Black–Scholes Formula and Risk-Neutral Valuation: Replication, PDEs, and Pricing

An option’s payoff is uncertain, but a portfolio can sometimes reproduce that payoff state by state. If so, the option and its replicating portfolio must have the same price in an arbitrage-free market. This is the key idea behind Black–Scholes and risk-neutral valuation.

These notes expand the supplied lecture outline with derivations and worked examples. Numbered reference fragments without a bibliography are omitted. Pricing examples are hypothetical and specify their conventions. Review Probability Theory, Brownian Motion, and Volatility Modeling for background.

Learning goals and notation

Replicate a two-state option using stock and cash.
Derive a risk-neutral probability without assuming investors are indifferent to risk.
Derive the Black–Scholes PDE through a self-financing strategy.
Calculate European call and put prices and check put–call parity.
Distinguish physical expectations, pricing expectations, and exercise probabilities.
Choose numerical methods appropriate to the payoff and exercise rights.

Let Sₜ be the stock price, K the strike, T maturity, and τ=T−t remaining time in years. The continuously compounded interest rate is r, volatility is σ, and Φ is the standard normal CDF. Unless a section says otherwise, the stock pays no dividends.

1. What is known about the price—and what remains uncertain?

Given a fully specified model, contract, current state, and calibrated inputs, a pricing function can return a definite value today. Knowing S₀ alone is not sufficient: strike, maturity, volatility, interest rates, dividends, and other contract or model state variables may matter.

At a future date the option value is random because the state changes. The terminal payoff remains random today, and uncertain parameters or competing models create additional valuation uncertainty. “No uncertainty in price” is therefore too broad: the correct statement is that the conditional model value is determined by the specified information and assumptions.

Exact replication also requires a suitable market. In a complete arbitrage-free model, attainable payoffs have a unique replication price. In incomplete markets, some claims cannot be replicated; no-arbitrage alone need not select one price for them. Investor preferences, risk premia, funding constraints, or an additional pricing model may then influence valuations.

2. The assumptions behind the classical result

The basic Black–Scholes market assumes:

A continuously tradable stock following geometric Brownian motion with constant σ>0.
A cash account growing at a known constant r, with matching borrowing and lending rates.
Frictionless trading, including feasible short selling and the required stock/cash financing.
No transaction costs, taxes, default, or binding trading and margin restrictions in the replication model.
A European payoff and the stated dividend convention.

Within this setting, the stock and cash span the single Brownian risk, and the standard model is complete. The derivative’s real-world drift input drops out through replication. That is a conditional theorem, not a claim that every derivative in every market is independent of risk preferences or all risk premia. MIT’s risk-neutral valuation lecture develops this model-based argument.

3. One-period binomial replication

At the end of one period, the stock is either Sᵤ or S_d, with Sᵤ>S_d. A derivative pays fᵤ or f_d in those states. Let R=exp(rΔt) be the cash account’s gross growth.

Choose Δ shares and an initial cash-account value B so that:

ΔSᵤ+BR=fᵤ, and ΔS_d+BR=f_d.

Subtracting the equations gives:

Δ=(fᵤ−f_d)/(Sᵤ−S_d),

B=(f_d−ΔS_d)/R.

The initial replicating cost is f₀=ΔS₀+B. B can be negative: borrowing is part of the strategy. B is today’s cash value; its terminal value is BR.

Worked one-year call

Let S₀=100, Sᵤ=120, S_d=80, K=100, and R=1.05. The equivalent continuous rate is r=log(1.05), not exactly 5%. Call payoffs are 20 and zero.

Δ=20/40=0.5, and B=−40/1.05≈−38.0952.

Buy half a share for 50 and borrow 38.0952. The initial cost is 11.9048. At maturity, the debt is 40: the portfolio pays 60−40=20 in the up state and 40−40=0 in the down state.

If the option were more expensive, sell it and buy the replicating strategy; if cheaper, take the reverse positions. Under the stated trading assumptions, the mismatched prices produce an arbitrage. The replication calculation did not require the real-world probability of an up move.

