0

EQτ [(A− qτ − wτ) qτ ]

]

subject to wτ qτ ≤ B, for all ω ∈ Ω.

The optimal solution (retailer’s reaction) is

q(wτ) = min

(Aτ − wτ

2

)+

,B

wτ

.

Step 3: The manufacturer problem is (Stackelberg leader)

ΠFM = EQ0

[maxwτ≥cτ

(wτ − cτ) qτ(wτ)]

.


Flexible Contract

Proposition. (Flexible Contract Solution)

Under Assumption , the equilibrium solution for the flexible contract is

wFτ =

Aτ + δ Fτ

2and q

Fτ =

Aτ − δ Fτ

4,

where

δFτ := max

cτ ,

√(A2

τ − 8 B)+

.

The equilibrium expected payoffs of the players are then given by

ΠFM|τ =

(Aτ + δ Fτ − 2 cτ) (Aτ − δ F

τ)

8and Π

FR|τ =

(Aτ − δ Fτ)

2

16.

Remarks:

-) δ Fτ is a modified production cost (≥ cτ) that is induced by a limited budget B.

-) w Fτ decreases in B and q F

τ , Π FM|τ and Π F

R|τ increase in B.


Flexible Contract vs. Simple Contract

Define wF:= EQ0 [w

Fτ ], q

F:= EQ0 [q

Fτ ], Π

FM := EQ0 [Π

FM|τ ] and Π

FR := EQ0 [Π

FR|τ ].

Proposition. Suppose that B ≤ A2τ−c2τ8 almost surely. Then

wF ≤ w

S, q

F ≥ qS, Π

FM ≤ Π

SM and Π

FR ≥ Π

SR.

However, if B ≥ max

A2

τ−c2τ8 ,

A2−c208

almost surely then

wF= w

S+

cτ − c0

2and q

F= q

S − cτ − c0

4

and ΠFM ≥ Π

SM and Π

FR ≥ Π

SR if and only if EQ0 [(Aτ − cτ)

2] ≥ (A− c0)

2.


Flexible Contract vs. Simple Contract

0 0.6 1.21

1.2

1.4

1.6

1.8

2Wholesale Price

0 0.6 1.20

0.1

0.2

0.3

0.4

0.5Ordering Level

0 0.6 1.2

0.5

1

1.5

2Flexible vs. Simple

0 0.6 1.21

1.2

1.4

1.6

1.8

2

0 0.6 1.20

0.1

0.2

0.3

0.4

0.5

Budget (B) 0 0.6 1.2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Case 1

Case 2

Flexible

Simple

Flexible

Simple

Flexible

Flexible

Simple

Simple

ΠRF/Π

RS

ΠRF/Π

RS

ΠPF/Π

PS

ΠPF/Π

PS

Aτ ∼ Uniform[1, 3], c0 = 0.3, cτ = 0.35 (case 1) and cτ = 0.7 (case 2).


Flexible Contract: Efficiency

0 0.5 10.3

0.35

0.4

0.45

0.5

0.55

0 0.5 12

2.5

3

3.5

0 0.5 10.2

0.3

0.4

0.5

0.6

0.7

0 0.5 10.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0 0.5 11.3

1.4

1.5

1.6

1.7

1.8

0 0.5 10.15

0.2

0.25

0.3

0.35

0.4

Budget (B)

QFτ WFτ PFτ

Aτ ≥ 3 cτ

Aτ ≤ 3 cτ

Aτ = 2, cτ = 0.6 (top) and cτ = 1.2 (bottom).


Flexible Contract with Financial Hedging

timet = 0 t = T

ττττ ττττ

t = τ

!" #$% &ττττ ' ( )$ * *

ττττ ' (


+% , , %- ,.// 0



timet = 0 t = T

ττττ ττττ

t = τ

!" #$% &ττττ ' ( )$ * *

ττττ ' (


+% , , %- ,.// 0

Step 1: At t = 0, and for a fixed τ ≤ T , the manufacturer offers a price menu wτ ∈ F Xτ .



timet = 0 t = T

ττττ ττττ

t = τ

!" #$% &ττττ ' ( )$ * *

ττττ ' (


+% , , %- ,.// 0


Step 2: In response, at t = 0, the retailer selects an optimal ordering menu q∗τ(wτ) ∈ F Xτ solving

ΠHR(wτ) = max

qτ≥0, GτEQ [(A− qτ) qτ − wτ qτ ]

subject to wτ qτ ≤ B + Gτ , for all ω ∈ Ω

EQ[Gτ ] = 0.



timet = 0 t = T

ττττ ττττ

t = τ

!" #$% &ττττ ' ( )$ * *

ττττ ' (


+% , , %- ,.// 0


Step 2: In response, at t = 0, the retailer selects an optimal ordering menu q∗τ(wτ) ∈ F Xτ solving

ΠHR(wτ) = max

qτ≥0, GτEQ [(A− qτ) qτ − wτ qτ ]

subject to wτ qτ ≤ B + Gτ , for all ω ∈ Ω

EQ[Gτ ] = 0.

