Options Pricing

🎯 Options Pricing: The Secret Language of Smart Money

Imagine you’re at a magical ticket booth where you can buy the RIGHT to do something later—but you don’t HAVE to do it. That’s what options are! Let’s decode how smart traders figure out what these magical tickets are really worth.

🎪 The Ticket Booth Story

Think of options like buying a ticket to a concert that’s happening in 3 months.

• The ticket gives you the RIGHT to go to the concert
• You DON’T HAVE to go if you change your mind
• The ticket has a price based on how popular the concert might be

Options work the same way for stocks!

💎 Intrinsic Value: The “Real Stuff” Inside

What Is It?

Intrinsic value is like asking: “If I used this ticket RIGHT NOW, would I make money?”

Think of it like a coupon:

• You have a coupon that lets you buy a toy for $10
• The toy now costs $15 in the store
• Your coupon is worth at least $5 RIGHT NOW (the difference!)

The Simple Math

For Call Options (right to BUY):

Intrinsic Value = Stock Price - Strike Price
(if positive, otherwise it's $0)

For Put Options (right to SELL):

Intrinsic Value = Strike Price - Stock Price
(if positive, otherwise it's $0)

Real Example 🍎

Apple stock is trading at $150.

You own a call option with strike price $140.

Your intrinsic value = $150 - $140 = $10

This is the “real stuff” your option is worth if you exercised it today!

⏰ Time Value: The “Hope and Possibility” Part

What Is It?

Time value is the extra money people pay for the POSSIBILITY that things might get even better!

Think of it like:

• A lottery ticket before the draw (lots of hope!)
• The same ticket after the draw (no hope left)

The Formula

Time Value = Option Price - Intrinsic Value

Why Does Time Matter?

More time = More possibilities = More value!

Real Example ⏰

Your option costs $15 total. Intrinsic value is $10.

Time value = $15 - $10 = $5

That $5 is what you’re paying for “hope and time”!

🎯 In-The-Money, At-The-Money, Out-Of-The-Money

Think of it Like a Dart Game! 🎯

Imagine throwing darts at a target:

graph TD
    A["🎯 Strike Price"] --> B["ITM: In-The-Money"]
    A --> C["ATM: At-The-Money"]
    A --> D["OTM: Out-Of-The-Money"]
    B --> E["Has real value NOW"]
    C --> F["Right on the edge"]
    D --> G["Only has hope value"]

For CALL Options (right to BUY)

For PUT Options (right to SELL)

The Golden Rule

• ITM options = More expensive (have real value)
• OTM options = Cheaper (only hope value)
• ATM options = Middle ground

⚖️ Put-Call Parity: The Balance Equation

The Big Idea

Put-call parity is like a see-saw that MUST stay balanced!

If calls and puts with the same strike and expiration get out of balance, smart traders rush in to fix it (and make free money doing it!).

The Magic Formula

Call + Cash = Put + Stock

Or more precisely:

C + K/(1+r)^t = P + S

Where:

• C = Call option price
• P = Put option price
• K = Strike price
• S = Stock price
• r = Interest rate
• t = Time to expiration

Why It Matters

If this equation is WRONG, there’s FREE MONEY (arbitrage)!

Example:

• Stock = $100
• Call ($100 strike) = $8
• Put ($100 strike) = $5
• Something’s off? Traders pounce!

🎛️ The Greeks: Your Dashboard of Risk

Meet the Greek Alphabet of Options!

The Greeks tell you HOW your option will react to changes. Think of them as dials on a control panel.

graph TD
    A["Option Price"] --> B["Delta Δ"]
    A --> C["Gamma Γ"]
    A --> D["Theta Θ"]
    A --> E["Vega ν"]
    A --> F["Rho ρ"]
    B --> G["Stock moves"]
    C --> H["Delta changes"]
    D --> I["Time passes"]
    E --> J["Volatility changes"]
    F --> K["Interest rates change"]

📊 Delta (Δ) - The Speed Meter

What it measures: How much your option moves when the stock moves $1

Simple rule:

• Calls have positive delta (0 to 1)
• Puts have negative delta (-1 to 0)

📈 Gamma (Γ) - The Acceleration

What it measures: How fast Delta itself changes

Think of it like:

• Delta is your SPEED
• Gamma is your ACCELERATION

High gamma = Delta changes quickly = More exciting (and risky)!

