Core Equations: Options, Futures & Other Derivatives#
John C. Hull (10th Edition)#
David Chan, Claude Sonnet 4.6 AI-Symbiosis Research · April 2026
Page 1 — Core Pricing Equations & Identities#
Foundation — Geometric Brownian Motion (Ch. 13)#
Stock price dynamics under GBM:
$$\frac{\Delta S}{S} \sim \mathcal{N}(\mu \Delta t,\ \sigma^2 \Delta t)$$$$\ln S_T \sim \mathcal{N}\!\left(\ln S_0 + \left(\mu - \frac{\sigma^2}{2}\right)T,\ \sigma^2 T\right)$$Note: Stock prices are lognormally distributed — not normally. The $-\sigma^2/2$ correction is Itô’s lemma in action: variance drags the expected log-price down. This is the single assumption that underpins everything in the book.
Identity 1 — Put-Call Parity (Ch. 10)#
For European options on a non-dividend-paying stock:
$$c + Ke^{-rT} = p + S_0$$With dividends (present value $D$):
$$c + D + Ke^{-rT} = p + S_0$$For American options — bounds only (no equality):
$$S_0 - K \leq C - P \leq S_0 - Ke^{-rT}$$Note: Put-call parity is a no-arbitrage identity. If it breaks, you can lock in a riskless profit by buying the cheap side and selling the expensive side. It holds regardless of the pricing model — even if BSM is wrong, parity holds.
Identity 2 — The BSM Differential Equation (Ch. 14)#
The PDE that any derivative price $\Pi$ must satisfy:
$$\frac{\partial \Pi}{\partial t} + rS\frac{\partial \Pi}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 \Pi}{\partial S^2} = r\Pi$$Note: This is the heart of options pricing. It says: the rate of change of the option’s value equals what you’d earn on a risk-free investment of the same value. Any security whose price depends on $S$ satisfies this equation — calls, puts, barriers, exotics.
Identity 3 — Black-Scholes-Merton Formulas (Ch. 14)#
European call and put on non-dividend-paying stock:
$$c = S_0 N(d_{1}) - Ke^{-rT} N(d_{2})$$$$p = Ke^{-rT} N(-d_{2}) - S_0 N(-d_{1})$$where:
$$d_{1} = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}, \qquad d_{2} = d_{1} - \sigma\sqrt{T}$$Note: $N(d_{2})$ is the risk-neutral probability the option expires in-the-money. $N(d_{1})$ is the delta — how much the option price moves per $1 move in the stock. The formula is just: (expected stock price × probability of exercise) minus (discounted strike × probability of exercise).
With continuous dividend yield $q$:
$$c = S_0 e^{-qT} N(d_{1}) - Ke^{-rT} N(d_{2})$$where $d_{1} = \dfrac{\ln(S_0/K) + (r - q + \sigma^2/2)T}{\sigma\sqrt{T}}$
Identity 4 — The Greeks (Ch. 18)#
For a European call on a non-dividend-paying stock:
| Greek | Formula | Meaning |
|---|---|---|
| Delta $\Delta$ | $N(d_{1})$ | $\partial c / \partial S$ — hedge ratio |
| Gamma $\Gamma$ | $\dfrac{N'(d_{1})}{S_0 \sigma \sqrt{T}}$ | $\partial^2 c / \partial S^2$ — convexity |
| Theta $\Theta$ | $-\dfrac{S_0 N'(d_{1})\sigma}{2\sqrt{T}} - rKe^{-rT}N(d_{2})$ | $\partial c / \partial t$ — time decay |
| Vega $\mathcal{V}$ | $S_0 \sqrt{T}\, N'(d_{1})$ | $\partial c / \partial \sigma$ — vol sensitivity |
| Rho $\rho$ | $KTe^{-rT}N(d_{2})$ | $\partial c / \partial r$ — rate sensitivity |
The BSM PDE restated in Greeks for a delta-neutral portfolio ($\Delta = 0$):
$$\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma = r\Pi$$Note: Theta and Gamma are always opposite signs for a delta-neutral portfolio. If you are long gamma (convex payoff), you pay theta (time decay). If you are short gamma (sold options), you collect theta but bleed when the market moves. Gamma is what you buy; theta is what you pay for it.
Identity 5 — Binomial Tree Risk-Neutral Pricing (Ch. 12)#
At each node, risk-neutral probability $p$:
$$p = \frac{e^{(r-q)\Delta t} - d}{u - d}$$with $u = e^{\sigma\sqrt{\Delta t}}$, $d = e^{-\sigma\sqrt{\Delta t}} = 1/u$
Option price at each node:
$$f = e^{-r\Delta t}\left[p f_u + (1-p) f_d\right]$$Alternative equal-probability parameterization ($p = 0.5$):
$$u = e^{(r-q-\sigma^2/2)\Delta t + \sigma\sqrt{\Delta t}}, \qquad d = e^{(r-q-\sigma^2/2)\Delta t - \sigma\sqrt{\Delta t}}$$Note: The binomial tree is the BSM formula in discrete time. As $\Delta t \to 0$, it converges to BSM exactly. The key insight: under risk-neutral probabilities, all assets grow at the risk-free rate $r$. You don’t need the real-world drift $\mu$ to price derivatives.
