+++
date = '2026-04-15T00:00:00-07:00'
draft = false
title = 'Core Equations: Options, Futures & Other Derivatives'
description = 'Key algorithms, equations and identities from Hull (10th ed.) — Black-Scholes-Merton, Greeks, binomial trees, put-call parity, and volatility smile.'
tags = ['options', 'derivatives', 'black-scholes', 'greeks', 'finance', 'quant']
+++

# Core Equations: Options, Futures & Other Derivatives
## John C. Hull (10th Edition)

**David Chan, Claude Sonnet 4.6**
*AI-Symbiosis Research · April 2026*

📄 [Download PDF](/files/hull_options_futures_core.pdf) · [Download MD](/files/hull_options_futures_core.md)

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## Page 1 — Core Pricing Equations & Identities

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### 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.

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### 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.

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### 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.

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### 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 $d_{1} = \dfrac{\ln(S_0/K) + (r - q + \sigma^2/2)T}{\sigma\sqrt{T}}$

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### 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.**

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### 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.

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## Page 2 — Deeper Analysis & Project Connections

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### 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.

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### 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.

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### 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.

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### 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:
1. You don't need to model investor utility
2. You don't need the real-world drift $\mu$
3. 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).

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### 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 |

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### 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.

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## References

1. Hull, J.C. (2018). *Options, Futures, and Other Derivatives* (10th ed.). Pearson.
2. Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. *Journal of Political Economy*, 81(3), 637–654.
3. Merton, R.C. (1973). Theory of rational option pricing. *Bell Journal of Economics*, 4(1), 141–183.
4. Cox, J., Ross, S., and Rubinstein, M. (1979). Option pricing: a simplified approach. *Journal of Financial Economics*, 7(3), 229–263.
5. Chan, D. (2026). Lab 4 — Baum HMM Regime Detector. *AI-Symbiosis Research.*
6. Chan, D. (2026). Lie Symmetry Analysis of the Merton HJB Equation. *AI-Symbiosis Research.*