SOPHIE Daddy Quant Blog - Stock & Options Analysis

The Laboratory Axioms

The frictionless environment required to derive the closed-form solution.

Geometric Brownian Motion

Assumes returns are Normally Distributed, meaning price levels follow a Lognormal Distribution. This prevents prices from dropping below zero (limited liability) and accounts for the compounding nature of financial growth. The mathematical form is dS = μS dt + σS dW, where μ is the drift rate and σ is the volatility parameter. This ensures that percentage changes, not absolute changes, are normally distributed—a crucial distinction that captures the multiplicative nature of financial returns.

Stochastic Constraint

Continuous Liquidity

The model assumes you can buy or sell any quantity of an asset instantly without moving the market price (zero slippage). It further assumes zero transaction costs and zero taxes, enabling infinitesimal re-hedging. This "perfect liquidity" assumption allows for continuous delta-hedging, where traders can adjust their positions instantaneously as the underlying moves. In reality, bid-ask spreads, market impact, and transaction costs create friction that professional traders must account for through more sophisticated models.

Execution Theory

Static Volatility

Volatility (σ) and interest rates (r) are assumed to be constant and known throughout the life of the option. This is the model's most famous simplification, leading to the creation of the "Volatility Surface" in practice. Real-world volatility clusters (high vol periods followed by high vol) and exhibits mean reversion. Interest rates also fluctuate, creating additional risk factors. Modern practitioners use stochastic volatility models (Heston, SABR) and term structure models to address these limitations.

Parameter Axiom

No Dividends (Original)

The original Black-Scholes model assumed no dividend payments during the option's life. This was later extended by Merton to include continuous dividend yields, adjusting the forward price to S₀e^(-qT) where q is the dividend yield. For discrete dividends, practitioners often use binomial trees or adjust the stock price by the present value of expected dividends.

Cash Flow Assumption

European Exercise

The model applies only to European-style options that can only be exercised at expiration. American options, which can be exercised early, require more complex numerical methods like binomial/trinomial trees or finite difference methods. Early exercise is optimal for American puts when they are deep in-the-money, and for American calls on dividend-paying stocks just before ex-dividend dates.

Exercise Constraint

Risk Neutrality & Martingales

The mathematical lens that ignores investor sentiment.

Risk-neutrality is a property of a "complete market." Because you can create a perfect hedge, the option's value depends only on the risk-free rate, not on how "bullish" or "bearish" the world is.

The Discounted Martingale

In this world, the discounted stock price is a "fair game." The best estimate of its future discounted value is today's price. This martingale property is the foundation of risk-neutral pricing.

S_0 = e^{-rT} \mathbb{E}^Q [ S_T ]

This equation states that today's stock price equals the discounted expected future price under the risk-neutral measure Q.

Market Completeness

A complete market allows any payoff to be replicated through dynamic trading of the underlying and risk-free bond. This replication argument eliminates arbitrage opportunities and uniquely determines option prices. The key insight: if two portfolios have identical payoffs, they must have identical prices to prevent risk-free profit.

Measure Transformation

Physical World (P)dS_t = \mu S_t dt + \sigma S_t dW_t

Drift (μ) includes the risk premium investors demand for holding the stock. This reflects actual investor preferences and risk aversion.

Risk-Neutral World (Q)dS_t = r S_t dt + \sigma S_t dW_t^Q

The drift is fixed to the risk-free rate (r). Preferences are deleted. All assets grow at the risk-free rate on average.

The Bridge: Girsanov\frac{dQ}{dP} = \exp\left(-\frac{\mu - r}{\sigma}W_T - \frac{1}{2}\left(\frac{\mu - r}{\sigma}\right)^2 T\right)

The Radon-Nikodym derivative transforms probabilities, making the discounted stock price a martingale.

The Stochastic Engine

The machinery that allows us to operate on random variables.

Girsanov Theorem

Girsanov allows us to change the probability measure. It provides the "Radon-Nikodym derivative," which acts as a filter that re-weights path probabilities so the weighted drift exactly equals r. This theorem is the mathematical foundation that transforms the physical measure (with risk premium μ) into the risk-neutral measure (with drift r).

