layout: default title: The Black–Scholes Model ——————————

The Black–Scholes Model

Sukrit Mittal Franklin Templeton Investments

Positioning of This Lecture

Students already know:

  • One-step binomial model
  • Replication and no-arbitrage pricing

This lecture answers:

What happens when time becomes continuous?

Black–Scholes is not a new idea.

It is the binomial model pushed to its logical limit.

Outline

  1. From binomial trees to continuous time
  2. Market model and assumptions
  3. Replication strategy in continuous time
  4. Self-financing portfolios
  5. Derivation of the Black–Scholes PDE
  6. Boundary and terminal conditions
  7. Black–Scholes formula for European options
  8. Interpretation and limitations
  9. Exercises

1. From Binomial to Black–Scholes

Recall the one-step binomial model:

  • Stock moves up or down
  • Option is replicated exactly
  • Price follows from no-arbitrage

Now increase the number of steps.

Let time steps shrink to zero.

This limit is the Black–Scholes world.

What Survives the Limit

From the binomial model we keep:

  • Replication
  • Self-financing strategies
  • No-arbitrage pricing

What changes:

  • Prices evolve continuously
  • Risk is driven by Brownian motion

The logic does not change.

Only the mathematics does.

2. Market Model and Assumptions

We assume:

  • One risky asset $S_t$
  • One risk-free asset $B_t$

Risk-free asset:

\[\frac{dB_t}{dt} = r B_t\]

Risky asset follows:

\[\frac{dS_t}{S_t} = \mu , dt + \sigma , dW_t\]

These are modeling choices.

Not truths.

Interpretation of Parameters

  • $\mu$: expected return (irrelevant for pricing)
  • $\sigma$: volatility (central)
  • $r$: risk-free rate

Notice:

Expected return disappears from option prices.

This is not an accident.

3. Derivative Pricing by Replication

Let $V(t,S)$ denote the option price.

Construct a portfolio:

  • Hold $\Delta_t$ shares of stock
  • Hold $\beta_t$ units of the bond

Portfolio value:

\[\Pi_t = \Delta_t S_t + \beta_t B_t\]

Choose $\Delta_t, \beta_t$ to replicate $V$.

Self-Financing Condition

The portfolio is self-financing if:

\[d\Pi_t = \Delta_t , dS_t + \beta_t , dB_t\]

No external cash is injected.

Replication relies on this constraint.

This mirrors the binomial model exactly.

4. Ito’s Formula (Used, Not Worshipped)

Apply Ito’s lemma to $V(t,S_t)$:

\[dV = \partial_t V , dt + \partial_S V , dS_t + \tfrac12 \sigma^2 S_t^2 \partial_{SS} V , dt\]

Ito’s lemma is bookkeeping.

Replication is the idea.

5. Derivation of the Black–Scholes Equation

Match the dynamics of $V$ and $\Pi$.

Choose:

\[\Delta_t = \partial_S V\]

The stochastic term disappears.

The portfolio becomes locally risk-free.

No-arbitrage implies:

\[\partial_t V + \tfrac12 \sigma^2 S^2 \partial_{SS} V + r S \partial_S V - r V = 0\]

This is the Black–Scholes PDE.

What Just Happened

  • Risk was eliminated by replication
  • Expected return $\mu$ vanished
  • Pricing became deterministic

This is no-arbitrage in action.

Not probability magic.

6. Boundary and Terminal Conditions

For a European call option:

Terminal condition:

\[V(T,S) = (S - K)^+\]

Boundary conditions:

  • $V(t,0)=0$
  • Linear growth as $S \to \infty$

A PDE without conditions is meaningless.

7. Black–Scholes Formula

Solving the PDE yields:

\[C(t,S) = S \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2)\]

where:

\[d_1 = \frac{\ln(S/K) + (r + \tfrac12 \sigma^2)(T-t)}{\sigma \sqrt{T-t}}, \quad d_2 = d_1 - \sigma \sqrt{T-t}\]

This is a solution—not an assumption.

Interpretation

  • $\Phi(d_1)$: delta-adjusted probability
  • $\Phi(d_2)$: risk-neutral exercise probability

Probabilities appear.

But pricing came first.

8. Why Risk-Neutral Valuation Works

Replication implies:

All equivalent martingale measures give the same price.

Risk-neutral pricing is a shortcut.

Replication is the foundation.

Never confuse the two.

9. Limitations of Black–Scholes

  • Constant volatility
  • Continuous trading
  • No transaction costs

Markets violate all three.

The model survives anyway.

Structure beats realism.

10. Exercises

Exercise 1

Derive the delta-hedging strategy explicitly for a European call.

Exercise 2

Show that the Black–Scholes price satisfies the PDE.

Exercise 3

Explain why $\mu$ does not appear in the pricing formula.

Relate this to no-arbitrage.

Final Takeaways

  • Black–Scholes is the continuous-time binomial model
  • Replication, not preferences, drives pricing
  • PDEs replace backward induction
  • Assumptions are strong—but explicit

Next: Greeks and hedging errors.

Where theory meets practice.