Risk Sensitive Inventory Management withSar‹yer, •Istanbul Games and Decisions in Reliability...

Risk Sensitive Inventory Management withFinancial Hedging1

Süleyman ÖzekiciKoç University

Department of Industrial EngineeringSar¬yer, ·Istanbul

Games and Decisions in Reliability and Risk WorkshopStatistical and Applied Mathematical Sciences Institute

16-20 May 2016, Durham, NC

1Joint work with F. Karaesmen and C. CanyakmazS. Özekici Financial Hedging in Inventory Management GDRR 1 / 24

Outline

Introduction and Motivation

Literature

Static Financial Hedging Model

Dynamic Financial Hedging Model

Numerical Illustration

Future Research

S. Özekici Financial Hedging in Inventory Management GDRR 2 / 24

Introduction and Motivation

Demand and supply uncertainties are primary sources of riskfor inventory managers

Price volatility is also major source of risk

Commodity based raw material prices, exchange rate�uctuations are typical examples

Exposure to these risks have signi�cant impacts on inputcosts, sales prices and volume

Successful inventory management can create even more valuein �uctuating price environments

One can use �nancial markets to cope with price andexchange rate �uctuations


Risk-Sensitive Inventory Management

Typical models are based on adjusting the replenishmentpolicy to optimize a certain measure such as

the mean of the cash �owthe variance of the cash �owutility of the decision makerprobability of achieving a certain target pro�t

Expected Utility Models: Boukazis and Sobel [OR�92],Agrawal and Seshandi [MSOM�00], Chen et al. [OR�07]

Mean-Variance Models: Lau [JORS�80], Berman andSchnabel [IJSS�86], Chen and Federgruen [WP�00], Wu et al.[OMEGA�09]

Value-at-Risk Models: Luciano et al. [IJPE�03], Tapiero[EJOR�05], Özler et al. [IJPE�09]

Minimum-Variance Models: Okyay et al. [ORS�14], Tekinand Özekici [IIETr�15]


Financial Hedging

The use of �nancial markets to reduce the price and inventoryrisk has gained importance

A �nancial hedge is an investment position to reduce thee¤ect of potential losses/gains incurred from anotherinvestment

Hedges can be constructed using stocks, indices, futures,options, swaps, etc.

Future contracts are most widely used to hedge the risks incommodity prices, energy prices, foreign currencies, interestrates, etc.

Gaur and Seshadri [MSOM�05], Caldentey and Haugh[MOR�06], Chod et al. [MS�10], Okyay et al. [ORS�14], Tekinand Özekici [IIETr�15]


The Model

Single-period inventory model with �uctuating sales prices andmodulated demandSales period is [0; T ]Stochastic price process P = fPt; t � 0g (compounded to time T )P0 : Purchase price at time 0Sales price at time t is f (Pt) (for example, f (p) = �p; where � > 1 isthe markup)The customer arrival process is N = fNt : t � 0g with intensity process� = f�t = � (Pt) ; t � 0g (Doubly stochastic Poisson process)h(p): holding cost, b(p): backordering costCash �ow at time T under order-up-to decision y is

CF (y;N ;P) = �P0y+NTXj=1

f�PTj��b (PT ) (NT � y)+ + h (PT ) (y �NT )+

�


Price and Arrival Processes


Security Prices and Trading Times

Price related risks: sales price and demand (also purchase price inmultiperiod inventory model)We assume that there are M �nancial securities which arecorrelated with the price process PS(i) =

nS(i)t ; t � 0

o: The price process for security i

(compounded to time T )S =

�S(1); S(2); :::; S(M)

�: The vector of security price processes

T = (t0; t1; t2; :::; tn�1) : Prespeci�ed trading times (t0 = 0,tn = T )

�k =��(1)k ; �

(2)k ; :::; �

(M)k

�: Portfolio decision at time tk

� = (�0; �1; :::; �n�1) : Financial hedging strategy or portfolio


Financial Cash Flow

The �nal payo¤ of the �nancial portfolio at time T as

G (�;S) =MXi=1

n�1Xk=0

�(i)k

�S(i)tk+1

� S(i)tk�=

n�1Xk=0

�Tk 4 Sk = �T 4 S

4Sk = Stk+1 � Stk : Vector of net payo¤s for holding one unitof each security during (tk; tk+1)

