3.3.3 The European Pricing Formulas (Closed Forms, Dividends, FX, Futures)

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma.

1. Intuition & Practical Objective

This is the complete closed-form lookup page for vanilla European options. The practical objective: one master formula (the generalized BSM with cost-of-carry $b$ ) that specializes to every vanilla model - plain stock, index with dividend yield $q$ , currency, and futures. Each specialization is a single change of $b$ . Every formula and every number below was verified against Haug’s Complete Guide to Option Pricing Formulas (numerically reproduced) and cross-checked against Shreve II Ch 4–5 and Hull Ch 15/17/18.

2. Mathematical Ground Truth & Derivations

2.1 The Generalized Black–Scholes–Merton formula (Haug §1.1.6)

c = S e^{(b - r) T} N (d_{1}) - X e^{- r T} N (d_{2})

p = X e^{- r T} N (- d_{2}) - S e^{(b - r) T} N (- d_{1})

d_{1} = \frac{ln ( S / X ) + ( b + \frac{1}{2} σ ^{2} ) T}{σ T}, d_{2} = d_{1} - σ T = \frac{ln ( S / X ) + ( b - \frac{1}{2} σ ^{2} ) T}{σ T} .

Cost-of-carry dictionary - the entire menu is one variable $b$ :

Model	$b$	Call $c$
BSM stock (1973), no div	$r$	$S N (d_{1}) - X e^{- r T} N (d_{2})$
Merton (1973), cont. yield $q$	$r - q$	$S e^{- qT} N (d_{1}) - X e^{- r T} N (d_{2})$
Black-76 futures/forward (1976)	$0$	$e^{- r T} [F N (d_{1}) - X N (d_{2})]$
Garman–Kohlhagen currency (1983)	$r - r_{f}$	$S e^{- r_{f} T} N (d_{1}) - X e^{- r T} N (d_{2})$
Asay margined futures (1982)	$0, r = 0$	$F N (d_{1}) - X N (d_{2})$

2.2 Put–Call Parity (Haug §1.2; Shreve II 4.5.29; Hull 18.1) - model-free

c - p = S e^{(b - r) T} - X e^{- r T} = e^{- r T} (S e^{b T} - X) .

Special cases: stock $c - p = S - X e^{- r T}$ ; continuous yield $c - p = S e^{- qT} - X e^{- r T}$ ; futures $c - p = (F - X) e^{- r T}$ ; currency $c - p = S e^{- r_{f} T} - X e^{- r T}$ . Parity is a pure no-arbitrage identity - it holds regardless of volatility and even for non-lognormal dynamics.

2.3 Price bounds (Hull 17, 18)

c \geq max (S e^{(b - r) T} - X e^{- r T}, 0), p \geq max (X e^{- r T} - S e^{(b - r) T}, 0) .

2.4 Interpretation

$N (d_{2})$ = risk-neutral probability that the option is exercised (Shreve II; Hull 15.8).
$S e^{(b - r) T} N (d_{1})$ = discounted expected stock price in the money under $Q$ .
As $σ \to 0$ , a call limits to the forward value $S e^{(b - r) T} - X e^{- r T}$ (Hull §15.8).
American call, no dividends: early exercise is never optimal, so the BSM formula is also the American price (Haug §1.2). American puts and dividend-paying calls need the tree/approximation machinery (Haug Ch 3).

3. Computational Implementation - the full pricing engine

A single function parameterized by $b$ , with every specialization verified against Haug’s published numbers. Stdlib only.

4. Failure Modes & First-Principles Breakdowns

Wrong $b$ for the instrument. Pricing a currency option with $b = r$ (instead of $r - r_{f}$ ) or an index with $b = r$ (instead of $r - q$ ) misprices by exactly the missing $e^{- qT}$ / $e^{- r_{f} T}$ factor. The cost-of-carry dictionary is the first thing to check.
Futures vs forward confusion. Black-76 uses $b = 0$ because under $Q$ a futures price is a martingale (drift zero; Hull 18.6; Shreve II 5.6.6). Conflating it with the forward (which has drift $r - q$ ) changes the $d_{1}$ drift term.
Dividend handling - yield vs discrete. The $e^{- qT}$ form is for continuous yield (index/FX); for known dollar dividends you replace $S_{0} \to S_{0} - I$ (PV of dividends; Hull 15.12), not use the yield form. Mixing them is a classic error.
Parity is only no-arbitrage, not a model statement. Parity holds for any dynamics; do not read it as evidence the lognormal formula is correct.

5. References

Haug, The Complete Guide to Option Pricing Formulas
Shreve, Stochastic Calculus for Finance II
Hull, Options, Futures, and Other Derivatives

6. Connected Graph Bridges

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