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Integrated Risk-Hedging Controland Production Planning
David Yao
Columbia University, CityU/IAS
Joint work with Liao Wang, Columbia/IEOR and HKU/Business
Aug 24/25, 2017
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 1 / 21
Classical Production Planning: Newsvendor Model
Payoff function (net return):
HT (Q) := p(Q ∧DT )− c(Q−DT )+ = pQ− (p+ c)(Q−DT )+,
p: unit profit, c: unit net cost (minus salvage value), ∧ := min, (x)+ := max{x, 0}.
NV solution:
QNV := arg maxQ
E[HT (Q)] = F−1( p
p+ c
).
F (·) denotes the distribution function of DT .
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 2 / 21
NV with a Shortfall (SF) Objective
Established studies on corporate finance show that meeting or beating earningstargets are primary concerns for executives; in case of missing targets, how muchbelow the target (i.e. shortfall) matters (more than a probability of suchevent.).
Minimizing shortfall: with m > 0 being a given target level,
QNV(m) := arg minQ
E[m−HT (Q)]+.
Solution:QNV(m) =
m
p∧QNV.
• Optimal production quantity will never exceed QNV.
• sNV(m) := minQ E[m−HT (Q)]+ = E[m−HT (QNV(m))]+ is increasing(and convex) in m, which constitutes an efficient frontier.
• These are readily verified by considering two cases: m ≤ pQ and m ≥ pQ.
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 3 / 21
Motivation: Demand Dependent on Financial Market
• “Deere story”: the firm manufactures equipment for planting orharvesting corn, which is a tradable commodity. As the corn pricefluctuates on the futures market, demand for corn, and hence for thefirm’s product, will change accordingly. (The New York Times, 23Nov 2016,“Wall St. Closes Mostly Higher.”)
• “Caterpillar story”: the firm produces construction and miningmachineries. Anticipating a booming commodity market, the firmincreases their capacity and production; then commodities slide andthe firm suffers from decreasing demand. (The Wall Street Journal,16 Oct 2016, “How Caterpillar’s Big Bet Backfired.”)
• “Wal-Mart story”: during the last recession, Wal-Mart experiencedincreasing demand.
“Car maker story”: coming out of recession, car makers predict ahuge increase in demand.
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 4 / 21
Motivation: Demand Dependent on Financial Market
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 5 / 21
Production Planning with Risk Hedging
• In addition to the production quantity decision Q made at t = 0,there’s a hedging/trading strategy ϑ := {θt}t∈[0,T ] to be carried outdynamically over the horizon.
(ϑ gives a cumulative wealth at T : χT =∫ T0θtdXt.)
• Hence, the terminal wealth at t = T is HT (Q) + χT (ϑ).
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 6 / 21
A New Demand Model
• Sources of uncertainty (building block for information dynamics): twoindependent standard BM’s, Bt and Bt, t ∈ [0, T ].
• Xt: the price of a tradable asset, or a broad market index as proxy forthe economy, with µ(t, x) and σ(t, x) being functions in (t,Xt),
dXt = µ(t,Xt)dt+ σ(t,Xt)dBt,
Note Xt is a general Markov diffusion process; special case: geometricBrownian Motion.
• Dt: (cumulative) demand up to t,
dDt = µ(Xt)dt+ σdBt,
where σ > 0; and µ(x) ≥ 0 is a non-negative function. Hence,
DT = AT + σBT , where AT :=∫ T0µ(Xt)dt. Note: µ(x) = constant reverts
back to the traditional demand model.
• Use D+T (instead of DT ) to enforce non-negativity.
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Rate Function: Deere versus Corn
“ Lower farm commodity prices directly affect farm incomes, which could
negatively affect sales of agricultural equipment.” — Disclosure of Risk Factors,
John Deere’s 2016 10k.
• Data: Monthly sales (in units) of Deere combines over 2011 - 2015; source:Deere’s data release. Daily prices (Xt) of CORN (an ETF tracking cornprice) over 2011 - 2015.
• Rate function: µ(x) = ax+ b.
• For i-th month (δt is one month; ξi are i.i.d. standard normals):
Di =
∫ ti+1
ti
µ(Xs)ds+ σ(Bti+1 −Bti) = a
∫ ti+1
ti
Xsds+ b · δt+ σ√δtξi
• To get a and b, regress Di (monthly sales) onto∫ ti+1
tiXsds (evaluated using
daily prices via trapezoidal rule).