Page 2 — Deeper Analysis & Project Connections#
Forward & Futures Pricing (Ch. 2-5)#
Cost-of-carry formula for forward price:
$$F_0 = S_0 e^{(r-q)T}$$For commodities with storage cost $u$ and convenience yield $y$:
$$F_0 = S_0 e^{(r+u-y)T}$$Optimal hedge ratio (minimum variance):
$$h^* = \rho \cdot \frac{\sigma_S}{\sigma_F}$$Number of futures contracts to hedge:
$$N^* = h^* \cdot \frac{V_A}{V_F}$$Note: $h^*$ is the regression coefficient of spot price changes on futures price changes. If $\rho = 1$ and $\sigma_S = \sigma_F$, you hedge 1:1. In practice, $\rho < 1$ introduces basis risk — the hedge is imperfect because spot and futures don’t move in lockstep.
Volatility Smile (Ch. 19)#
Implied volatility $\hat{\sigma}$ is the $\sigma$ that makes BSM match the market price. The volatility smile plots $\hat{\sigma}$ vs. strike $K$:
- Equity options (post-1987): Volatility skew — implied vol decreases as $K$ increases. Deep OTM puts are expensive. Market prices in crash risk (fat left tail).
- FX options: Symmetric smile — both deep OTM calls and puts have elevated implied vol. Market prices in jump risk in both directions.
The implied probability distribution inferred from the smile:
$$g(S_T) = e^{rT} \frac{\partial^2 c}{\partial K^2}\Bigg|_{K=S_T}$$Note: The shape of the volatility surface tells you what the market believes about the tail distribution of the underlying. A steep skew = market fears crashes. A flat smile = market thinks moves are symmetric. BSM assumes a flat smile (constant $\sigma$) — which is why it misprices tails.
Gamma-Vega Neutrality (Ch. 18)#
To make a portfolio simultaneously gamma and vega neutral using two traded options with quantities $w_1, w_2$:
$$\Gamma_{\text{portfolio}} + w_1 \Gamma_1 + w_2 \Gamma_2 = 0$$$$\mathcal{V}_{\text{portfolio}} + w_1 \mathcal{V}_1 + w_2 \mathcal{V}_2 = 0$$Solve the $2 \times 2$ system for $w_1, w_2$, then rebalance delta.
Note: Gamma-vega hedging is the practical core of options market-making. You delta-hedge continuously (cheap), but gamma and vega require additional options (expensive). The trade-off: gamma/vega neutral = expensive but stable. Delta-only = cheap but fragile to large moves or vol changes.
The Risk-Neutral Valuation Principle#
The single most important idea in the book:
In a risk-neutral world, all assets earn the risk-free rate $r$. Expected payoffs discounted at $r$ give the correct no-arbitrage price — regardless of investor risk preferences.
This means:
- You don’t need to model investor utility
- You don’t need the real-world drift $\mu$
- The only inputs are: $S_0, K, r, \sigma, T$ (and $q$ for dividends)
This is why BSM is tractable. It’s also why it fails when the risk-neutral measure doesn’t exist or isn’t unique — i.e., in incomplete markets (see volatility smile).
Relevance Map to Our Projects#
| Hull Topic | Our Project Connection |
|---|---|
| GBM / lognormal prices | BTC data in Lab 4 — log returns as HMM observations |
| BSM differential equation | Connects to Bluman-HJB paper — BSM PDE has Lie symmetry group |
| Risk-neutral valuation | Pham’s stochastic control book — HJB equation is the continuous-time version |
| Volatility smile | Regime-dependent volatility → HMM detects vol regimes (Lab 4 Choppy state = high vol) |
| Delta-gamma hedging | Fortuna strategy layer — delta-neutral positions within each regime |
| Optimal hedge ratio $h^*$ | Pairs trading in Regime 1 (Choppy) — $h^*$ is the cointegration coefficient |
| Binomial tree | Discrete-time version of continuous HMM emissions |
The Key Insight Hull Gives Us for Lab 4#
The BSM PDE and the HMM are solving the same problem from different directions:
- BSM: given the price process, derive the no-arbitrage derivative price
- HMM: given the price process, infer the hidden regime driving it
Both assume GBM. Both require $\sigma$. The difference: BSM treats $\sigma$ as constant; HMM treats $\sigma$ as regime-dependent. The volatility smile is the market’s empirical evidence that $\sigma$ is not constant — i.e., that the HMM (or some regime model) is closer to the truth than flat BSM.
Lab 4’s three regimes have measured volatilities: Bull 2.5%, Bear 3.0%, Choppy 5.3%. These map directly to regime-dependent BSM pricing — a different implied vol surface for each regime.
References#
- Hull, J.C. (2018). Options, Futures, and Other Derivatives (10th ed.). Pearson.
- Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.
- Merton, R.C. (1973). Theory of rational option pricing. Bell Journal of Economics, 4(1), 141–183.
- Cox, J., Ross, S., and Rubinstein, M. (1979). Option pricing: a simplified approach. Journal of Financial Economics, 7(3), 229–263.
- Chan, D. (2026). Lab 4 — Baum HMM Regime Detector. AI-Symbiosis Research.
- Chan, D. (2026). Lie Symmetry Analysis of the Merton HJB Equation. AI-Symbiosis Research.