Black-Scholes
Introduction to Financial Options / Lecture 05
The binomial hedge, rebalanced continuously.
Last time, the hedge moved
Introduction to Financial Options / Lecture 05
At the first node
We held 0.55 share against the option.
After an up move
We needed about 0.917 share.
The option price was supported by a strategy, not by one static stock position.
Make the steps smaller
Introduction to Financial Options / Lecture 05
Imagine rehedging more often while each stock move gets smaller.
One-step tree
→
Many small steps
→
Continuous hedge
Black-Scholes is a continuous-time version of the replication argument.
What we assume
Introduction to Financial Options / Lecture 05
Contract
European option. Fixed strike and expiry. No dividends.
Market
Trade stock and cash continuously, without costs. One constant interest rate r.
Stock process
Continuous paths and one known constant volatility \sigma.
These assumptions make the hedge exact inside the model.
A model for a tiny stock move
Introduction to Financial Options / Lecture 05
dS=\mu S\,dt+\sigma S\,dW^P
Drift \mu
The real-world average directional growth rate.
Volatility \sigma
The size of the unpredictable part of a short move.
P is the real-world probability law. Under P, the Brownian increment dW^P has standard deviation \sqrt{dt}.
Time scales differently from noise
Introduction to Financial Options / Lecture 05
Predictable move
Drift over a short interval is proportional to dt.
Random move
Its standard deviation is proportional to \sqrt{dt}.
\operatorname{Var}(dS)\approx \sigma^2S^2\,dt
That variance term will survive even when we hedge away the first-order stock move.
An option price moves for two reasons
Introduction to Financial Options / Lecture 05
C=C(S,t)
The share moves
C_S measures the first-order change per dollar of stock movement.
Time passes
C_t measures the change at fixed share price as calendar time advances.
Subscripts denote partial derivatives. The remaining time to expiry shrinks as t increases.
First-order Taylor is incomplete
Introduction to Financial Options / Lecture 05
dC\approx C_t\,dt+C_S\,dS+\tfrac12 C_{SS}(dS)^2
The extra term is curvature: the option’s slope changes as the share moves.
We cannot discard (dS)^2 merely because the time step is tiny.
Why the square stays
Introduction to Financial Options / Lecture 05
dS\sim \sigma S\sqrt{dt}\quad\Longrightarrow\quad(dS)^2\sim\sigma^2S^2dt
A random move is of order \sqrt{dt}, so its square is of order dt: the same order as time decay and interest.
In the diffusion limit, (dW^P)^2=dt in the quadratic-variation sense, not as an ordinary pointwise identity.
The option’s local change
Introduction to Financial Options / Lecture 05
dC=\left(C_t+\tfrac12\sigma^2S^2C_{SS}\right)dt+C_S\,dS
Time + curvature
Proportional to dt.
Directional exposure
Proportional to the actual stock move dS.
This is Itô’s formula applied to C(S,t) under the assumed continuous stock model.
Build a self-financing replica
Introduction to Financial Options / Lecture 05
V=\Delta S+B
Hold \Delta shares and a bank balance B. Pay for stock purchases out of the bank account when the hedge changes.
dV=\Delta\,dS+rB\,dt
Self-financing means no fresh outside money is added as \Delta changes.
Match the stock shock
Introduction to Financial Options / Lecture 05
dC=\left(C_t+\tfrac12\sigma^2S^2C_{SS}\right)dt+C_S\,dS
dV=\Delta\,dS+rB\,dt\quad\Longrightarrow\quad\Delta=C_S
Option and replica now have the same unpredictable stock move. The hedge must be updated as C_S changes.
Turn the hedge knob
Introduction to Financial Options / Lecture 05
This local example has option delta 0.50. Change the number of shares sold against one long call.
Illustrative local Taylor approximation. A delta hedge removes slope at the current share price, not curvature or jump risk.
Rebalancing is financed, not free
Introduction to Financial Options / Lecture 05
If delta increases
Buy more shares. The cash account pays for them.
If delta decreases
Sell shares. The cash account receives the proceeds.
Self-financing means no new outside money is added during rehedging. It does not mean no funding or trading frictions in reality.
Match the remaining cashflows
Introduction to Financial Options / Lecture 05
With \Delta=C_S and V=C, the bank balance is B=C-SC_S. The option’s remaining local change must equal its financing gain.
C_t+\tfrac12\sigma^2S^2C_{SS}=r(C-SC_S)
The random dS terms matched. Only predictable dt terms remain to be equated.
The Black–Scholes pricing equation
Introduction to Financial Options / Lecture 05
C_t+\tfrac12\sigma^2S^2C_{SS}+rSC_S-rC=0
Time change + curvature benefit + financed stock exposure = 0.
This equation came from hedging and no arbitrage, not from guessing the stock’s average return.
Where did expected return go?
Introduction to Financial Options / Lecture 05
The stock model contained
dS=\mu Sdt+\sigma SdW^P
The pricing equation contains
r and \sigma, but not \mu.
The hedge removes the stock shock and its associated drift exposure. Different views of \mu do not change this model price.
