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An option is a right, not an obligation. Pricing that right cleanly — without arbitrage, without gut feeling — is what Black–Scholes achieved. If you are tackling stochastic calculus and martingale pricing in a London MSc, the model is your starting point. It is also the one a quantitative finance tutor London programmes expect you to rebuild from scratch.

1 · The underlying and the risk-free asset

The model begins with two tradable assets. One is risky: a stock whose price S_t follows geometric Brownian motion. The other is risk-free: a bond or money-market account growing at the constant continuously compounded rate r.

The stock’s dynamics are the heart of the model. Under the physical measure ℙ, the stochastic differential equation is:

dS_t = μ S_t dt + σ S_t dW_t

μ is the expected return. σ is volatility — constant, known, and the parameter that drives every option price. dW_t is a standard Brownian increment. The solution, via Itô’s lemma, is the familiar log-normal path:

S_t = S₀ exp((μ − σ²/2)t + σ W_t)

The bond is simpler. If B_t is the price of a risk-free zero-coupon bond paying 1 at time T, then B_t = e^−r(T−t). Equivalently, a money-market account starting at 1 grows to e^rt by time t. No randomness, no credit risk, no term structure. This is the baseline, and every extension — stochastic volatility, jumps, stochastic rates — relaxes one of these assumptions.

2 · The replicating portfolio and the Black–Scholes PDE

The breakthrough is the idea that an option can be replicated by trading the stock and the bond dynamically. Hold Δ_t shares of stock and invest V_t − Δ_tS_t in the risk-free asset. If the portfolio value matches the option payoff at expiry and is self-financing, then by no-arbitrage the option price must equal the portfolio value at every earlier time.

Set up a portfolio Π_t = V(S_t, t) − Δ_tS_t. Choose Δ_t = ∂V/∂S — the delta — to eliminate the dW_t term. The portfolio becomes instantaneously risk-free. Since it is risk-free, it must earn the risk-free rate, or arbitrage exists. Equating the return on Π_t to rΠ_t dt and applying Itô’s lemma to dV yields the Black–Scholes PDE:

∂V/∂t + ½σ²S² ∂²V/∂S² + r S ∂V/∂S − r V = 0

Notice what is absent. μ, the stock’s expected return, does not appear. The PDE is preference-free. Any two investors who agree on σ and r will agree on the option price, regardless of their views on the stock’s direction. That is the model’s most powerful — and most examined — property.

3 · Risk-neutral valuation

The PDE has a second interpretation. By the Feynman–Kac theorem, its solution is the expected discounted payoff under a different probability measure ℚ — the risk-neutral measure — where the stock’s drift is r, not μ. Under ℚ:

dS_t = r S_t dt + σ S_t dW̃_t

where W̃_t is a ℚ-Brownian motion. The option price is then:

V(S_t, t) = e^−r(T−t) 𝔼^ℚ[Payoff(S_T) | ℱ_t]

This is the martingale pricing formula. The discounted option price is a ℚ-martingale. For a European call with strike K and expiry T, the payoff is max(S_T − K, 0). Evaluating the expectation under the log-normal ℚ-distribution of S_T gives the Black–Scholes formula:

C = S₀ Φ(d₁) − K e^−rT Φ(d₂)

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

Φ(·) is the standard normal CDF. The formula is closed-form, which is rare in finance and one reason the model endures. Every term has an economic interpretation: S₀ Φ(d₁) is the stock component of the replicating portfolio, K e^−rT Φ(d₂) is the bond component, and Φ(d₂) is the risk-neutral probability that the option finishes in the money.

4 · The Greeks — what moves the price

The partial derivatives of the Black–Scholes formula are the Greeks. They measure risk exposures and are the language of any derivatives desk. Five matter most.

Delta (Δ = ∂C/∂S = Φ(d₁)): the number of shares in the replicating portfolio. For a call, 0 < Δ < 1. Deep in-the-money calls have Δ ≈ 1; deep out-of-the-money calls have Δ ≈ 0.

Gamma (Γ = ∂²C/∂S² = φ(d₁) / (S₀σ√T)): the rate of change of delta. Highest near the money, falling as the option moves deep in or out. Gamma measures how often a delta-hedged portfolio needs rebalancing.

Vega (ν = ∂C/∂σ = S₀ φ(d₁) √T): sensitivity to volatility. Always positive for vanilla calls and puts. Vega is not a Greek letter, but the name stuck. It peaks near the money and decays with time.

Theta (Θ = ∂C/∂t): time decay. For a call, Θ is usually negative — the option loses value as expiry approaches, all else equal. Theta and gamma pull in opposite directions: a delta-hedged position earns theta but pays gamma when the underlying moves.