4. Risk-neutral probability emerges from the same algebra

Define:

p_Q=(S₀R−S_d)/(Sᵤ−S_d).

Then the replication price can be written:

f₀=R⁻¹[p_Qfᵤ+(1−p_Q)f_d].

For strict no-arbitrage with both modeled states possible and unconstrained stock/cash trading, require S_d<S₀R<Sᵤ. This puts p_Q strictly between zero and one. If that condition fails, the two-state inputs do not describe the intended arbitrage-free model; a negative computed “probability” is not a valid pricing workaround.

For the example, p_Q=(105−80)/40=0.625, so f₀=(0.625×20)/1.05=11.9048. Also:

S₀=R⁻¹E_Q[S_T].

Discounted stock value is a martingale under Q. This does not assert a 62.5% physical chance of an up move, nor that investors are psychologically risk-neutral. Q is the probability representation implied by the stock-and-cash prices in this model.

If an investor instead believes P(up)=0.8, its discounted physical expected call payoff is 16/1.05≈15.2381, but the replication cost remains 11.9048. The physical forecast can affect a trading preference without changing the model’s arbitrage price.

5. From geometric Brownian motion to Itô’s formula

Under the physical probability measure P, suppose:

dSₜ=μSₜdt+σSₜdWₜ^P.

For a sufficiently smooth value f(t,S), Itô’s formula gives:

df=[f_t+μSf_S+½σ²S²f_SS]dt+σSf_SdW^P.

The derivatives are time sensitivity f_t, delta f_S, and gamma f_SS. The second-order term survives because Brownian quadratic variation is dt. This formula describes the derivative’s value changes given f; it does not yet determine which f is an arbitrage-free price.

6. Self-financing delta replication and the PDE

Let a replicating strategy have value V=ΔS+B, where B is the cash-account value. Self-financing means stock rebalancing is paid for by changes in cash, with no external money added or withdrawn. Its gain equation is:

dV=ΔdS+rBdt.

Set V=f and choose Δ=f_S, so B=f−Sf_S. The Brownian terms in dV and df match. Equating their drift terms:

f_t+μSf_S+½σ²S²f_SS=μSf_S+r(f−Sf_S).

The physical drift cancels, leaving:

f_t+½σ²S²f_SS+rSf_S−rf=0.

This is the Black–Scholes PDE. It follows from matching the gains of a financed strategy, not from treating a changing delta as a constant position without accounting for its cost. The familiar short-option/long-delta-stock argument has the same local hedging intuition; its rebalancing still needs a cash ledger.

To price a European call, impose f(T,S)=max(S−K,0). A put uses max(K−S,0). Suitable boundary and growth conditions are part of the pricing problem; the PDE alone without the payoff does not select one derivative value.

7. Why discounted Q-expectations solve the PDE

In the basic model, under the equivalent pricing measure Q:

dSₜ=rSₜdt+σSₜdWₜ^Q.

The risk-neutral stock evolves with drift r, so its discounted value has zero drift. Under appropriate integrability and regularity conditions, the pricing PDE and conditional expectation agree:

f(t,Sₜ)=E_Q[exp(−r(T−t))g(S_T) | ℱₜ].

The conditioning matters. At t=0 it may be suppressed when all current state information is fixed; at a later date a bare unconditional E_Q[g(S_T)] would answer a different question.

With stochastic interest rates, the general form involves E_Q[exp(−∫ₜᵀr_sds)g(S_T) | ℱₜ] in an appropriate model. The discount factor cannot generally be pulled outside the expectation. Dividends and other paid cash flows require the corresponding gains-based treatment.

The terminal stock law

For constant r and σ in the no-dividend model:

S_T=Sₜ exp[(r−½σ²)τ+σ√τ Z], with Z∼N(0,1) under Q conditional on current information.

The term −½σ² is the Itô correction. It ensures E_Q[S_T | ℱₜ]=Sₜexp(rτ), despite the log-return mean being (r−½σ²)τ. The stock level is lognormal; its log return is normal.

8. Deriving the European call formula

For a call payoff, write:

C=exp(−rτ){E_Q[S_T 1{S_T>K}]−KQ(S_T>K)}.