A terminal condition completes the problem
Introduction to Financial Options / Lecture 05
C(S,T)=(S-K)^+
Solve the pricing equation backward from the known expiry payoff, just as we worked backward through the binomial tree.
For a put, replace the terminal payoff by (K-S)^+.
Same stock. Two probability laws.
Introduction to Financial Options / Lecture 05
P · real-world law
\frac{dS}{S}=\mu\,dt+\sigma\,dW^P
Use this law to describe or forecast actual stock returns.
Q · pricing law
\frac{dS}{S}=r\,dt+\sigma\,dW^Q
Use this law to price payoffs consistent with the hedge.
Q reweights possible paths for pricing. It does not claim that the stock actually earns r. Here we assume no dividends.
An equivalent way to solve it
Introduction to Financial Options / Lecture 05
C_0=e^{-rT}\mathbb E^{Q}\!\left[(S_T-K)^+\right]
Under the pricing distribution Q, the stock grows on average at r, not at its real-world drift \mu.
This expectation is a way to solve the same no-arbitrage equation. It is not a forecast of actual frequencies.
The stock’s terminal distribution
Introduction to Financial Options / Lecture 05
S_T=S_0\exp\!\left[(r-\tfrac12\sigma^2)T+\sigma\sqrt{T}Z\right],\quad Z\sim N(0,1)
Log returns are normal in this model. Stock prices are lognormal, so they cannot become negative.
The -\tfrac12\sigma^2 adjustment keeps the pricing average at S_0e^{rT}.
Split the call’s payoff
Introduction to Financial Options / Lecture 05
C_0=e^{-rT}\mathbb E^Q[S_T\mathbf1_{S_T>K}]-Ke^{-rT}Q(S_T>K)
Stock part
Receive the share in the in-the-money states.
Strike part
Pay K only in those states.
The strike-payment probability
Introduction to Financial Options / Lecture 05
d_2=\frac{\ln(S_0/K)+(r-\tfrac12\sigma^2)T}{\sigma\sqrt T}
Q(S_T>K)=N(d_2)
N is the standard-normal cumulative distribution function. N(d_2) is a model pricing probability, not necessarily a real-world probability.
The stock part gets a different weight
Introduction to Financial Options / Lecture 05
d_1=d_2+\sigma\sqrt T
e^{-rT}\mathbb E^Q[S_T\mathbf1_{S_T>K}]=S_0N(d_1)
High stock-price states count more in the stock part of the payoff than in the simple exercise probability.
The European call formula
Introduction to Financial Options / Lecture 05
\boxed{C_0=S_0N(d_1)-Ke^{-rT}N(d_2)}
d_1=\frac{\ln(S_0/K)+(r+\tfrac12\sigma^2)T}{\sigma\sqrt T},\qquad d_2=d_1-\sigma\sqrt T
The formula is the solution. The hedging argument explains why that solution prices the option.
A four-dollar example
Introduction to Financial Options / Lecture 05
S_0=K=\$100, r=0, T=\tfrac14 year, \sigma=20\%.
\sigma\sqrt T=0.10,\qquad d_1=0.05,\qquad d_2=-0.05
C_0=100[N(0.05)-N(-0.05)]\approx\boxed{\$3.99}
At zero rates and no dividends, the matching at-the-money put also costs about $3.99 by parity.
The hedge ratio returns
Introduction to Financial Options / Lecture 05
\Delta=C_S=N(d_1)
For the four-dollar example, N(0.05)\approx0.520. The model hedge is about 0.52 share per call.
The formula and the replicating hedge agree. Delta changes as S and time change.
At zero rates, time and curvature balance
Introduction to Financial Options / Lecture 05
r=0\quad\Longrightarrow\quad C_t=-\tfrac12\sigma^2S^2C_{SS}
Gamma C_{SS}
The option benefits from curvature under price movement.
Calendar time C_t
Holding the share fixed, the option loses time value.
This is a model balance at fixed stock and volatility, not a guaranteed realized trading P&L.
What the formula does not promise
Introduction to Financial Options / Lecture 05
Real hedging
Trades are discrete and costly. Jumps cannot be hedged continuously through.
Real volatility
The future \sigma is not known and is not generally constant.
Real contracts
Dividends and early exercise change the inputs or problem.
Black-Scholes is a disciplined pricing benchmark, not a claim that every market follows one lognormal distribution.
Check your understanding
Introduction to Financial Options / Lecture 05
Two traders disagree sharply about the stock’s real-world drift \mu, but agree on S_0, K, T, r, and the model volatility \sigma. Do they obtain different Black-Scholes call prices?
No. The hedged pricing equation contains r and \sigma, not \mu. They may still disagree about whether buying the call is an attractive investment.
Summary
Introduction to Financial Options / Lecture 05
- The continuous stock model has drift, volatility, and \sqrt{dt}-sized shocks.
- Option curvature contributes at the same time scale as interest and decay.
- A self-financing portfolio holding C_S shares matches the option’s stock shock.
- Matching the remaining financing cashflows gives the Black-Scholes pricing equation.
- The call formula solves that equation. Its Q expectation prices the payoff; P describes real-world outcomes.
Next: invert the formula to quote an option price as implied volatility.