Rho (ρ = ∂C/∂r): sensitivity to the risk-free rate. For calls, ρ > 0; higher rates reduce the present value of the strike, making the call more valuable. For short-dated options, rho is small.

Examiners test the Greeks in two ways: algebraic derivation from the Black–Scholes formula, and qualitative reasoning about how a position’s risk changes with the underlying. Both require fluency with the normal density φ(x) = (1/√(2π)) e^−x²/2 and the chain rule.

5 · Put-call parity and the put formula

European puts are priced by the same PDE with boundary condition P(S_T, T) = max(K − S_T, 0). The put formula is:

P = K e^−rT Φ(−d₂) − S₀ Φ(−d₁)

But you rarely need to compute it from scratch. Put-call parity links the two:

C − P = S₀ − K e^−rT

This is a no-arbitrage relationship. A portfolio of long call, short put, and a risk-free bond paying K at T replicates the stock. If parity fails, there is an arbitrage. The relationship holds for European options on non-dividend-paying stocks regardless of the distribution of S_T — it does not depend on the Black–Scholes assumptions at all. That makes it a powerful check on any pricing model.

Worked example — pricing a European call

Consider a stock trading at S₀ = £100. A six-month European call has strike K = £105. The risk-free rate is r = 5% per annum continuously compounded. Volatility is σ = 20% per annum. No dividends.

Step 1 — Gather parameters. S₀ = 100, K = 105, T = 0.5, r = 0.05, σ = 0.20.

Step 2 — Compute d₁ and d₂. First, σ√T = 0.20 × √0.5 = 0.1414. Then:

d₁ = [ln(100/105) + (0.05 + 0.20²/2) × 0.5] / 0.1414
= [−0.04879 + (0.05 + 0.02) × 0.5] / 0.1414
= [−0.04879 + 0.035] / 0.1414
= −0.01379 / 0.1414 = −0.0975

d₂ = d₁ − σ√T = −0.0975 − 0.1414 = −0.2389

Step 3 — Evaluate Φ(d₁) and Φ(d₂). From standard normal tables or a calculator:

Φ(−0.0975) ≈ 0.4612
Φ(−0.2389) ≈ 0.4056

Step 4 — Compute the call price.

C = 100 × 0.4612 − 105 × e^−0.05×0.5 × 0.4056
= 46.12 − 105 × 0.97531 × 0.4056
= 46.12 − 41.54
= £4.58

Step 5 — Verify with put-call parity. First compute the put price:

P = 105 × e^−0.025 × Φ(0.2389) − 100 × Φ(0.0975)
= 105 × 0.97531 × 0.5944 − 100 × 0.5388
= 60.87 − 53.88 = £6.99

Check parity: C − P = 4.58 − 6.99 = −2.41. S₀ − K e^−rT = 100 − 102.41 = −2.41. The two sides agree. ✓

Step 6 — Interpretation. The call is out of the money (S₀ < K), so its price is entirely time value and volatility value. The delta of 0.4612 means the replicating portfolio holds roughly 46 shares per option. The risk-neutral probability of finishing in the money is Φ(d₂) = 40.6%, but the option is worth more than that probability times the expected payoff because of the skewness of the log-normal distribution — the stock can finish far above K, and those states contribute disproportionately to the expectation.

Figure 1 — The worked example, drawn exactly: the Black–Scholes value with six months to run (σ = 0.20, r = 0.05) lies strictly above the expiry payoff at every price; at S = 100 the call is worth £4.58, all of it time value.

Comfortable with Itô’s lemma but lose the thread at the change of measure? That is exactly the gap a one-on-one session closes — working from the PDE derivation through risk-neutral pricing to the Greeks, in your course’s notation. Book a trial session.

Practice

Q1. A stock trades at £50. A three-month European call with strike £52 has Black–Scholes price £1.86. The risk-free rate is 4% and volatility is 25%.
(a) Compute d₁ and d₂.
(b) Verify the call price using the Black–Scholes formula.
(c) What is the put price?

Q2. A European call on a non-dividend-paying stock has S₀ = £80, K = £80, T = 0.25, r = 3%, σ = 30%.
(a) Price the call.
(b) Compute delta, gamma, and vega.
(c) If the stock price rises by £1, what is the approximate new call price using delta alone? Using delta and gamma?

Q3. A put option has S₀ = £200, K = £180, T = 1, r = 5%, σ = 40%.
(a) Price the put using Black–Scholes.
(b) Verify put-call parity by computing the corresponding call price.
(c) The stock pays a dividend yield of q = 2%. How does the put formula change? Re-price the put.

