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Chapter 6: Stochastic Models & Capital Pricing

CAPM, Geometric Brownian Motion, Black-Scholes, and Real Options

This chapter introduces the quantitative foundations of asset pricing as they apply to supply chain finance. We move from deterministic models to stochastic ones, building intuition through derivations that connect differential equations to the financial instruments covered in previous chapters.

6.1 Time Value of Money Review

Before entering stochastic territory, we anchor on the deterministic principle that a dollar today is worth more than a dollar tomorrow. This is the foundation of all pricing in finance.

Present Value: PV = FV / (1 + r)^t

Continuous compounding: PV = FV × e^−rT

The discount rate r reflects two things: (1) the time preference for consumption today over tomorrow, and (2) compensation for risk. In supply chain finance, the discount rate determines the price at which receivables are purchased, the interest charged on inventory loans, and the hurdle rate for supply chain investments.

Continuous vs. Discrete Compounding

Finance uses continuous compounding (e^rT) because it simplifies the mathematics of stochastic models. The conversion is:

r_continuous = ln(1 + r_discrete)

Example: A bank offers a 6% annual rate, compounded quarterly. The effective annual rate is (1 + 0.06/4)⁴ − 1 = 6.136%. The equivalent continuous rate is ln(1.06136) = 5.955%.

6.2 The Capital Asset Pricing Model (CAPM)

CAPM answers the question: what return should an investor demand for holding a risky asset? The model separates total risk into systematic risk (market-wide, non-diversifiable) and idiosyncratic risk (firm-specific, diversifiable). Only systematic risk is compensated in equilibrium.

E[R_i] = R_f + β_i (E[R_m] − R_f)

Where:

E[R_i]: Expected return on asset i
R_f: Risk-free rate (e.g., US Treasury yield)
β_i: Sensitivity of asset i to market movements. β = Cov(R_i, R_m) / Var(R_m)
E[R_m] − R_f: Market risk premium (historically ~5-7% for US equities)

Derivation Sketch: CAPM from Portfolio Theory

Step 1 (Mean-Variance Optimization): An investor choosing among N risky assets minimizes portfolio variance for a given expected return. The solution traces out the efficient frontier in (risk, return) space.

Step 2 (Two-Fund Separation): Introducing a risk-free asset, every optimal portfolio is a combination of the risk-free asset and a single tangency portfolio. In equilibrium, this tangency portfolio is the market portfolio (all risky assets weighted by market capitalization).

Step 3 (Security Market Line): For the market portfolio to be the tangency portfolio, every asset must lie on the Security Market Line: E[R_i] = R_f + β_i(E[R_m] − R_f). If an asset offers a return above the SML, investors buy it (pushing the price up and return down) until it falls back on the line.

CAPM in Supply Chain Finance

CAPM determines the cost of equity for supply chain investments. When a firm evaluates whether to build a new warehouse, invest in a supplier, or launch a reverse factoring program, the discount rate for the NPV calculation should reflect the project's systematic risk.

Example: A logistics firm (β = 0.85) considers investing in an automated warehouse. R_f = 4.5%, E[R_m] − R_f = 6%.

Cost of equity = 4.5% + 0.85 × 6% = 9.6%

If the warehouse project has an IRR of 8%, it destroys value (IRR < cost of equity). Only projects exceeding 9.6% should be undertaken.

6.3 Geometric Brownian Motion

Most asset prices are modeled as following Geometric Brownian Motion (GBM), a continuous-time stochastic process. GBM captures two features of real asset prices: an upward drift and random fluctuations.

dS = μS dt + σS dW

Where:

S: Asset price at time t
μ: Drift (expected rate of return)
σ: Volatility (standard deviation of returns)
dW: Wiener process increment (dW ~ N(0, dt))

Intuition: Over a tiny interval dt, the asset price changes by a deterministic amount (μS dt, the trend) plus a random shock (σS dW, the noise). The shocks are proportional to the current price, ensuring that a $100 stock is twice as volatile in dollar terms as a $50 stock, while percentage volatility remains constant.

Derivation: Solving the GBM Stochastic Differential Equation

Step 1: Let Y = ln(S). Apply Ito's Lemma to f(S) = ln(S):
df = (1/S) dS − (1/2)(1/S²)(dS)²

Step 2: Substitute dS = μS dt + σS dW and note (dS)² = σ²S² dt (by Ito's rule: (dW)² = dt):
d(ln S) = (μ − σ²/2) dt + σ dW

Step 3: Integrate from 0 to T:
ln S_T − ln S₀ = (μ − σ²/2)T + σ W_T

Step 4: Exponentiate:
S_T = S₀ exp[(μ − σ²/2)T + σ√T × Z] where Z ~ N(0,1)

Implication: Log-returns are normally distributed: ln(S_T/S₀) ~ N((μ − σ²/2)T, σ²T). This is the basis for the lognormal price model used in options pricing.