Step 3: The manufacturer selects the optimal wholesale price menu w∗τ solving

ΠHM(wτ) = max

wτEQ

[wτ q

∗τ(wτ)− cτ q

∗τ(wτ)

].



Proposition. (Retailer’s Optimal Strategy)

Let Qτ , X and X c be defined as follows

Qτ ,(

Aτ − wτ

2

)+

, X , ω ∈ Ω : B ≥ Qτ wτ , and X c , Ω− X .





Qτ ,(

Aτ − wτ

2

)+


Case 1: Suppose that EQ [Qτ wτ ] ≤ B. Then q∗τ(wτ) = Qτ and there are infinitely many

choices of the optimal claim, Gτ . One natural choice is to take

Gτ = [Qτ wτ − B] ·

δ if ω ∈ X1 if ω ∈ X c

δ ,∫Xc [Qτ wτ − B] dQ∫X [B −Qτ wτ ] dQ

.





Qτ ,(

Aτ − wτ

2

)+


Case 1: Suppose that EQ [Qτ wτ ] ≤ B. Then q∗τ(wτ) = Qτ and there are infinitely many

choices of the optimal claim, Gτ . One natural choice is to take

Gτ = [Qτ wτ − B] ·

δ if ω ∈ X1 if ω ∈ X c

δ ,∫Xc [Qτ wτ − B] dQ∫X [B −Qτ wτ ] dQ

.

Remark: In this case, it is possible to completely eliminate the budget constraint by trading in the

financial market.



Proposition. (Continuation)

Case 2: Suppose that B < EQ [Qτ wτ ]. Then

qτ(wτ) =

(Aτ − wτ (1 + λ)

2

)+

where λ ≥ 0 solves EQ[

wτ

(Aτ − wτ (1 + λ)

2

)+]

= B.





qτ(wτ) =

(Aτ − wτ (1 + λ)

2

)+


wτ

(Aτ − wτ (1 + λ)

2

)+]

= B.

Proposition. (Producer’s Optimal Strategy and the Stackelberg Solution)

Let φ∗ , inf

φ ≥ 1 such that EQ

[(A2

τ−(φ cτ )2

8

)+]≤ B

.

Then, w∗τ = Aτ+φ∗ cτ

2 and q∗τ =(

Aτ−φ∗ cτ4

)+

and the players’ expected payoffs satisfy

ΠHM|τ =

(Aτ + φ∗ cτ − 2cτ) (Aτ − φ∗ cτ)+

8and Π

HR|τ =

((Aτ − φ∗ cτ)+)2

16.





qτ(wτ) =

(Aτ − wτ (1 + λ)

2

)+


wτ

(Aτ − wτ (1 + λ)

2

)+]

= B.

Proposition. (Producer’s Optimal Strategy and the Stackelberg Solution)

Let φ∗ , inf

φ ≥ 1 such that EQ

[(A2

τ−(φ cτ )2

8

)+]≤ B

.

Then, w∗τ = Aτ+φ∗ cτ

2 and q∗τ =(

Aτ−φ∗ cτ4

)+

and the players’ expected payoffs satisfy

ΠHM|τ =

(Aτ + φ∗ cτ − 2cτ) (Aτ − φ∗ cτ)+

8and Π

HR|τ =

((Aτ − φ∗ cτ)+)2

16.

Remark: When q∗τ = 0, the manufacturer decides to overcharge the retailer making the supply chain

non-operative. This is never the case if the retailer does not have not access to the financial market.


Flexible Contract with Financial HedgingProposition. The manufacturer always prefers the H-contract to the F-Contract. On the other hand,

the retailer’s preferences are

!"#$ %"&"'($


Flexible Contract with Financial HedgingProposition. The manufacturer always prefers the H-contract to the F-Contract. On the other hand,

the retailer’s preferences are

!"#$ %"&"'($

0 0.5 1 1.5 2

1.2

1.4

1.6

1.8

2

2.2

2.4

Wholesale Price

Budget (B)0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4Ordering Level

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35Producer’s Payoff

0 0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18Retailer’s Payoff

Budget (B)

Budget (B)Budget (B)

H−Contract

F−Contract

H−Contract

F−Contract

F−Contract F−Contract

H−Contract

H−Contract


Flexible Contract with Financial HedgingEfficiency

On path-by-path basis, the Centralized system is not necessarily more efficient thanthe Decentralized Supply Chain!

