Options Pricing

Options Pricing: The Secret Recipe Behind Every Option’s Price Tag

The Big Picture: What Determines an Option’s Price?

Imagine you’re at a carnival, and there’s a booth selling “lottery tickets” for prizes. The price of each ticket depends on:

• How likely you are to win (the game’s difficulty)
• How much time is left before the booth closes
• How wild the prizes might change in value

Options work the same way! The price of an option (called the premium) isn’t random. It’s calculated using special ingredients called The Greeks, plus factors like time and how crazy the market is acting.

Meet The Greeks: Your Option Pricing Superheroes

Think of The Greeks as five different superpowers that tell you HOW your option’s price will change. Each one measures something different.

graph TD
    A["Option Price"] --> B["Delta Δ"]
    A --> C["Gamma Γ"]
    A --> D["Theta Θ"]
    A --> E["Vega V"]
    A --> F["Rho ρ"]
    B --> G["Stock Movement"]
    C --> H[Delta's Speed]
    D --> I["Time Decay"]
    E --> J["Volatility"]
    F --> K["Interest Rates"]

Delta (Δ): The Direction Detective

Simple Idea: Delta tells you how much your option price moves when the stock moves $1.

The Lemonade Stand Story

Imagine you have a coupon that lets you buy lemonade for $1 (when it normally costs $2).

• If lemonade prices go UP to $3, your coupon becomes MORE valuable
• If lemonade prices go DOWN to $0.50, your coupon becomes LESS valuable

Delta measures this relationship!

Delta Numbers Explained

Call Options: Delta is positive (0 to +1)

• Stock goes UP → Call value goes UP

Put Options: Delta is negative (-1 to 0)

• Stock goes DOWN → Put value goes UP

Real Example

You own a call option with Delta = 0.60

• Stock price goes UP $2
• Your option goes UP: $2 × 0.60 = $1.20

Gamma (Γ): The Acceleration Expert

Simple Idea: Gamma tells you how FAST Delta itself changes.

The Bicycle Story

Imagine riding a bicycle down a hill:

• Delta = Your current speed (10 mph, 20 mph, etc.)
• Gamma = How fast you’re speeding up (acceleration)

When you’re near the bottom of the hill (option is “at-the-money”), your acceleration is highest. That’s when Gamma is biggest!

Why Gamma Matters

Real Example

Your call has Delta = 0.50 and Gamma = 0.05

• Stock goes UP $1
• New Delta = 0.50 + 0.05 = 0.55
• Now your option is MORE sensitive to the next $1 move!

Theta (Θ): The Time Thief

Simple Idea: Theta tells you how much value your option loses each day just from time passing.

The Ice Cream Cone Story

You buy an ice cream cone on a hot day. Every minute that passes:

• The ice cream melts a little
• It becomes less valuable
• Eventually, it’s just a soggy cone

Options melt too! This is called Time Decay.

graph TD
    A["Option Value"] --> B["30 Days Left: $5.00"]
    B --> C["15 Days Left: $3.50"]
    C --> D["7 Days Left: $2.00"]
    D --> E["1 Day Left: $0.50"]
    E --> F["Expiration: Maybe $0"]

Theta Numbers

• Theta = -0.05 means your option loses 5 cents per day
• Theta = -0.15 means your option loses 15 cents per day

Important: Theta is ALWAYS negative for option buyers (you LOSE value). Time decay SPEEDS UP as expiration approaches!

Real Example

You buy a call option for $3.00 with Theta = -0.10

• After 10 days (if nothing else changes)
• Option value = $3.00 - (10 × $0.10) = $2.00
• You lost $1 just from time!

Time Decay: The Sneaky Value Stealer

Time Decay is the practical effect of Theta. Let’s understand it deeper.

The Countdown Clock

Think of your option like a parking meter:

• Full meter = Lots of time = Valuable
• Low meter = Little time = Less valuable
• Expired = No time = Worthless (if out of the money)

The Time Decay Curve

Time decay isn’t constant. It’s like a ball rolling downhill:

Why This Matters

Option Buyers: Time is your ENEMY

• Every day costs you money
• Need the stock to move FAST

Option Sellers: Time is your FRIEND

• Every day puts money in your pocket
• Want the stock to stay still

Implied Volatility (IV): The Fear & Excitement Meter

Simple Idea: IV measures how much the market EXPECTS the stock to move in the future.

The Weather Forecast Story

Imagine you’re planning a picnic:

• Calm forecast (Low IV): You’re confident, umbrella prices are cheap
• Storm warning (High IV): Everyone’s worried, umbrella prices SKYROCKET

IV works the same way! When people expect big moves, option prices go UP.

IV Percentages

Real Example

Same stock, same strike, same expiration:

• IV = 20%: Call option costs $2.00
• IV = 40%: Call option costs $4.00

Double the IV = Much higher premium!

IV vs. Historical Volatility

IV Crush: The Morning-After Surprise

Simple Idea: IV Crush is when implied volatility drops suddenly, causing option prices to collapse.