�k;4Sk are M � 1 column vectors�;4S are Mn� 1 column vectors


The Hedged Cash Flow

Total hedged cash �ow at time T is

HCF (�; y;N ;P; S) = CF (y;N ;P) +G (�; S)

The objective of the inventory manager is to solve

maxy�0

E [HCF (� (y) ; y;N ;P; S)]

subject to� (y) = argmin

�V ar (HCF (�; y;N ;P; S))


Static Model: Minimum-Variance Portfolio

Portfolio chosen once at the beginning only (n = 1; t0 = 0)Covariance matrix

Cij = Cov�S(i)T ; S

(j)T

�Covariance vector

�i (y) = Cov�CF (y;N ;P) ; S(i)T

�= Cov

0@NTXj=1

f�PTj�; S

(i)T

1A� Cov �h (PT ) (y �NT )+ ; S(i)T ��Cov

�b (PT ) (NT � y)+ ; S(i)T

�Theorem

V ar (�; y;N ;P; S) is convex in � and minimum-variance portfolio for order quantityy is given by

�� (y) = �C�1� (y)


Static Model: Optimal Base-Stock Level

Assumption

The function

E�(h (PT ) + b (PT )) 1fNT�yg

��Cov

�(h (PT ) + b (PT )) 1fNT�yg; ST

�TC�1E [4S]

is increasing in y:

TheoremThe optimal order quantity that maximizes the expected cash �ow using theminimum-variance portfolio �� (y) = �C�1� (y) is

y� = inf�y � 0;E

�(h (PT ) + b (PT )) 1fNT�yg

��Cov

�(h (PT ) + b (PT )) 1fNT�yg; ST

�TC�1E [4S]

o� E [b (PT )]� P0 � Cov (b (PT ) ; ST )T C�1E [4S]

o


Static Model: Zero Expected Financial Gain

If E [4S] = 0 (or security prices are martingales), thenAssumption 1 is always satis�edIn this case, optimal order-up-to level is

y�= inf�y � 0;E

�(h (PT ) + b (PT )) 1fNT�yg

�� E [b (PT )]� P0

This is not the newsvendor solution since N and P are dependentWe obtain the newsvendor solution

y�= inf

�y � 0;PfNT � yg �

b� ch+ b

�if P0 = c; h (p) = h and b (p) = b


Static Model: Demand is Independent of Prices

If N is a Poisson process with rate � and independent of P , then Assumption 1 is always satis�ed.In this case

� (y) = �

TZ0

Cov (f (Pt) ; ST ) dt� E�(y �NT )+

�Cov (h (PT ) ; ST )

�E�(NT � y)+

�Cov (b (PT ) ; ST )

and the optimal order-up-to level is

y� = inf

(y � 0;P fNT � yg �

E [b (PT )]� P0 � Cov (b (PT ) ; ST )T C�1E [4S]E [h (PT )] + E [b (PT )]� Cov (h (PT ) + b (PT ) ; ST )T C�1E [4S]

)

If E [4S] = 0; then

y� = inf

�y � 0 : P fNT � yg �

E [b (PT )]� P0E [h (PT )] + E [b (PT )]

�


Static Model: Single Financial Security (Future)

If S is a future written on PT , then S0 = P0 and ST = PTLet us assume that b (PT ) = b+ PT ; h (PT ) = h� PTIn this case, optimal order-up-to level is

y� = inf

�y � 0 : P fNT � yg �

b+ (E [PT ]� P0)b+ h+ (1� )P0

�and the minimum-variance portfolio is

�� = E�(NT � y)+

�� E

�(y �NT )+

��

TZ0

�tdt

where

�t =Cov (f (Pt) ; PT )

V ar (PT )


Dynamic Model

Trading times are t0 = 0; t1; t2; :::; tn�1Assumption: Security prices are martingales