These expectations use the current state. Define:

d₁=[log(S/K)+(r+½σ²)τ]/(σ√τ),

d₂=d₁−σ√τ.

The lognormal threshold gives Q(S_T>K)=Φ(d₂). Completing the square in the truncated normal integral gives E_Q[S_T 1{S_T>K}]=S exp(rτ)Φ(d₁). Substituting yields:

C=SΦ(d₁)−K exp(−rτ)Φ(d₂).

The European put formula is:

P=K exp(−rτ)Φ(−d₂)−SΦ(−d₁).

Here S is the current stock price. At expiry, use the intrinsic payoff directly. At σ=0, the d formulas divide by zero, but the deterministic-path limiting call value is max(S−K exp(−rτ),0) under these no-dividend assumptions.

d₁ is not the exercise probability

Under this Q model, Φ(d₂) is the probability the call finishes in the money. Φ(d₁) is the no-dividend call delta and arises from a stock-weighted expectation; it is not the same probability. Neither quantity should automatically be interpreted as the real-world probability of exercise.

Finishing in the money is also different from making a net profit on the original option purchase: premium, financing, and fees affect the latter threshold.

9. A numerical Black–Scholes price and hedge

Take S=100, K=100, τ=1 year, r=5% continuously compounded, σ=20%, and no dividends. Then:

d₁=0.35; d₂=0.15,

Φ(d₁)≈0.636831; Φ(d₂)≈0.559618.

The prices are:

European call: C≈10.4506.
European put: P≈5.5735.

The call’s model delta is approximately 0.636831 shares per option on one share. Its cash-account component is C−SΔ≈−53.2325, so the initial replication borrows about 53.2325 and holds stock worth about 63.6831.

These positions must change as S, time, and the model delta change. A standard exchange-traded option may cover multiple shares; apply the actual contract multiplier rather than treating one quoted option as one share of exposure.

10. Put–call parity: the payoff argument comes first

For European options with the same underlying, strike, maturity, settlement, and no dividends:

max(S_T−K,0)−max(K−S_T,0)=S_T−K.

The payoff identity and replication with stock and a payment of K at T imply:

C−P=S−K exp(−rτ).

For the worked example, 10.4506−5.5735≈4.8771, equal to 100−100exp(−0.05). Unlike the Black–Scholes formula, this parity relation does not require geometric Brownian motion or constant volatility. It still assumes compatible European payoffs and a feasible no-arbitrage trading and financing setup.

With a continuous dividend yield q, parity becomes C−P=S exp(−qτ)−K exp(−rτ), and the call formula becomes S exp(−qτ)Φ(d₁)−K exp(−rτ)Φ(d₂) with r−q in the d₁ numerator. American exercise rights, discrete dividends, and market frictions need their appropriate treatment rather than blind use of the no-dividend equality.

Useful no-dividend European call bounds are max(S−K exp(−rτ),0)≤C≤S under the stated frictionless setup. They are quick checks, not substitutes for validating the contract and quote.

11. Greeks and the limits of actual hedging

For the no-dividend European call:

Delta=Φ(d₁),

Gamma=φ(d₁)/(Sσ√τ),

Vega=Sφ(d₁)√τ, where φ is the standard normal density.

Vega here is per one unit of decimal volatility. A one-percentage-point volatility change is 0.01 units, so its small-change price effect is approximately 0.01×Vega. Delta hedges small stock moves; gamma describes how delta changes, and vega describes sensitivity to the volatility input.

In the ideal model, continuous self-financing trading with correct inputs replicates the payoff. Real hedges are discrete and face costs, jumps, uncertain volatility, borrowing differences, and market constraints. Their errors can be economically significant. Replacing historical σ with implied σ can align one option’s price, but does not automatically make a constant-volatility model reproduce the whole surface or hedge it perfectly.

Risk aversion and supply/demand can affect implied volatility and other market inputs. Their absence from the conditional Black–Scholes formula should not be overstated as their absence from real markets. Scholes’s Nobel lecture discusses the framework’s development and practical extensions.

12. Numerical valuation methods

Not every exotic or early-exercise problem lacks every possible analytic simplification, but many require numerical valuation. The method must match the payoff’s state dependence and exercise rights.