The Concert Ticket Story

Before a big concert:

• Tickets are expensive (high demand, uncertainty)
• This is HIGH IV

The day AFTER the concert:

• Tickets are worthless
• This is IV CRUSH

When Does IV Crush Happen?

graph TD
    A["Big Event Coming"] --> B["IV Goes UP"]
    B --> C["Event Happens"]
    C --> D["Uncertainty Gone"]
    D --> E["IV CRASHES Down"]
    E --> F["Option Prices DROP"]

Common IV Crush Events:

• Earnings announcements
• FDA drug approvals
• Election results
• Major economic data releases

Real Example

Before earnings:

• Stock at $100
• Call option: $8.00 (IV = 60%)

After earnings (stock moved to $102):

• You expected profit!
• BUT IV dropped to 25%
• Call option now: $4.50
• You LOST money even though you were RIGHT!

Protecting Against IV Crush

• Sell options before big events (collect high IV)
• Buy options AFTER events (IV already low)
• Use spreads to reduce IV exposure

Options Assignment: When You Actually Have to Do Something

Simple Idea: Assignment is when an option seller MUST fulfill their obligation.

The Promise Story

You sell a “promise ticket” saying:

“I promise to sell you my bicycle for $50 anytime this week”

If your bicycle is now worth $70, the buyer will USE that promise!

• You MUST sell for $50
• You lose $20

That’s assignment!

How Assignment Works

For CALL Sellers:

• You must SELL 100 shares at the strike price
• Happens when stock price > strike price

For PUT Sellers:

• You must BUY 100 shares at the strike price
• Happens when stock price < strike price

Assignment Risk Factors

Real Example

You sold a $50 call for $2 premium:

• Stock rises to $55
• You get ASSIGNED
• Must sell shares at $50 (miss out on $5 gain)
• But you keep the $2 premium
• Net result: Lost opportunity of $3

Options Expiration: The Final Countdown

Simple Idea: Expiration is the last day your option exists. Use it or lose it!

The Milk Carton Story

Your option is like a milk carton:

• Has an expiration date stamped on it
• After that date, it’s no good
• Must use it before it expires!

What Happens at Expiration

graph TD
    A["Expiration Day Arrives"] --> B{Is Option In-The-Money?}
    B -->|Yes| C["Auto-Exercised"]
    B -->|No| D["Expires Worthless"]
    C --> E["Shares Bought/Sold"]
    D --> F["Premium Lost"]

ITM vs OTM at Expiration

The Three Choices Before Expiration

1. Sell the option - Cash out your profit/loss
2. Exercise early - Use your right to buy/sell shares
3. Let it expire - If worthless, it just disappears

Real Example

You own a $50 call, stock is at $53 at expiration:

• Option is $3 in-the-money
• If you don’t sell it, you’ll automatically buy 100 shares at $50
• Cost: $5,000 (but shares worth $5,300)
• Make sure you have the money!

Vega (V): The Volatility Sensor

Simple Idea: Vega tells you how much your option price changes when IV moves 1%.

The Insurance Story

During hurricane season:

• Home insurance prices JUMP
• More uncertainty = Higher prices

Vega measures how sensitive your option is to these “uncertainty changes.”

Vega Numbers

Real Example

Your option has Vega = 0.15, current IV = 30%

• IV rises to 35% (+5%)
• Option price increases: 5 × $0.15 = $0.75

Rho (ρ): The Interest Rate Indicator

Simple Idea: Rho tells you how option prices change when interest rates move.

Why It Matters (Usually Doesn’t Much!)

Rho is the least important Greek for most traders because:

• Interest rates change slowly
• Changes are usually small
• Effect is minimal for short-term options

Putting It All Together: The Option Price Formula

Your option’s price moves based on ALL these factors:

Option Price Change = (Delta × Stock Move) + (Theta × Days) + (Vega × IV Change) + (Gamma Effects) + (Rho × Rate Change)

A Day in the Life of Your Option

Morning: You buy a call for $5.00

• Delta = 0.50
• Theta = -0.10
• Vega = 0.20

During the day:

• Stock goes UP $2 → Add $1.00 (2 × 0.50)
• One day passes → Subtract $0.10
• IV drops 2% → Subtract $0.40 (2 × 0.20)

End of day value: $5.00 + $1.00 - $0.10 - $0.40 = $5.50

Quick Summary: The Greeks at a Glance

Your Confidence Checklist

After reading this, you should feel confident about:

• [ ] Delta tells you direction sensitivity
• [ ] Gamma shows how fast Delta changes
• [ ] Theta is the daily cost of holding options
• [ ] Time Decay speeds up near expiration
• [ ] IV measures expected future movement
• [ ] IV Crush happens after big events
• [ ] Assignment means fulfilling your obligation
• [ ] Expiration is your option’s final day
• [ ] All Greeks work TOGETHER to determine price

You now understand the secret recipe behind every option’s price tag!