In this case, the minimum-variance problem for every ydecision is reduced to

min�Eh�CF (y;N;P) + �T 4 S

�2iThis objective is separable in terms of dynamic programming


Dynamic Programming: State Transitions

We use four states X;W;P; S

Inventory level transitions

Xk+1 = Xk �N[tk;tk+1]X0 = y

Wealth level transitions

Wk+1 = Wk +R[tk;tk+1] + �Tk 4 Sk

W0 = 0

Operational revenue during [tk; tk+1]

R[tk;tk+1] =

N[tk;tk+1]Xj=1

f�PTj+tk

�S. Özekici Financial Hedging in Inventory Management GDRR 17 / 24

Dynamic Programming Formulation

Objective function is

E

24 CF (y;N ;P) + n�1Xk=0

�Tk 4 Sk

!235 = E h�Wn ��b (Ptn) (�Xn)+ + h (Ptn)X+

n

��2iDP formulation is

Vk (x;w; p; s) = min�kE�Vk+1

�Xtk �N[tk;tk+1];Wtk +R[tk;tk+1] + �k 4 Sk; Ptk+1 ; Stk+1

�jXtk = x;Wtk = w;Ptk = p; Stk = s]

with boundary condition

Vn (x;w; p; s) =�w � b (p) (�x)+ � h (p)x+

�2Here Vk is the value function at trading time tk


Notation

Ck (s)ij = Cov�S(i)tk+1

; S(j)tk+1

j S(i)tk = s(i); S

(j)tk= s(j)

�R[tk;tn] =

n�1Xj=k

R[tj ;tj+1]

�k (x; p; s)j = Cov�R[tk;tn] � b (Ptn)

�N[tk;tn] � x

�+ � h (Ptn) �x�N[tk;tn]�+ ; S(j)tk+1 jPtk = p; S(j)tk = s(j)�

gk (x;w; p) = E

��w +R[tk;tn] � b (Ptn)

�N[tk;tn] � x

�+ � h (Ptn) �x�N[tk;tn]�+�2 jPtk = p�

hk (x; p; s) = ��k (x; p; s)T Ck (s)�1 �k (x; p; s)+E�hk+1

�x�N[tk;tk+1]; Ptk+1 ; Stk+1

�j Ptk = p; Stk = s

�


The Optimal Solution

TheoremValue function for period k is

Vk (x;w; p; s) = gk (x;w; p) + hk (x; p; s)

and the minimum-variance portfolio is

��k (x; p; s) = �Ck (s)�1 �k (x; p; s)

TheoremOptimal order-up-to level that maximizes the expected hedgedcash �ow is

y� = inf�y � 0;E

�(h (PT ) + b (PT )) 1fNT�yg

�� E [b (PT )]� P0

Risk-neutral solutionMartingale security price processes


Numerical Illustration

Schwartz and Smith [MS�00] uses a price model that describeslong and short term behaviours of commodities

Pt = e�t+�t

whered�t = ��tdt+ ��dW

(�)t

is an Ornstein-Uhlenbeck process that models the short-termdeviations (mean reverting to zero) and

d�t = ��dt+ ��dW(�)t

is a geometric Brownian motion that models the long-termequilibrium (dW (�)

t dW(�)t = �dt)

We use the risk-neutral version where

dPt = (�� + ��)PtdW1t + ��

p1� �2PtdW 2

t


Financial Securities and Parameters

T = 1; f (p) = 2p, b (p) = 4 + p, h (p) = 1,�t = (90� 1:4Pt)+

P0 = 20; �� = 0:25; �� = 0:15, � = 0:3

S(1)t = Pt : Future

S(2)t = E

�(PT � 20)+ j Pt

�: Call option


E¤ect of Financial Hedging on Risk Reduction

Mean � 2000


Conclusion and Future Work

Financial hedging of an inventory system where a stochasticprice process modulates the demand and sales prices

We characterize the static and dynamic �nancial hedgingpolicies

We also analyzed the multi-period inventory version

Di¤erent objective functions (mean-variance, utility functions)

Continuous-time versions

Budget constraint


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