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 8 / 21
Learning the Rate Function: Deere versus Corn
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 9 / 21
Learning the Rate Function: GM versus SP500
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 10 / 21
Production Planning with Risk Hedging
• In addition to the production quantity decision Q made at t = 0,there’s a hedging/trading strategy ϑ := {θt}t∈[0,T ] to be carried outdynamically over the horizon.
(ϑ gives a cumulative wealth at T : χT =∫ T0θtdXt.)
• Hence, the terminal wealth at t = T is HT (Q) + χT (ϑ).
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 11 / 21
From Variance to Shortfall
• Wang and Yao (Operations Research, 2016) studied mean-variancehedging:
infQ,ϑ
Var[HT (Q) + χT (ϑ)]
s.t. E[HT (Q) + χT (ϑ)] = m′
• Here, we use shortfall minimization as objective. WithW := HT + χT denoting the total terminal wealth and m′ := E(W ):
min E[(m′ −W )2] → min E[(m−W )+]
where m is any given target.
• Same demand model, but with partial information and a budgetconstraint (for hedging).
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 12 / 21
Formulation: Shortfall Minimization
• Two decisions: Q, production quantity; and ϑ := {θt}t∈[0,T ], with θtbeing the hedging position: number of shares of Xt held at t.
• Q induces HT (Q) (payoff from production), and ϑ induces
χT (ϑ) :=∫ T0 θtdXt (terminal wealth from hedging).
• Shortfall-minimization under partial information:
infQ,ϑ
E[m−HT (Q)− χT (ϑ)
]+s.t. χt :=
∫ t
0θsdXs ≥ −C, θt ∈ FX
t , t ∈ [0, T ].
• First fix Q, solve for the optimal hedging strategy to obtain theminimal shortfall value, s(m,Q). Then,
s(m,Q∗(m)) = infQs(m,Q) ⇒ Q∗(m).
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 13 / 21
Duality: Solution via a Lower-Bound Problem
The hedging problem (given Q) is:
infϑ
E(m−HT − χT (ϑ)
)+s.t. χt :=
∫ t
0
θsdXs ≥ −C, θt ∈ FXt , t ∈ [0, T ].
Applying Jensen’s inequality, we first turn the above hedging problem into a staticoptimization problem:
minVT
E(m−HT − VT
)+s.t. VT ≥ −C, EM (VT ) ≤ 0 ,
where, the additional constraint follows from χt being a PM -supermartingale, andit serves the purpose of closing the duality gap (for o.w., a strategy that attains
the lower bound may not be primal feasible). Then, V ∗T = χ∗T =∫ T0θ∗t dt.
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 14 / 21
Solving the Hedging Problem
The dual (lower-bound) problem is solved by a standard Lagrangian multiplierapproach; we focus on the optimal terminal payoff χ∗T , as the strategy θ∗t isdeduced by standard technique using Ito’s Lemma.
Proposition
For given m and Q satisfying m− pQ+ C ≥ 0, the optimal hedging is:
χ∗T = V ∗T = (m− pQ+ C)1{λ∗ZT ≤ 1}+ (p+ c)[Q− D+T (λ∗)]+ − C,
where ZT := dPM
dP is the R-N derivative with representation:
ZT = exp{−∫ T0ηtdBt − 1
2
∫ T0η2t dt} with ηt := µ(t,Xt)
σ(t,Xt), (“Market Price of
Risk”);DT (λ∗) := AT + σ
√TΦ−1(λ∗ZT ), a surrogate for DT = AT + σ
√Tξ; and λ∗
satisfies the constraint EM (V ∗T ) = 0.(The solution exists and is unique. Note: V ∗T decreases in λ∗.)
• Turns out we don’t need to consider the range m+ C − pQ < 0, in whichcase the shortfall will increase in Q.
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 15 / 21
Optimal Hedging = Put Option + Digital Option
V ∗T = (p+ c)(Q− D+T )+ + (m− pQ+ C)1{λ∗ZT ≤ 1} − C
• m−HT (Q) = (p+ c)(Q−D+T )+ + (m− pQ) is the remaining gap
(from the target) after payoff from production.
• The “put option,” (p+ c)(Q− D+T )+, tries to close the first part of
the gap, (p+ c)(Q−D+T )+, but needs to use DT as a surrogate for
DT due to partial information.