Simulation Example: Crude oil at S₀ = $75, μ = 5% (annual drift), σ = 30% (annual volatility).

After T = 0.5 year, the expected log price is:
E[ln S] = ln(75) + (0.05 − 0.09/2) × 0.5 = 4.317 + 0.0025 = 4.320
Std dev of log price = 0.30 × √0.5 = 0.212

95% confidence interval: exp(4.320 ± 1.96 × 0.212) = [$49.67, $114.71]

This wide range illustrates why commodity price risk matters for supply chain planning.

6.4 Black-Scholes Formula

The Black-Scholes model prices European options (options exercisable only at expiry) under the assumption that the underlying asset follows GBM. The derivation constructs a riskless portfolio and applies the no-arbitrage condition.

Call: C = S₀ N(d₁) − K e^−rT N(d₂)

Put: P = K e^−rT N(−d₂) − S₀ N(−d₁)

d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T)
d₂ = d₁ − σ√T

Where N(·) is the standard normal CDF, K is the strike price, r is the risk-free rate, and σ is the volatility.

Derivation Sketch: The Black-Scholes PDE

Step 1 (Portfolio Construction): Form a portfolio Π = V − ΔS, where V is the option value and Δ = ∂V/∂S. This portfolio is locally risk-free if Δ is chosen correctly (delta hedging).

Step 2 (Apply Ito's Lemma to V(S,t)):
dV = (∂V/∂t) dt + (∂V/∂S) dS + (1/2)(∂²V/∂S²) σ²S² dt

Step 3 (Risk-Free Portfolio): Substituting Δ = ∂V/∂S eliminates the dW term:
dΠ = [∂V/∂t + (1/2)σ²S² ∂²V/∂S²] dt

Step 4 (No-Arbitrage): A risk-free portfolio must earn the risk-free rate: dΠ = rΠ dt. Substituting:
∂V/∂t + (1/2)σ²S² ∂²V/∂S² + rS ∂V/∂S − rV = 0

Step 5: This is the Black-Scholes PDE, a second-order parabolic partial differential equation. With boundary condition V(S,T) = max(S − K, 0) for a call, the solution gives the Black-Scholes formula above.

Key insight: The drift μ does not appear in the PDE. Under risk-neutral pricing, we replace μ with r and price options as discounted expected payoffs under the risk-neutral measure.

Numerical Example: Price a 6-month European call on copper. S₀ = $8,500/ton, K = $9,000, r = 5%, σ = 25%.

d₁ = [ln(8500/9000) + (0.05 + 0.0625/2) × 0.5] / (0.25 × √0.5)
= [−0.0572 + 0.0406] / 0.1768 = −0.094
d₂ = −0.094 − 0.1768 = −0.271

N(d₁) = N(−0.094) = 0.4625
N(d₂) = N(−0.271) = 0.3932

C = 8500 × 0.4625 − 9000 × e^−0.025 × 0.3932
= 3,931.25 − 8,777.50 × 0.3932 = 3,931.25 − 3,452.43 = $478.82 per ton

For a 25-ton contract: option premium = $11,970.50. This is the cost of insuring against copper prices exceeding $9,000/ton over the next 6 months.

6.5 The Greeks

The "Greeks" measure the sensitivity of option prices to changes in underlying parameters. They are essential for risk management.

Greek	Symbol	Definition	Interpretation
Delta	Δ	∂V/∂S	Change in option value per $1 change in the underlying. Used for hedging.
Gamma	Γ	∂²V/∂S²	Rate of change of delta. High gamma near the strike means delta changes rapidly.
Theta	Θ	∂V/∂t	Time decay. Options lose value as expiry approaches (all else equal).
Vega	ν	∂V/∂σ	Sensitivity to volatility. A 1% increase in σ changes the option price by vega.
Rho	ρ	∂V/∂r	Sensitivity to interest rates. Typically small for short-dated options.

6.6 Real Options in Supply Chains

Real options extend financial option theory to physical investment decisions. A supply chain investment with embedded flexibility (the option to expand, contract, delay, or abandon) is worth more than its static NPV suggests.

