Quantitative Finance SocietyOptions, explained

State of the MarketsWhat’s been going on?

State of the MarketsWhat’s been going on?

• U.S. homebuilders- confidence drops– Seasonal or something

else?

• Greece – No to bailout extension– Closer to euro exit?

• Oil– BP’s report

Equities

• What do you own?

• How do you make money?

• Preferred vs. Common Stock

• Equity Fundamentals Equity

Preferred Equity

Mezzanine

Unsecured Bonds

Secured Bonds

Bank Debt

Options

• What does it mean to have an option?• Who has the obligation to perform a duty,

who has the option to take action?• What is a derivative?

Options

• Right to buy (or sell) something at a certain point in time

• But you don’t have to if you don’t want to

• Drivers?– Underlying/Spot– Strike– Dividends– Interest Rates– Time– Volatility

Difference between Stocks and Options

• Regular equities can be held indefinitely… options have expiration dates– If an OTM option is not exercised on or before

expiration, it no longer exists and expires worthless

• No Physical certificates for stock options• No ownership – owning options doesn’t confer

voting rights, dividends, ownership, etc.– Unless option is exercised

• Fixed number of stocks issued by company• Opportunity for Leverage

Naked Options

• A “naked” option position is a portfolio consisting only of options of a given type (i.e. calls or puts)

• 4 kinds of naked options positions– Long Call– Short Call– Long Put– Short Put

Purpose

• Options offer one-sided protection against price moves– Instruments of *financial insurance*– Call: provides protection against an increase

in price– Put: provides protection against a decrease in

price

• Can be used to take positions on market direction and market volatility– Bullish on vol: long options,– Bearish on vol: short options

Risk Management

Options Strategies

• Presence of non-linearity in their payoffs– Options can be combined into portfolios to

produce precise and targeted payoff patterns

• Two components of an option premium: – Intrinsic Value: ITM portion of the option

premium• Alternatively, value that any given option would have

if expired today

– Time Value = Option Premium – Intrinsic Value

Factors Affecting Option Prices Strike Price Stock Price Implied Volatility Time to expiry Risk-free rate

Exotics

• Any option different from vanilla options• Not necessarily more complex– Digital options have simple structures– Can be complex – barriers, Asians, quantos

• Why use exotics?– Richer payoff patterns/costs

• Insurance contracts with greater flexibility

Advanced Options

Put-Call Parity

The equation follows as such: c + PV(x) = p + s

This relationship tells us that going a long a call (+C) and shorting a put (–P) will yield a function that resembles a Forward

C - P = F

Put-Call Parity

The equation follows as such: c + PV(x) = p + s

This relationship tells us that going a long a call (+C) and shorting a put (–P) will yield a function that resembles a Forward

C - P = F

Some interesting exercise tactics When do you exercise an American

call early? When do you exercise an American

put early?

The Greeks: Delta

Mathematical Definition dV/dS

What does it mean? The change in the option’s value per $1 change in

stock price How do we think about it intuitively?

Probability of the option finishing in the money Important Graphs

Delta v. Spot (at different times) Delta v. Time to Expiry (with different moneyness)

The Greeks: Gamma

Mathematical Definition dΔ/dS = d2V/dS2

What is it? The change in the option’s delta per $1 change

in stock price How do we think about it intuitively?

How convex is your option price? How fast does your option value accelerate?

Graphs Gamma vs. Spot (at different times)

The Greeks: Theta

Mathematical Definition dV/d(T-t)

What is it? The change in the option’s value for 1 day

passing How do we think about it?

How much am I paying to hold this option for a day?

Graph Theta decay

The Greeks: Vega

Mathematical Definition dV/dσ

What is it? The change in the option’s value per 1%

change in implied volatility How do we think about it?

What is the size of my volatility position? Graphs

Vega vs. Spot (at different levels of IV)

The Greeks: Vanna

Mathematical Definition dVega/dS = dΔ/dσ = d2V/dSdσ

What is it? The change in the option’s Vega per $1

change in spot How do we think about it?

What is the size of my skew position?

The Greeks: Volga

Mathematical Definition dVega/dσ = d2V/dσ2

What is it? The change in the option’s Vega per 1%

change in implied volatility How do we think about it?

How much does my option benefit from vol on vol?

Volatility Skew

Definition The difference between OTM and ATM IV

How do we think about it? How much does my IV change with a change

in spot? Three main positions

ATM Straddle 25-delta Risk-Reversal 25-delta Butterfly

Volatility Skew (Continued)

How to compare IV across term structures and skews? Use square root of time rule to normalize

You can use this to make relative value vol trades Can stay vega-neutral and just take advantage

of the pricing discrepancy

Gamma Scalping

You think implied volatility is very low relative to historical/realized volatility. How do you take advantage of this situation?

Calendar Spreads Trading

If a surface rises with power < .5: Short calendar

If a surface falls with power < .5: Long calendar

If a surface rises with power > .5: Long calendar

If a surface falls with power > .5: Short calendar

Trading Situation

You are looking at the term structure and skew of implied volatility. You notice that options with longer maturities have higher IVs than shorter dated options. You also notice that there is some pretty strong skew (OTM put IVs are much higher than ATM put IVs).

You think that realized vol will pick up soon, while long-term vol will be lower than the OTM puts suggest. What do you do?

Questions?