∃ω ∈ Ω such that q HC|τ = 0 and q H

τ > 0.

Remarks: This is never the case under a Flexible Contract without Hedging.


Flexible Contract with Financial HedgingEfficiency

On path-by-path basis, the Centralized system is not necessarily more efficient thanthe Decentralized Supply Chain!

∃ω ∈ Ω such that q HC|τ = 0 and q H

τ > 0.

Remarks: This is never the case under a Flexible Contract without Hedging.

On average, the Centralized solution is more efficient than the Decentralizedsolution.

EQ0[qHC|τ ] ≥ EQ0[q H

τ ].


Summary

Simple extension to the traditional wholesale contract.


Summary


The proposed procurement contracts uses the Financial Market as:

– A source of public information upon which contracts can be written.

– A means for financial hedging to mitigate the impact of the budget constraint.


Summary





Consistent with the notions of production postponement and demand forecast.


Summary






Managerial Insights:

– Manufacturer and Retailer incentives are not always aligned as a function of B.

– Manufacturer prefers retailers that have access to the financial market.

– With hedging, the supply chain might not operate in some states ω ∈ Ω.

– In some cases, financial hedging eliminates the budget constraint.

– Optimal time τ of the contract balances Var(Aτ) and cτ .


Summary






Managerial Insights:

– Manufacturer and Retailer incentives are not always aligned as a function of B.

– Manufacturer prefers retailers that have access to the financial market.

– With hedging, the supply chain might not operate in some states ω ∈ Ω.

– In some cases, financial hedging eliminates the budget constraint.

– Optimal time τ of the contract balances Var(Aτ) and cτ .

Extensions:

– Other types of contracts: quantity discount, buy-back, etc.

– Include other sources of uncertainty: exchange rates, interest rates, credit risk.


Part III

Models with Tail Constraints

Example: A Procurement Model


Problem Formulation

maxI,θ

EQ[H(I)]

subject to EP[R(H(I) + G(θ))] ≥ H.


Problem Formulation

maxI,θ

EQ[H(I)]


1−η1−η1−η1−η

η

!""# $%&'


Problem Formulation

maxI,θ

EQ[H(I)]


1−η1−η1−η1−η

η

!""# $%&'

Value-at-Risk (VaRη): Hη ≥ H.


Problem Formulation

maxI,θ

EQ[H(I)]


1−η1−η1−η1−η

η

!""# $%&'


Conditional Value-at-Risk (CVaRη): EP[H(I) |H(I) ≤ Hη] ≥ H.


Problem Formulation

maxI,θ

EQ[H(I)]


1−η1−η1−η1−η

η

!""# $%&'


Conditional Value-at-Risk (CVaRη): EP[H(I) |H(I) ≤ Hη] ≥ H.

Mean & Standard-Deviation (MStdη): EP[H(I)]−√

η1−η

√Var(H(I)) ≥ H.


Tail Constraints: Bounds

Rockafellar and Ursayev (2002):

CVaRη(X) = minζ

ζ +

1

1− ηEP[(X − ζ)

+]

.




CVaRη(X) = minζ

ζ +

1

1− ηEP[(X − ζ)

+]

.

Define EP[X] = µX and Var(X) = σ2X then Gallego (1992):

EP[(X − ξ)

+]≤

√σ2

X + (ξ − µX)2 − (ξ − µX)

2.




CVaRη(X) = minζ

ζ +

1

1− ηEP[(X − ζ)

+]

.


EP[(X − ξ)

+]≤

√σ2

X + (ξ − µX)2 − (ξ − µX)

2.

Combining these results we get

CVaRη(X) ≤ µX +

√η

1− ησX , MStdη(X).




CVaRη(X) = minζ

ζ +

1

1− ηEP[(X − ζ)

+]

.


EP[(X − ξ)

+]≤

√σ2

X + (ξ − µX)2 − (ξ − µX)

2.

Combining these results we get

CVaRη(X) ≤ µX +

√η

1− ησX , MStdη(X).

Tail Constraint Ordering

VaRη ≥ CVaRη ≥ MStdη.