Types of Real Options

Option Type	Description	Supply Chain Example
Option to Defer	Wait for more information before investing	Delay building a new distribution center until demand trends are clearer
Option to Expand	Scale up if conditions are favorable	Design a factory with extra floor space for a second production line
Option to Contract	Scale down if demand falls	Use flexible labor contracts that allow workforce reduction
Option to Abandon	Exit and recover salvage value	Lease rather than buy equipment, allowing return if the project fails
Option to Switch	Change inputs or outputs	A dual-fuel power plant that can switch between natural gas and oil

Why NPV Undervalues Flexibility: Standard NPV discounts expected cash flows at a fixed rate and ignores managerial flexibility to respond to new information. A project with an NPV of −$1M might actually be worth +$2M once you account for the option to abandon if things go badly (limiting downside) and expand if things go well (capturing upside). Real options valuation uses the Black-Scholes framework (or binomial trees) to price this flexibility.

Real Option Valuation: Expansion Example

Setup: A firm considers a $5M warehouse investment. In 2 years, it can expand capacity by investing an additional $3M. The present value of expansion cash flows is currently $2.5M but is uncertain (σ = 40%). r = 5%.

Map to Black-Scholes:
S₀ = $2.5M (PV of expansion cash flows)
K = $3M (expansion investment cost)
T = 2 years, σ = 40%, r = 5%

Calculate:
d₁ = [ln(2.5/3) + (0.05 + 0.16/2) × 2] / (0.4 × √2) = [−0.182 + 0.26] / 0.566 = 0.138
d₂ = 0.138 − 0.566 = −0.428
C = 2.5 × N(0.138) − 3 × e^−0.10 × N(−0.428)
= 2.5 × 0.555 − 2.714 × 0.334 = 1.388 − 0.907 = $0.48M

Interpretation: The option to expand is worth $480K. The total project value is the base NPV plus the expansion option value. If the base warehouse NPV is −$200K, the adjusted NPV is −$200K + $480K = +$280K. The project should proceed because the embedded flexibility creates value that static NPV ignores.

6.7 Connecting the Models

The quantitative tools in this chapter form a coherent framework for supply chain financial decisions:

CAPM sets the discount rate (cost of capital) for evaluating supply chain investments. It answers: "What return must this project earn to compensate investors for systematic risk?"
GBM models the stochastic evolution of commodity prices, demand, and other uncertain inputs. It provides the probabilistic foundation for simulation and option pricing.
Black-Scholes prices the financial options (calls, puts, collars) used to hedge commodity risk in Chapters 3 and 4. The same framework, applied to real assets, values the flexibility embedded in supply chain investments.
Real Options bridge operations and finance. They show that supply chain flexibility (dual sourcing, modular capacity, postponement) has quantifiable financial value that traditional NPV ignores.

The Central Lesson: Supply chain decisions are financial decisions. Every inventory order, capacity investment, and hedging strategy involves pricing risk under uncertainty. The models in this chapter provide the language and tools to make those decisions rigorously. The mathematics may look intimidating, but the core ideas are straightforward: time has value, risk requires compensation, and flexibility is worth paying for.

6.8 Interactive: Black-Scholes Calculator

Adjust the parameters below to price European call and put options using the Black-Scholes model. The calculator also computes the Greeks for the call option.

Black-Scholes Option Pricing

Spot Price S₀ $100

Strike Price K $100

Risk-free Rate r 5.0%

Volatility σ 25%

Time to Expiry T 1.0 yr

d₁

0.3500

d₂

0.1000

Call Price

$12.34

Put Price

$7.45

Greeks (Call)

Delta Δ

0.637

Gamma Γ

0.016

Theta Θ

-6.41

per year

Vega ν

39.58

per 100%

Rho ρ

53.23

per 100%

Put-Call Parity check: C - P = S - Ke^(-rT) → verified

6.9 Interactive: GBM Price Path Simulator

Generate random price paths following Geometric Brownian Motion. Each click of "Simulate" draws new random paths, showing the mean trajectory and 95% confidence interval.

Geometric Brownian Motion Simulator

Initial Price S₀ $100

Drift μ 8%

Volatility σ 25%

Time Horizon T 2.0 yr

Number of Paths 20

Chapter 6 Takeaways

CAPM separates diversifiable from non-diversifiable risk. Only systematic risk (β) determines the required return on supply chain investments.
GBM (dS = μS dt + σS dW) models asset prices as continuous-time stochastic processes. Ito's Lemma is the chain rule for stochastic calculus.
The Black-Scholes PDE arises from constructing a risk-free portfolio via delta hedging. The drift μ drops out; pricing depends only on r, σ, and time.
The Greeks (Δ, Γ, Θ, ν, ρ) quantify option sensitivities and guide hedging strategies.
Real options value managerial flexibility (defer, expand, contract, abandon, switch) using the same framework as financial options. Standard NPV systematically undervalues projects with embedded optionality.