SOPHIE Daddy Quant Blog - Stock & Options Analysis

01Module

The Epistemology of Price

The Illusion of the Scalar.

In classical economics, price is a scalar—a single number representing the intersection of supply and demand ($100). In reality, price is a vector field. The current spot price tells you where the market is, but it tells you nothing about the texture of the market's beliefs.

"The Spot Price is the collapsed wave function. The Option Chain is the uncollapsed probability cloud. To trade effectively, you must study the cloud, not just the lightning strike."

The Hidden Variables

The Spot (S_t)

The current consensus value.

The Volatility (σ)

The speed of change.

The Skew

The fear of the downside.

The Kurtosis

The risk of extreme events.

The Tale of Two Stocks

Stock A: Utility Co.

$100.00

RND Shape

Tall & Narrow (Leptokurtic)

Stock B: BioTech

$100.00

RND Shape

Bimodal (Camel Humps)

Both cost $100. But Stock B implies a 50% chance of $0 and a 50% chance of $200. The spot price masks the risk.

The Blind Spot of Linearity

Standard "Linear Analysis" (Chart patterns, Moving Averages) operates in 2D. It ignores the Z-axis (Implied Volatility).

The Butterfly Advantage: By constructing a butterfly, you are essentially taking a core sample of the Z-axis at a specific price point, allowing you to profit from the shape of the distribution rather than the direction of the price.

02Module

Theoretical Foundations

From Market Prices to Probability Distributions.

The Transformation Pipeline

How raw market data becomes a probability map.

Input

Option Prices

C(K)

Discrete data points from the option chain (bid/ask quotes).

The Engine

Differentiation

∂²C / ∂K²

Calculating the curvature (convexity) of the price curve.

Output

Probability Density

f(K)

The implied likelihood of the asset price ending at K.

The Breeden-Litzenberger Theorem (1978)

The mathematical link between curvature and probability.

f(K)=e^rT·∂²C(K, T)∂K²

In Plain English:The probability density f(K) at a specific price K is exactly proportional to the convexity (second derivative) of the Call Option pricing function.

The Intuitive Proof (Digital Link)

Step 1: The Spread

Buy Call(K) and Sell Call(K+1). This creates a Call Spread.

Limit → Binary Option (Pays $1 if S > K)

Step 2: The Butterfly

Buy Call Spread(K) and Sell Call Spread(K+1). This difference of differences creates the Butterfly.

Limit → Density (Pays $1 if S = K)

P-Measure (Physical)

Real World

• Includes Risk Premium (Drift = μ)
• Subjective & Hard to Estimate
• Used for: Risk Management (VaR)

Q-Measure (Risk-Neutral)

Pricing World

• Risk Premium Removed (Drift = r)
• Implied directly from Prices
• Used for: Derivatives Pricing

03Module

The Butterfly Spread

The 'Atomic Unit' of Probability.

Market Neutral • High Kurtosis

The Sharpshooter's Strategy

While a Straddle buys the entire market variance (betting on movement), a Butterfly Spread targets a specific price outcome (betting on location). It is a limited-risk, limited-profit strategy that combines a Bull Spread and a Bear Spread.

The Body

Sold options. The "Pin" target. High Theta decay.

The Wings

Bought options. The Protection. Caps risk.

The Payoff

Very High Reward-to-Risk ratio (often 5:1 or 10:1).

Why Use It?

Low Capital Requirement
Defines Maximum Risk
Exploits High Volatility
Precise Probability Tool

Strategy Variants

+1 CallStrike K - ΔK

-2 CallsStrike K (Center)

+1 CallStrike K + ΔK

Best for: Typical RND extraction. Uses Call liquidity. Debit spread.

The Profit Equation

Max ProfitΔK - Debit Paid

Max RiskDebit Paid

Break EvenK ± (ΔK - Debit)

Payoff Diagram

At Expiry

The Pin Risk: Maximum profit is achieved only if the stock closes exactly at K. This is statistically rare.

Finite Difference:

V_fly ≈ P(S_T = K)

The Greeks of the Fly

How risk behaves across the "Tent".

Delta

Neutral. It forms an "S-Curve". Positive on the left wing, Negative on the right wing. Zero exactly at K.

Gamma

Explosive. Massive positive Gamma at the center (price instability), negative Gamma at the wings.

Theta

Positive. Time is your best friend. Theta is highest at the body (Strike K) and accelerates near expiry.

Vega

Usually Short. You generally want volatility to collapse (IV Crush) so the stock settles at K.

The Finite Difference Approximation

We approximate the continuous second derivative using the Central Difference method with a step size of ΔK. This mathematical bridge allows us to translate option prices directly into probability mass.

∂²C∂K²≈C(K+ΔK) - 2C(K) + C(K-ΔK)(ΔK)²

04Module

Trading Applications

Alpha Generation via Distribution Analysis

Directional Volatility

Trading the "Smirk"

Equity markets typically exhibit a "Skew" where OTM Puts trade at higher IV than OTM Calls (Crash protection is expensive). When this skew gets too steep or inverts, opportunities arise.

The Trade Setup: Risk Reversal

Bullish Skew: Sell Expensive Puts (Short Vol) / Buy Cheap Calls (Long Vol).
Funded Play: The premium from selling the put often finances the call completely (Zero-Cost Collar).

RND Logic

The market's RND is heavily "Left Skewed" (fat left tail). You believe the distribution is actually more symmetric or right-skewed.

Puts (Overpriced)Calls (Cheap)

Subjective vs. Risk-Neutral Density (SPD vs RND)

Alpha comes from the divergence between RND (What option prices imply) and SPD (Your proprietary forecast). If your model shows a higher probability density at Strike K than the RND implies, the Butterfly at K is statistically cheap.

Comparative Analysis

Feature	Long Butterfly	Iron Condor	Straddle
View	Precision / Specific Target	Range / "Somewhere inside"	Directionless Volatility
Cost	Very Low (Debit)	Credit Received	Very High (Debit)
Max Risk	Debit Paid	Wing Width - Credit	Unlimited (Short) / Debit (Long)

05Module

Extraction Methodologies

The art of smoothing the smile.

Shimko's Method (1993)

Invert Black-Scholes: Convert market prices C(K) into Implied Volatility points σ(K).
Interpolate: Fit a quadratic or cubic spline to the σ(K) smile. This creates a smooth continuous function.
Re-Price: Feed the smoothed σ(K) back into Black-Scholes to get a dense set of Call prices.
Differentiate: Apply the Breeden-Litzenberger formula to the smoothed prices.

Malz's Delta Space (FX)

Common in Forex markets where strikes are quoted in Delta (Δ) rather than price.

Risk Reversal (RR)

Measures Skew (Slope at ATM).

Strangle (Butterfly)

Measures Kurtosis (Curvature at ATM).

Continue Learning

Read Full Research Paper

Calculations assume European options and frictionless markets.

Unlocking the Volatility Surface