L_T = \frac{dQ}{dP} = \exp\left( -\int \theta dW_t - \frac{1}{2} \int \theta^2 dt \right)

Where θ = (μ-r)/σ is the market price of risk

Feynman-Kac Identity

The link between finance and physics. It proves that the solution to a Heat Equation (PDE) is the same as the expectation of a random process. This is why Monte Carlo simulation works—we can either solve the PDE numerically or simulate thousands of random paths and average the payoffs.

f(x,t) = \mathbb{E}^Q \left[ e^{-r(T-t)} \Phi(S_T) \mid S_t = x \right]

Where Φ(S_T) is the option payoff at expiration

Itô's Lemma

In the stochastic world, change happens in the second order. Because (dW)² = dt, we get an extra term representing the "convexity" of the payoff—this is the source of Gamma. This quadratic variation term is what distinguishes stochastic calculus from ordinary calculus.

Key Insight

The ½σ²S²∂²f/∂S² term captures the "convexity premium" - the value created by the option's non-linear payoff structure. This is why options have time value even when at-the-money.

df = \left( \frac{\partial f}{\partial t} + rS\frac{\partial f}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 f}{\partial S^2} \right)dt + \sigma S \frac{\partial f}{\partial S} dW_t

Theta
Time decay

Delta
Price sensitivity

Gamma
Convexity

The Black-Scholes Partial Differential Equation

Applying Itô's Lemma to the option value f(S,t) and using the no-arbitrage condition yields the famous PDE:

\frac{\partial f}{\partial t} + rS\frac{\partial f}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 f}{\partial S^2} = rf

Subject to boundary conditions: f(S,T) = max(S-K,0) for a call option

The Expectation Derivation

Calculating the fair price through weighted path averaging.

The Lognormal Integral

We define the Call price as the discounted average of payoffs above strike K. Integrating against the density f(S_T) reveals the internal weights N(d₁) and N(d₂). The key insight is that we're computing a conditional expectation over the region where S_T > K.

The Core Expectation

C = e^{-rT} \int_{K}^{\infty} (S_T - K) f(S_T) dS_T

We perform a change of variable to transform this into a Standard Normal Integral using the fact that ln(S_T) ~ N(ln(S_0) + (r-σ²/2)T, σ²T).

Step 1: Substitution

Let X = ln(S_T/S₀). Then X ~ N((r-σ²/2)T, σ²T), and we can rewrite the integral in terms of the standard normal distribution.

X = \ln\left(\frac{S_T}{S_0}\right) \sim N\left(\left(r-\frac{\sigma^2}{2}\right)T, \sigma^2 T\right)

Step 2: Integration by Parts

The integral splits into two parts: the stock-weighted probability N(d₁) and the strike-weighted probability N(d₂). This decomposition reveals the economic meaning of each term.

\int S_T \phi(z) dz - K \int \phi(z) dz

The N(d₁) Weight

Representing the stock-weighted probability of exercise. Physically, this is the Delta (Δ), the amount of stock required to replicate the option. It's the probability of finishing ITM, weighted by the stock price.

d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}

The N(d₂) Weight

The simple, risk-neutral probability that the option finishes in-the-money. This is the likelihood you will actually pay the strike price K. Note that d₂ = d₁ - σ√T.

d_2 = \frac{\ln(S_0/K) + (r - \sigma^2/2)T}{\sigma\sqrt{T}}

The Black-Scholes-Merton Result

C = S_0 N(d_1) - K e^{-rT} N(d_2)

This elegant formula encapsulates the entire theory of option pricing in a single equation.

Put-Call Parity

The relationship between call and put prices is not derived from Black-Scholes but from pure arbitrage arguments. It must hold regardless of the pricing model used.

C - P = S_0 - Ke^{-rT}

This means: Long Call + Short Put = Long Forward Contract

The Sensitivity Gallery

Measuring the vital signs of a derivative position.

\Delta

Delta

The speed. Sensitivity to stock price changes. For calls: 0 < Δ < 1, for puts: -1 < Δ < 0.