Method	Core operation	Strength and limitation
Finite differences	Approximate the pricing PDE on time and state grids	Useful in low-dimensional problems; boundaries, stability, and grid convergence matter
Binomial or trinomial tree	Work backward through states	Handles exercise decisions naturally; state growth and convergence need checking
Monte Carlo	Average discounted Q-payoffs over simulated paths	Flexible for path dependence and many factors; sampling error and exercise treatment matter

In a Cox–Ross–Rubinstein-style step, set u=exp(σ√Δt), d=1/u, and p_Q=[exp(rΔt)−d]/(u−d) in the no-dividend case. At each node, discount the probability-weighted next values. Check that the step’s no-arbitrage condition gives a valid probability. More steps should approach the intended continuous-model price under the construction’s assumptions.

For an American option, compare immediate exercise value with continuation value at each node and keep the larger. A no-dividend American call with a nonnegative interest rate has no early-exercise benefit in the standard frictionless setting, but that result should not be applied to every option or dividend scenario.

Monte Carlo for a European call

Draw independent standard normals Zᵢ and use the exact terminal-stock expression above. Estimate:

Ĉ=(1/M)Σᵢ exp(−rτ)max(S_T⁽ⁱ⁾−K,0).

The estimated Monte Carlo standard error is the sample standard deviation of the discounted payoffs divided by √M, assuming independent draws with finite variance. Four times as many paths roughly halves that sampling error. It does not fix model error, incorrect inputs, or time-discretization bias in an approximate path simulation.

Path-dependent contracts require the relevant path information, not just S_T. Discrete grids can miss continuously monitored barriers; bridge corrections or finer monitoring may be needed under suitable models.

American Monte Carlo needs a stopping rule

Simply averaging terminal payoffs prices a European-style payoff, not a general American option. American valuation involves an optimal stopping problem. Methods such as least-squares Monte Carlo estimate continuation values and exercise decisions; they also require validation and careful handling of fitting bias.

Selecting each simulated path’s best future exercise time with hindsight is not a feasible adapted strategy and can overstate value. A numerical method must respect the information available when exercise occurs.

13. A practical pricing checklist

Specify the payoff, exercise rights, underlying unit, settlement, maturity, and dividend convention.
Identify the model, current state, pricing measure, curve inputs, and volatility assumptions.
Confirm whether replication and uniqueness apply, or whether unhedgeable risks require further assumptions.
Check signs, rate compounding, year fractions, percentage units, and contract multipliers.
Validate against payoff limits, monotonicity where appropriate, parity, and known special cases.
For numerical methods, test grid or path convergence and report sampling uncertainty separately from model uncertainty.
If implementing a hedge, include rebalancing cash flows, transaction costs, liquidity, and residual risk under plausible model failures.

14. Practice questions

In the one-period example, what are the call’s stock holding and initial cash balance?
Why must S_d<S₀R<Sᵤ hold in the intended two-state no-arbitrage model?
Does Q(up)=0.625 state the real-world frequency of an up move?
Which term cancels from the self-financing PDE derivation: μ, σ, or r?
In the numerical Black–Scholes example, is Φ(d₁) or Φ(d₂) the Q-probability of finishing above K?
Does put–call parity require normally distributed log returns?
If independent Monte Carlo paths increase from 10,000 to 40,000, what happens approximately to sampling standard error?
Why does taking the best exercise time after seeing each entire simulated path fail to value a feasible American strategy?

Answers

0.5 shares and B≈−38.0952; the initial cost is approximately 11.9048.
It prevents a stock–cash arbitrage and makes both pricing-state probabilities strictly positive.
No. It is implied by the model’s stock and cash prices for valuation, not automatically a physical forecast.
The physical drift μ. Volatility and the financing rate remain.
Φ(d₂)≈0.559618, under the stated Q model.
No. It follows from matching European payoffs and the appropriate stock/cash replication assumptions.
It approximately halves, other conditions unchanged.
It uses future information unavailable at the exercise date. Exercise must be a valid stopping decision based on information then available.