Black-Scholes Derivation
Understanding Through the Two Fundamental Principles
π― The Two Fundamental Principles
Principle 1: Construct a Riskless Portfolio
Value must be agnostic to outcome
- Hold Ξ = βC/βS shares of stock
- Short 1 option
- This eliminates all randomness!
In the math: The random term ΟS(βC/βS)dWt gets cancelled out when we hold the right number of shares.
Principle 2: Riskless Investments Earn Risk-Free Rate
No arbitrage condition
- Since the portfolio is riskless, it must earn rate r
- Otherwise, arbitrage exists!
In the math: The drift term must equal rC (enforced by the martingale property).
1
Start: Option Price Function
We begin with the option price as a function of time and stock price:
$$C = C(t, S_t)$$
Where:
- C = option price
- t = time
- S_t = stock price at time t
2
Find dC Using ItΓ΄'s Lemma
Since $S_t$ is stochastic, we use ItΓ΄'s lemma to find $dC$:
$$dC = \frac{\partial C}{\partial t}dt + \frac{\partial C}{\partial S}dS_t + \frac{1}{2}\frac{\partial^2 C}{\partial S^2}(dS_t)^2$$
Key insight: The extra $(dS_t)^2$ term appears because we're dealing with stochastic processes. In regular calculus, $(dx)^2 = 0$, but in stochastic calculus, $(dW_t)^2 = dt$!
3
Model Stock Price Dynamics (dS)
π― PRINCIPLE 2 APPEARS HERE
Under the risk-neutral measure, stock prices follow:
$$dS_t = rS_t dt + \sigma S_t dW_t$$
β‘ Principle 2 Connection: Notice the expected return is r (the risk-free rate), not some higher rate ΞΌ! This is risk-neutral valuation in action - we're already assuming riskless investments earn the risk-free rate.
Where:
- r = risk-free interest rate (not ΞΌ!)
- Ο = volatility
- dW_t = Brownian motion increment (the random part)
And importantly: $(dS_t)^2 = \sigma^2 S_t^2 dt$
Critical Assumption: We're treating Ο as a constant! If volatility were stochastic (i.e., Ο = Ο(t, S_t)), we'd need to include dΟ terms, which opens a whole Pandora's box of complexity. This constant volatility assumption is one of the key limitations of the basic Black-Scholes model.
4
Substitute dS into dC
Now we substitute our stock dynamics into the $dC$ expression:
$$dC = \frac{\partial C}{\partial t}dt + \frac{\partial C}{\partial S}(rS_t dt + \sigma S_t dW_t) + \frac{1}{2}\frac{\partial^2 C}{\partial S^2}(\sigma^2 S_t^2 dt)$$
Collecting terms:
$$dC = \left(\frac{\partial C}{\partial t} + rS_t\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S_t^2\frac{\partial^2 C}{\partial S^2}\right)dt + \sigma S_t\frac{\partial C}{\partial S}dW_t$$
$$dC = \underbrace{\left(\text{drift terms}\right)dt}_{\text{Deterministic part}} + \underbrace{\sigma S_t\frac{\partial C}{\partial S}dW_t}_{\text{Random part - this is what we eliminate!}}$$
β‘ Principle 1 Connection: See that random term $\sigma S_t\frac{\partial C}{\partial S}dW_t$? This is the uncertainty in the option value. By holding exactly Ξ = βC/βS shares of stock, we can eliminate this randomness and create a riskless portfolio!
How? The stock also has a random component: $\sigma S_t dW_t$. When we hold βC/βS shares, the random movements in the stock perfectly offset the random movements in the option!
5
Apply Martingale Property
π― BOTH PRINCIPLES UNITE HERE
Under the risk-neutral measure, the discounted option price $e^{-rt}C(t,S_t)$ must be a martingale.
What this means: The expected return on the option must equal the risk-free rate. This eliminates arbitrage opportunities!
β‘ Principle 2 Connection: This is where we enforce that the riskless portfolio must earn the risk-free rate! The martingale property is the mathematical way of saying "no arbitrage."
For a martingale, the drift term (after discounting) must equal zero. This gives us:
$$\frac{\partial C}{\partial t} + rS_t\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S_t^2\frac{\partial^2 C}{\partial S^2} - rC = 0$$
Rearranging:
$$\frac{\partial C}{\partial t} + rS_t\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S_t^2\frac{\partial^2 C}{\partial S^2} = rC$$
See it? The right side is rC - the option earning the risk-free rate! This is Principle 2 enforced mathematically.