Sensitivity: Hedge Ratio

\Gamma

Gamma

The acceleration. Sensitivity of Delta to stock price. Always positive for long options, highest ATM.

Sensitivity: Path Risk

\Theta

Theta

The time bleed. Daily loss of value due to expiry. Accelerates as expiration approaches.

Sensitivity: Time Decay

\nu

Vega

The uncertainty risk. Sensitivity to market fear (IV). Highest for ATM options with time remaining.

Sensitivity: Uncertainty

\rho

Rho

Interest rate sensitivity. More important for longer-dated options and deep ITM calls.

Sensitivity: Rate Risk

\frac{\partial^2 V}{\partial S \partial \sigma}

Vanna

Cross-derivative: sensitivity of Delta to volatility changes. Second-order Greek.

Sensitivity: Vol-Spot Risk

\frac{\partial^2 V}{\partial S \partial t}

Charm

Delta decay: how Delta changes with time passage. Critical for dynamic hedging.

Sensitivity: Delta Decay

Greek Relationships & Hedging

Delta-Gamma Hedging

Delta hedging requires continuous rebalancing. Gamma measures how much your Delta hedge will be "wrong" for a given move. High Gamma positions need frequent rehedging.

\text{P&L} \approx \Delta \cdot \Delta S + \frac{1}{2} \Gamma \cdot (\Delta S)^2

Theta-Gamma Relationship

For a delta-neutral position, Theta and Gamma are related through the Black-Scholes PDE. You "pay" Theta to "own" Gamma.

\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma + rS\Delta = r \cdot \text{Portfolio Value}

Trader Heuristics & Pit Wisdom

The mental models and 'oral traditions' of the options pits.

The Rule of 16

Normalizing Volatility

Market makers think in daily moves, not annual percentages. Since there are roughly 256 trading days in a year and √256 = 16, the conversion is simple:

\text{Daily \% Expected Move} \approx \frac{\sigma_{\text{annual}}}{16}

32% IV2% Daily Move

ATM Straddle Rule

The Linear Approximation

For an At-The-Money (ATM) straddle, the Black-Scholes complex math collapses into a linear function of Price (S) and Volatility (σ). This provides instant position sizing and risk assessment.

\text{Straddle Price} \approx 0.8 \cdot S \cdot \sigma \cdot \sqrt{T}

"This provides the cost of the 'Uncertainty Envelope' instantly."

Example: $100 stock, 25% IV, 30 days → Straddle ≈ $100 × 0.25 × √(30/365) × 0.8 ≈ $5.70

The Greek Rent

Theta vs. Gamma

A delta-hedger who is "Long Gamma" (expecting moves) is "paying rent" via Theta (decay). In an efficient market, they balance out perfectly:

\text{Theta Loss} \approx \text{Gamma Gain} \times \frac{\sigma^2 S^2}{2}

"You pay for the privilege of being random."

The 20-80 Rule

Delta as Probability

Traders use Delta (Δ) as a raw probability proxy. A 25-delta call is treated as having a 25% chance of finishing in-the-money. This heuristic works because N(d₂) ≈ Delta for ATM options.

16 Delta Call1-Std Dev move

50 Delta CallAt-The-Money

84 Delta Call1-Std Dev ITM

Pro Tip: The 25-delta put and 25-delta call define the "1-standard deviation" wings used in risk reversal strategies.

The Universal Standard

Implied Volatility (IV)
as a "Ruler"

"We don't use Black-Scholes because it's correct; we use it to see where the market thinks it is currently wrong."

Market Skew Analysis

If OTM Puts have higher IV than OTM Calls, the market is pricing in "Crashophobia." This deviation from the flat-vol axiom tells you more about investor fear than any raw price chart ever could.

Weekend Effect

Calendar Time vs. Trading Time

Options decay over calendar time, not trading time. A Friday-to-Monday passage costs 3 days of Theta, but the market is only open 1 day. This creates predictable patterns in option pricing around weekends and holidays.

