FINM6041 · Applied Derivatives

Black-Scholes-Merton & Volatility

Black-Scholes-Merton & Volatility (Lecture 6 with Workshop 6) takes the risk-neutral idea of the binomial tree to its continuous-time limit. The model assumes the stock price is lognormal — ln S_T is normally distributed — and delivers a closed-form European option price: c = S₀N(d₁) − Xe^{−rT}N(d₂), with d₁ = [ln(S₀/X) + (r + σ²/2)T]/(σ√T) and d₂ = d₁ − σ√T; the put follows from put-call parity. N(·) is the standard-normal CDF, read from the tables provided in the exam using linear interpolation and the symmetry rule N(−d) = 1 − N(d). Discrete cash dividends are handled by replacing S₀ with S₀ − PV(D). The workshop half makes σ the star: implied volatility (the σ that makes the model price equal the market price), the volatility smile, and the VIX 'fear index'.

What this chapter covers

• 01The lognormal model: ln S_T is normal, and volatility scales with √T
• 02The BSM call formula c = S₀N(d₁) − Xe^{−rT}N(d₂), and the put via parity
• 03Computing d₁ = [ln(S₀/X) + (r + σ²/2)T]/(σ√T) and d₂ = d₁ − σ√T
• 04Reading N(d₁), N(d₂) by linear interpolation; the symmetry N(−d) = 1 − N(d)
• 05Discrete cash dividends: replace S₀ with S₀ − PV(D)
• 06The eight BSM assumptions and the boundary checks that sanity-test a price
• 07Volatility: historical vs implied volatility, the volatility smile, and the VIX

European call by Black-Scholes-Merton, reading N(d) from the table

Key terms

Lognormal price

The BSM assumption that S_T has a lognormal distribution — equivalently ln S_T is normal with mean ln S₀ + (μ − σ²/2)T and variance σ²T. It keeps prices positive and makes continuously-compounded returns normal.

d₁ and d₂

The two standardised inputs to BSM: d₁ = [ln(S₀/X) + (r + σ²/2)T]/(σ√T) and d₂ = d₁ − σ√T. N(d₂) is the risk-neutral probability the call finishes in-the-money.

N(x)

The standard-normal cumulative distribution function, supplied as a table in the exam. Values between rows are found by linear interpolation, and negatives via the symmetry N(−d) = 1 − N(d).

Discrete-dividend adjustment

For cash dividends with ex-dates during the option's life, replace S₀ by S₀ − PV(D) everywhere in the BSM formula; a continuous dividend yield is handled instead by S₀e^{−qT} (Lecture 7).

Implied volatility

The volatility σ that makes the BSM price equal the option's observed market price, solved numerically. It is forward-looking, which is why traders often quote options in volatility terms rather than dollars.

Volatility smile / VIX

Plotting implied volatility against strike for one expiry gives a U-shaped 'smile' — higher for deep in- and out-of-the-money options than at-the-money. The VIX is the implied volatility of S&P 500 options, popularly the market's 'fear index'.

Black-Scholes-Merton & Volatility FAQ

Exam move

Make the d₁/d₂ pipeline muscle memory: write d₁ = [ln(S₀/X) + (r + σ²/2)T]/(σ√T), compute the numerator and σ√T separately, then d₂ = d₁ − σ√T, then read N(d₁) and N(d₂) by interpolation, then assemble c = S₀N(d₁) − Xe^{−rT}N(d₂) and get the put from parity. Practise the table technique explicitly — interpolate between rows and use N(−d) = 1 − N(d) — because a small slip here moves the final price. Learn the one substitution that unifies the whole options block: S₀ − PV(D) for discrete dividends, and S₀e^{−qT} for a yield, an index, a currency or a futures underlying. Finish on the conceptual layer the recap flags as examinable — the eight assumptions, and implied volatility, the smile and the VIX — since these show up as short 'explain' prompts worth easy marks.