Problem Formulation

For some η ∈ [0, 1] and H, we would like to solve

maxI,G(θ)

EQ [H(I)]

subject to EP[H(I) + G(θ)]−√(

η

1− η

)Var (H(I) + G(θ)) ≥ H,

EQ[G(θ)] = 0 and θt ∈ FXt︸︷︷︸

Incomplete Inf.

or θt ∈ Ft︸︷︷︸Complete Inf.

.


Problem Formulation


maxI,G(θ)

EQ [H(I)]


η

1− η

)Var (H(I) + G(θ)) ≥ H,


Incomplete Inf.


.

Two-step Optimization:

Step 1: (Hedging Policy) For a fixed operating policy, I, we solve the hedging problem.

V (I) , maxG(θ)

MStdη(H(I) + G(θ)) subject to EQ[G(θ)] = 0.


Problem Formulation


maxI,G(θ)

EQ [H(I)]


η

1− η

)Var (H(I) + G(θ)) ≥ H,


Incomplete Inf.


.

Two-step Optimization:

Step 1: (Hedging Policy) For a fixed operating policy, I, we solve the hedging problem.

V (I) , maxG(θ)

MStdη(H(I) + G(θ)) subject to EQ[G(θ)] = 0.

Step 2: (Operational Policy) Then, we compute the optimal operational policy solving

maxIEQ[H(I)] subject to V (I) ≥ H.


Incomplete Information Model: Step 1

Theorem. (Incomplete Information Hedging)

Let us denote by π the Radon-Nikodym derivative of Q with respect to P. Suppose that the hedging

strategy θ is restricted to be FXT -measurable. Then, the following two cases are possible:

Case 1: If η ≥ EP[π2]−1

EP[π2]then

V (I) = EQ[H(I)]−[(

η − (1− η) (EP[π2]− 1)

1− η

)Var(H(I)− EP[H(I)|FX

T ])

]12

.

Case 2: If η < EP[π2]−1

EP[π2]then

V (I) = +∞.

Remark: In Case 2, financial trading has removed the risk management constraint.


Incomplete Information Model: Step 2

Case 1: If η ≥ EP[π2]−1

EP[π2], then the (financially hedged) operational problem is

maxI

EQ[H(I)]

s.t. EQ[H(I)]−[(

η − (1− η) (EP[π2]− 1)

1− η

)Var(H(I)− EP[H(I)|FX

T ])

]12

≥ H.

Case 2: If η < EP[π2]−1

EP[π2], then the (financially hedged) operational problem is

maxI

EQ[H(I)]

Remarks:

X Case 2 is the standard risk neutral operational model but under the “appropriate” EMM.

X If H(I) ∈ FXT then Case 1 reduces to

maxI

EQ[H(I)] s.t. EQ[H(I)] ≥ H.


A Procurement Example

!

0 T

The Retailer’s profit is H(I) = (P (I)− c) I.

I : Ordering Level at time 0.

P (I) : Clearance Price at time T as a function of I.

c : Per unit purchasing cost.

Problem:maxI,G

EQ[H(I)] subject to MStdη(H(I) + G) ≥ H.

Assumptions: Xt ∼ (µ, σ)-GBM : dXt = µ Xt dt + σ Xt dW1t.

Linear Demand Model: P (I) = A− I, A = ζ + γ ln(XT ) + α W2T .


A Procurement Example (contd’)

Solution without Financial Hedging

0 800 1600−2

0

10

20x 10

5

Objective: EQ[H(I)]

MStdη Constraint

No Hedging

I*

H

Ic



Solution with Financial Hedging if η ≥ EP[π2]−1

EP[π2]

0 800 1600−2

0

10

20x 10

5

Objective: EQ[H(I)]

MStdη Constraint

No Hedging

I*

H

MStdη Constraint

With Incomplete Hedging

MStdη Constraint

With Complete Hedging



0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

Hoptc

/Hoptu

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

T

Hoptc

/Hoptu

No Hedging

Incomplete

Complete

Incomplete

No Hedging

Complete


Conclusions

A modelling framework for selecting operating and financial hedging strategies

– Consistent with maximizing economic value of the firm.

– Models risk preferences using risk management constraints.

– The only motivation for financial trading is hedgingin particular, no profit motivation for hedging.

– Structural properties result in considerable tractability

– Considers different information structures.

Dynamic trading

– Reduces, and in some cases, eliminates the negative effects of risk constraints.

– Improves the overall performance/efficiency of the operations.

Further Research


Documents

The Martingale Approach to Operational and Financial Hedgingpages.stern.nyu.edu/~rcaldent/papers/Presentation-Duke.pdf · The Martingale Approach to Operational and Financial Hedging