⏳ Theta (Θ) - The Time Thief

What it measures: How much value you LOSE each day

Example: Theta = -$0.05 means your option loses 5 cents every day!

The cruel truth: Time decay speeds up near expiration!

🌊 Vega (ν) - The Excitement Meter

What it measures: How your option reacts to volatility changes

High Vega = Your option LOVES when markets get wild!

Example: Vega = $0.15 means if volatility goes up 1%, your option gains $0.15!

💰 Rho (ρ) - The Interest Rate Friend

What it measures: How your option reacts to interest rate changes

Usually the SMALLEST effect—most traders ignore it!

📊 Implied Volatility: The Market’s Fear Gauge

What Is It?

Implied volatility (IV) is the market’s GUESS about how crazy the stock will move in the future.

Think of it like:

• Weather forecast for stock movement
• The “fear gauge” of the market

High IV vs Low IV

Real Example 🌪️

Before a company announces earnings:

• IV jumps to 60% (everyone expects drama!)
• Options become VERY expensive

After earnings are announced:

• IV crashes to 25% (mystery solved)
• This is called “IV crush”

📜 Historical Volatility: Looking in the Rearview Mirror

What Is It?

Historical volatility (HV) measures how much a stock ACTUALLY moved in the past.

Think of it like:

• Checking a roller coaster’s past rides
• The FACTS, not predictions

How It’s Calculated

HV = Standard deviation of past returns
     (usually 20-30 days)

IV vs HV: The Key Comparison

Trading Insight:

• IV > HV = Options might be OVERPRICED
• IV < HV = Options might be UNDERPRICED

🧮 Black-Scholes Model: The Nobel Prize Formula

The Story

In 1973, two brilliant minds (Fischer Black and Myron Scholes) cracked the code for pricing options. They won the Nobel Prize for it!

What It Does

The Black-Scholes model calculates the “fair price” of an option using 5 ingredients:

graph TD
    A["Black-Scholes"] --> B["Stock Price S"]
    A --> C["Strike Price K"]
    A --> D["Time to Expiration t"]
    A --> E["Risk-Free Rate r"]
    A --> F["Volatility σ"]
    B & C & D & E & F --> G["Option Price!"]

The 5 Magic Ingredients

The Formula (Don’t Panic!)

Call = S × N(d1) - K × e^(-rt) × N(d2)

Where:
d1 = [ln(S/K) + (r + σ²/2)t] / (σ√t)
d2 = d1 - σ√t
N() = Normal distribution

What This Means in Plain English

The formula basically says:

“The option price equals the expected value of what you’d get, minus what you’d pay, adjusted for time and probability.”

Real Example 🎲

Inputs:

• Stock = $100
• Strike = $100
• Time = 30 days
• Volatility = 25%
• Interest rate = 5%

Black-Scholes says: Call option ≈ $3.50

🎓 Quick Summary: The Big Picture

graph TD
    A["Option Price"] --> B["Intrinsic Value"]
    A --> C["Time Value"]
    B --> D["Real value NOW"]
    C --> E["Hope for LATER"]
    C --> F["Affected by Greeks"]
    F --> G["Delta: Stock moves"]
    F --> H["Theta: Time decay"]
    F --> I["Vega: Volatility"]
    F --> J["Gamma: Delta changes"]

The Golden Rules

1. Intrinsic value = What it’s worth RIGHT NOW
2. Time value = Extra for hope and possibility
3. ITM/ATM/OTM = Where you are vs the strike
4. Put-Call Parity = The balance that must exist
5. Greeks = Your risk dashboard
6. IV = What market EXPECTS
7. HV = What actually HAPPENED
8. Black-Scholes = The master formula

🚀 You’ve Got This!

Options pricing might seem complex, but it’s really just:

The value of having choices + the value of having time

Now you understand:

• ✅ Why options cost what they cost
• ✅ How to read the Greeks
• ✅ Why volatility matters so much
• ✅ The Nobel Prize-winning formula behind it all

You’re now speaking the secret language of smart money! 💰🎯