• The “digital option,” (m− pQ+ C)1{λ∗ZT ≤ 1}, aims to close theother part of the gap (after subtracting C).
• Under optimal hedging, the corresponding shortfall is
s(m,Q) = (p+ c)E[Q ∧ D+T −D
+T ]+ + (m− pQ+ C)P(λ∗ZT ≥ 1).
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 16 / 21
Optimal Production Quantity
Recall, given m, the optimal production quantity is denotedQ∗(m) = arg minQ s(m,Q), and we know Q∗(m) ∈ [0, m+C
p ].
Proposition
• s(m,Q) = (p+ c)E[Q ∧ D+T −D
+T ]+ + (m− pQ+ C)P(λ∗ZT ≥ 1)
is convex in Q ∈ [0, m+Cp ]. (Thus, finding Q∗(m) is a convex
minimization problem, solvable by a simple line search.)
• s(m,Q∗(m)) is increasing in m, hence constitutes an efficient frontier.
• Q∗(m)→ 0 as m→ 0.
• Q∗(m)→ QNV as m→∞.
Furthermore, a finite upper-bound Qu ≥ supmQ∗(m) is readily identified
via a line search.
Hence, for large m, the importance of the “NV+” strategy: optimal hedgingcombined with the NV production quantity.
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 17 / 21
Shortfall Reduction
Recall, given m, Q∗(m) is the optimal solution integrated with the optimal
hedging, and QNV(m) := mp ∧Q
NV is the optimal solution to the NV shortfallobjective.
Proposition
For a given target, m, the following range of Q guarantees shortfall reductionover the production-only decision (i.e., the minimum NV shortfall):
Q ∈ [Q∗(m) ∧QNV(m), Q∗(m) ∨QNV(m)];
and the reduction is at least (i.e., lower-bounded by):
β∗(m− pQNV)+ + Cψ∗,
where β∗ = E(ZT − λ∗)+ and ψ∗ = E(λ∗ − ZT )+. Recall, ZT = dPM
dP is theR-N derivative, which follows a log-normal distribution.
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 18 / 21
Impact of Budget
Proposition
• The shortfall is decreasing and convex in the budget C (given Qand m);specifically, ∂s
∂C = −ψ. (ψ := E(λ− ZT )+ as defined in the previousslide.)
• When C → 0, Q∗(C)→ mp ∧Q
NV, the NV optimal solution .
• When C →∞, the shortfall s(Q)→ 0 for any Q.
• When C →∞, Q∗(C)→ 0.
• Furthermore, when C →∞, we have s(Q)s(0) →
m+cQm > 1 for any Q > 0
(give m).
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 19 / 21
Production and Shortfall Risk
Data: p = 1, c = 0.5, C = 400, µ = 0.2, σ = 0.15, σ = 500;X0 = 1550, T = 63, N = 250, dt = 1
252 .
Results: production, shortfall and upper bound on production.
µ(x), m model Q∗ (Qu) shortfall
µ(x) = 10xm = 4103
NV 4103 295
NV+ 4103 250 (-15%)
optimal 3878 (4303) 213 (-28%)
µ(x) = 1.2×106√1+x
m = 7659
NV 7659 291
NV+ 7659 172 (-41%)
optimal 7600 (7868) 170 (-42%)
Table: The target is set at m = pQNV; “shortfall” reports the objective value,and the percentage in parentheses is the reduction from the NV case;Qu ≥ supmQ
∗(m).
Recall dDt = µ(Xt)dt+ σdBt.
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 20 / 21
Efficient Frontier
3700 3800 3900 4000 4100
shor
tfall
50
100
150
200
250
300
k0
= 10
NV
NV+
optimal
1700 1800 1900 2000 21000
50
100
150
200
250
k0
= 5
NV
NV+
optimal
900 1000 1100 1200 13000
50
100
150
200
250
k0
= 3
NV
NV+
optimal
7250 7350 7450 7550 76500
50
100
150
200
250
300
k1= 1.2 × 106
NV
NV+
optimal
target
3500 3600 3700 3800 39000
50
100
150
200
250
k1= 0.6 × 106
NV
NV+
optimal
1600 1700 1800 1900 20000
50
100
150
200
250
k1= 0.3 × 106
NV
NV+
optimal
Figure: First row: µ(x) = k0x; second row: µ(x) = k1√1+x
.
LW/DY (Columbia, CityU/IAS) Production Planning with Risk Hedging Aug 24/25, 2017 21 / 21