Option Values in Capacity and Inventory Decisions

John R. Birge (U. of Michigan) joint with Rachel Zhang

INFORMS Fall 98 Seattle

\underline{Outline}

• Capacity and financial models

• Basic real option theory

• Options in inventory models

• Application to News Vendor

• Risk-adjusted news vendor solution • Conclusions

Capacity and Financial Models

• Problems of Discounted Cash Flow

– Must find a discount rate

– Assumes that risk level is known

– Production and inventory decisions determine risk • Catch 22

– Evaluation not possible before decision

– Decision not possible without evaluation

Option in Inventory Page 1

Remedy from Financial Theory

• Use Capital Asset Pricing Model or Option Pricing • Consistent Evaluation and Consideration of Risk • Can Include Risk Before Making Decision

• Assumptions:

– Complete market

– No transaction cost

– For options often, Ito process

Options in Inventory Page 3

Standard Pricing-CAPM

Given: cash flow S_{t} at time t = 1 Find: present value S_{0} by

S_{0} = (E(S_{1}) − λCov(S_{1}, r_{m}))/(1 + r_{f}),  (1)

where:

• r_{m} is the random return of the market over a single time period • r_{m} is the expected value of this return, r_{f} is the corresponding risk-

free rate

• λ = (r_{m} − r_{f})/σ_{m}^{2} is the market price of a unit of risk

• σ_{m}^{2} the market return variance,

• Cov(S_{1}, r_{m}) is the covariance of S_{1}, with the market return,

• E(S_{1}) denotes expectation

Standard Pricing-Option Theory

Given: call option - right (but not the obligation) to buy one share at a fixed strike price, K, at a given time, T. Find: premium C_{t} at t when the share price is S_{t}. Process: use risk neutral evaluation

• Construct equivalent riskfree probability measure

• Evaluate as if all investors neutral to risk (under riskfree measure)

• Same value in real world Result:

C_{t} = e^{−r_{f} (T −t)} ∫_{K}^{∞}(S_{T} − K)dF_{f}(S_{T}).  (2)

Application to Inventory Decisions

Inventory Decisions:

• Inventory is option to sell

• Demand above inventory is call sold to customer • Evaluate inventory using the market minus call • Basic process: Singhal (1988) Parameters:

• Purchase cost c

• Selling price p

• No salvage value

• Demand distribution F with density f

Results using Risk Neutral Valuation

Steps:

• Transform to risk neutral equivalent measure • Evaluate as if risk free

• Return result to real world Result:

F (e^{δ}x^{∗}) = (p − c)/p,  (3)

or

x^{∗} = e^{−δ}F^{−1}((p − c)/p).  (4)

Equivalence of Two Approaches

Assume: Lognormal distribution on demand (equivalent to Ito process). Substituting result with risk neutral equivlance, Singhal result implies:

F (e^{δ}x^{∗}) = Φ((ln(e^{δ}x^{∗}) − ln(P V (s)) − r + σ^{2}/2)/σ)
= Φ((ln(x^{∗}) + β(r_{m} − r_{f}) − ln(P V (s)) − r_{f} − β(r_{m} − r_{f}) + σ^{2}/2)/σ)
= Φ((ln(x^{∗}) − r_{f} − ln(P V (s)) + σ^{2}/2)/σ)
= Φ((ln(x^{∗}) − ln((E(s) − λCov(s, r_{m}))/e^{r_{f}}) − r_{f} + σ^{2}/2)/σ)
= Φ(−(ln((E(s) − λCov(s, r_{m}))/e^{r_{f}}x^{∗}) + (r_{f} − σ^{2}/2))/σ)
= Φ(−a_{2}).  (5)

Implications

• Simple adjustment possible to compensate for risk • Close to normal result in CAPM (as in Singhal)

• Can extend to other inventory problems

• Applies directly in capacity

\underline{Conclusions}

• Need for risk evaluation before decision • General value of CAPM and options • Application to news vendor

• Extensions possible