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.

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.

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.

Parameter Axiom

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.

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

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.

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.

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.

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

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.

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

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.

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

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 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.

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.

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.

The Black-Scholes-Merton Result

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

The Sensitivity Gallery

Measuring the vital signs of a derivative position.

\Delta

Delta

The speed. Sensitivity to stock price changes.

Sensitivity: Hedge Ratio

\Gamma

Gamma

The acceleration. Sensitivity of Delta to stock price.

Sensitivity: Path Risk

\Theta

Theta

The time bleed. Daily loss of value due to expiry.

Sensitivity: Time Decay

\nu

Vega

The uncertainty risk. Sensitivity to market fear (IV).

Sensitivity: Uncertainty

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).

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

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

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.

16 Delta Call1-Std Dev move

50 Delta Call50/50 Coin Flip

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.

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.

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.

Execution Failure

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.

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."

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