Answers: Q1 (a) d₁ = −0.1713, d₂ = −0.2963; (b) C = 50×0.4320 − 52×e⁻⁰·⁰¹×0.3835 = 21.60 − 19.74 = £1.86 ✓; (c) P = C − S₀ + Ke⁻ʳᵀ = 1.86 − 50 + 51.48 = £3.34. Q2 (a) d₁ = 0.125, d₂ = −0.025; C = 80×0.5497 − 80×e⁻⁰·⁰⁰⁷⁵×0.4900 = 43.98 − 38.91 = £5.07; (b) Δ = 0.5497, Γ = φ(0.125)/(80×0.30×√0.25) = 0.3956/12 = 0.0330, ν = 80×0.3956×0.5 = £15.82 per 100% vol change (so £0.1582 per 1% vol change); (c) Δ-only: 5.07 + 0.5497 = £5.62; Δ+Γ: 5.62 + ½×0.0330×1² = £5.64. Q3 (a) d₁ = 0.5884, d₂ = 0.1884; P = 180×e⁻⁰·⁰⁵×Φ(−0.1884) − 200×Φ(−0.5884) = £17.19; (b) C = P + S₀ − Ke⁻ʳᵀ = 17.19 + 200 − 171.22 = £45.97; (c) Replace S₀ with S₀e⁻qT = 200×e⁻⁰·⁰² = 196.04 in both d₁ and the formula. d₁ = 0.5384, d₂ = 0.1384; P = 180×e⁻⁰·⁰⁵×Φ(−0.1384) − 196.04×Φ(−0.5384) = £18.33.

Key takeaways

Black–Scholes prices a European option by replication: a dynamically adjusted portfolio of stock and risk-free bonds that exactly matches the option payoff.
The PDE ∂V/∂t + ½σ²S² ∂²V/∂S² + r S ∂V/∂S − r V = 0 is preference-free. The stock’s expected return μ does not appear.
Risk-neutral valuation rewrites the price as e^−rT 𝔼^ℚ[Payoff], where the stock drifts at r under ℚ. The formula is a discounted expectation, not a forecast.
The Greeks — delta, gamma, vega, theta, rho — measure risk exposures and are the practical output of the model for hedging and risk management.
Put-call parity is a no-arbitrage condition independent of the Black–Scholes distributional assumptions. It is a consistency check on any European option pricing model.

Why London students choose our quantitative finance tutoring

Postgraduate-level specialists: our tutors hold PhDs in mathematical finance, stochastic calculus, or financial economics and teach the Black–Scholes model at the level your MSc demands — from Itô’s lemma through the Feynman–Kac theorem to the Greeks.
One-on-one format: every session is built around your lecture notes, your notation, and your problem sets, whether you are at LSE, Imperial, UCL, King’s, or joining online from another London institution.
Coding integration: if your course requires Python, R, or MATLAB implementations of the Black–Scholes formula, Monte Carlo pricing, or finite-difference PDE solvers, sessions can include live coding review alongside the mathematics.

FAQ

Q: What is the Black–Scholes model?
A: A continuous-time model for pricing European options. It assumes the underlying stock follows geometric Brownian motion with constant volatility, trading is frictionless, and the risk-free rate is constant. The model derives a PDE whose solution is the no-arbitrage option price.

Q: Why doesn’t the expected return μ appear in the Black–Scholes formula?
A: Because the option is priced by replication, not by forecasting. The delta-hedged portfolio eliminates the dW_t term, making the portfolio instantaneously risk-free. The return on a risk-free portfolio must be r, so μ cancels out. The formula depends only on σ, r, S₀, K, and T.

Q: What is risk-neutral valuation?
A: A pricing technique where the expected payoff is computed under a probability measure ℚ in which all assets earn the risk-free rate. The true probabilities and expected returns are irrelevant. The option price is the discounted ℚ-expectation of the payoff.

Q: What are the limitations of the Black–Scholes model?
A: Constant volatility is the biggest — real markets show volatility smiles and skews. The model also assumes continuous trading, no transaction costs, a constant risk-free rate, and log-normal stock prices with no jumps. Each limitation has spawned an extension: stochastic volatility (Heston), jump-diffusion (Merton), and local volatility (Dupire).

Q: Do I need stochastic calculus for Black–Scholes?
A: At postgraduate level, yes. You need Itô’s lemma to derive the PDE, Girsanov’s theorem to justify the change of measure, and the Feynman–Kac theorem to link the PDE to the risk-neutral expectation. If your course is more applied, a quantitative finance tutor London programmes recommend can adjust the mathematical depth to match your syllabus.

Q: How do dividends affect the Black–Scholes formula?
A: For a continuous dividend yield q, replace S₀ with S₀e^−qT in the formula for C, and adjust d₁ to [ln(S₀/K) + (r − q + σ²/2)T] / (σ√T). The stock grows more slowly under ℚ because dividends leak value to shareholders rather than remaining in the stock.

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