π THE BLACK-SCHOLES PDE! π
$$\frac{\partial C}{\partial t} + rS\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 C}{\partial S^2} = rC$$
This is the famous Black-Scholes partial differential equation!
How the Two Principles Appear:
Principle 1 (Riskless Portfolio): By holding Ξ = βC/βS shares, we eliminated the random dWt term
Principle 2 (Earns Risk-Free Rate): The equation states the drift = rC, meaning the portfolio earns rate r
6
Set Up Terminal & Boundary Conditions
Now we need to solve this PDE! We know the option value at expiry and the boundaries:
For a European Call Option:
- At expiry (t = T): $C(T, S) = \max(S - K, 0)$ - the "hockey stick" payoff
- At S = 0: $C(t, 0) = 0$ - worthless if stock is worthless
- At S β β: $C(t, S) \approx S - Ke^{-r(T-t)}$ - behaves like stock minus discounted strike
For a European Put Option:
- At expiry (t = T): $P(T, S) = \max(K - S, 0)$
- At S = 0: $P(t, 0) = Ke^{-r(T-t)}$ - maximum value when stock worthless
- At S β β: $P(t, S) = 0$ - worthless when stock very expensive
7
The Mathematical Magic: Analytical Solution
Here's where Black, Scholes, and Merton performed mathematical wizardry. Using transform methods and the special structure of the PDE, they found an exact solution!
European Call Option:
$$C = S_0 N(d_1) - Ke^{-rT}N(d_2)$$
European Put Option:
$$P = Ke^{-rT}N(-d_2) - S_0 N(-d_1)$$
Where:
$$d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$$
$$d_2 = d_1 - \sigma\sqrt{T}$$
And $N(\cdot)$ is the cumulative standard normal distribution function.
Connection to Binomial: Remember from Chapter 13 appendix? As you increase the number of steps in a binomial tree to infinity, you get these exact formulas! Same principles, different mathematical framework.
π
Binomial Trees vs. Black-Scholes: Same Principles!
| Aspect |
Binomial Trees |
Black-Scholes |
| Time |
Discrete steps |
Continuous |
| Principle 1 |
Hold Ξ shares to make portfolio value same in up/down |
Hold Ξ = βC/βS shares to eliminate dWt term |
| Principle 2 |
Portfolio earns erΞt |
Drift = rC (martingale property) |
| Probabilities |
p = (erΞt - d)/(u - d) |
Risk-neutral measure (Q-measure) |
| Result |
f = e-rΞt[pfu + (1-p)fd] |
C = SβN(dβ) - Ke-rTN(dβ) |
|
As n β β in binomial trees, you get Black-Scholes!
|
π
The Revolutionary Impact
This changed everything!
Before Black-Scholes: "How much is this option worth?" β Educated guesswork and intuition
After Black-Scholes: "How much is this option worth?" β Plug numbers into formula, get exact answer!
Why it was revolutionary:
- Accessibility: Any trader could use it - no need to solve PDEs!
- Standardization: Markets could agree on "fair value"
- Foundation: Launched quantitative finance as a field
- Same principles as binomial: Just expressed in continuous time!
The Nobel Prize (1997): Myron Scholes and Robert Merton won for developing "a new method to determine the value of derivatives" - both the PDE derivation AND the analytical solution!
β οΈ
When the Magic Breaks
The closed-form solution only works under specific conditions:
- European exercise only (no early exercise allowed)
- Constant volatility (Ο is fixed)
- Constant interest rates (r is fixed)
- No barriers or path dependence
- No dividends (or constant dividend yield)
Reality Check: Once you relax these assumptions (American options, stochastic volatility, barriers), you lose the analytical solution and must use numerical methods like finite differences, Monte Carlo, or... go back to binomial trees!
But the foundation remains: Even complex models like Heston and SABR build on the Black-Scholes framework - they just require more computational firepower!
β
The Journey Complete: Two Principles, Two Methods
π― Summary: The Same Two Principles Throughout
Binomial Trees (Discrete):
- Hold Ξ shares β portfolio value same regardless of up/down
- Riskless portfolio earns risk-free rate β solve for option price
Black-Scholes-Merton (Continuous):
- Hold Ξ = βC/βS shares β eliminate random dWt term
- Enforce drift = rC β arrive at Black-Scholes PDE
The Beautiful Unity
Whether you use discrete binomial trees or continuous Black-Scholes, you're applying the SAME fundamental no-arbitrage logic!
As n β β, binomial converges to Black-Scholes. They're not different theories - they're the same theory at different levels of granularity.