Trading Insight: Short-dated options often see accelerated decay on Fridays as weekend risk is priced in.

Pin Risk

Expiration Magnetism

Stock prices tend to "pin" to strike prices at expiration due to delta hedging flows. Market makers who are short options will buy stock as it approaches the strike (to hedge their short delta), creating support/resistance.

Gamma Squeeze: When large amounts of call options are in-the-money, dealers must buy stock to hedge, amplifying upward moves.

Modern Extensions & Practical Applications

How Black-Scholes evolved to meet real-world trading demands.

Stochastic Volatility Models

Modern models treat volatility as a random variable with its own dynamics. The Heston model adds a second stochastic process for volatility, while SABR models the forward price and volatility jointly. These capture the volatility smile and term structure more accurately.

Heston, SABR, Local Vol

Jump Diffusion Models

Merton's extension adds sudden price jumps to the geometric Brownian motion. This better captures crash risk and explains why out-of-the-money puts trade at higher implied volatilities. The model includes jump frequency, jump size distribution, and jump timing.

Merton Jump-Diffusion

Numerical Methods & Implementation

Binomial Trees

Cox-Ross-Rubinstein trees discretize the continuous process into up/down moves. Perfect for American options and path-dependent payoffs.

u = e^(σ√Δt), d = 1/u

Monte Carlo

Simulate thousands of price paths and average the payoffs. Excellent for exotic options and high-dimensional problems.

S(T) = S₀ × e^((r-σ²/2)T + σ√T×Z)

Finite Difference

Solve the Black-Scholes PDE directly using numerical grids. Handles complex boundary conditions and early exercise features.

Explicit, Implicit, Crank-Nicolson

The Practitioner's Toolkit

Market Making

• Delta-neutral inventory management
• Gamma scalping for profit extraction
• Volatility surface interpolation
• Risk limits and position sizing

Portfolio Management

• Volatility forecasting models
• Correlation trading strategies
• Tail risk hedging programs
• Dynamic hedging algorithms

The Map vs. The Territory

Recognizing the structural failure points of the mathematical model.

Fat Tails (Kurtosis)

Real market returns have "Heavy Tails." The model assumes a 10-sigma crash happens once every 10 billion years; in the real "Territory," these events happen almost every decade. The normal distribution underestimates extreme events by orders of magnitude. This is why VIX spikes to 80+ during crises, not the 20-30 that normal distributions would predict.

Statistical Bias

Gap & Liquidity Risk

The model assumes prices move continuously. In reality, markets Gap overnight from $100 to $80. A delta-hedger cannot adjust their position mid-gap, leading to "Jump Risk" bankruptcy. This is particularly dangerous for short option positions during earnings announcements or geopolitical events.

Execution Failure

Stochastic Volatility

Volatility is not constant—it clusters, mean-reverts, and correlates with price movements. The "volatility smile" exists because traders know the model is wrong and adjust prices accordingly. Modern models like Heston and SABR attempt to capture this stochastic nature.

Parameter Instability

Transaction Costs

Real trading involves bid-ask spreads, commissions, and market impact. High-frequency delta hedging becomes prohibitively expensive. This creates "hedging bands" where traders only rehedge when Delta moves beyond certain thresholds, introducing basis risk.

Friction Reality

Interest Rate Risk

Interest rates are not constant, especially for longer-dated options. Changes in the yield curve affect option values through Rho, and the correlation between rates and stock prices creates additional complexity that the basic model ignores.

Rate Sensitivity

The Volatility Smile

Professional traders don't quote options in dollars; they quote them in "Volatility points." The Smile is the map of how much the model is currently underestimating the probability of extreme events.

OTM PutsHigher IV (Crash Protection)

ATM OptionsBaseline IV

OTM CallsLower IV (Momentum Bias)

Masterclass Takeaway

"Black-Scholes is the first map of a random world. It is flawed, elegant, and essential. It doesn't tell you the price; it tells you the language of value."

Modern Usage

Today, Black-Scholes serves as the "numeraire" for option pricing—a common language that allows traders to compare the relative cheapness of different options by converting prices to implied volatilities.

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