Stock prices are volatile

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Volatility means that a stock's future price path is uncertain

The more volatile a stock, the more uncertain its future value

An option can make you a ton of money or you can lose it all

A forecast for a stock is a bell-shaped curve

You can translate your estimate of possible future prices into a forecast

You are 99.7% certain the outcome will be within the curve

One chance in ten that price will be in any given decile

You can translate a forecast into potential price paths

Monte Carlo simulations show relationship between paths and forecast

From stock's historical returns, calculate historical standard deviation

Continuously compounded returns are normally distributed

Stock-price changes are lognormally distributed

Price paths are characterized by geometric Brownian motion

Volatility is constant over the investment horizon

May not be your customary way of thinking

Lose all your money, rate of return is negative infinity

Continuously compounded return on portfolio?

Convert simple interest to continuously compounded

Find the present value of a future dollar amount

Expected return is average of all returns in probability distribution

Stock's expected return is median plus half standard deviation squared

Expected return varies with time

Uncertainty varies with square root of time

Is your portfolio manager talking holding-period returns?

Lock Random Seed lets you create the same price path with variations

Dividend payments reduce the price of a stock

Dividends shift price probability distribution down

A dividend yield shifts price probability distribution down

A call gives you the right to buy a stock at a pre-set price

Simulate potential outcomes of investing in a call

Histogram approximates probability distribution for option

Option's expected return is average of returns in probability distribution

Color deciles link stock forecast to option forecast

At extremes of probability distributions, divide into more intervals

Put gives you right to sell a stock at a pre-set price

Simulate potential outcomes of investing in put

Calculate put's probability of profit and expected return

If you're thinking and counting trading days, set days per year to 252

Does the expression probability density function make your brain hurt?

An option's probability-weighted net present value

It's like doing discounted cash-flow analysis in corporate finance

Animation calculates cost of setting up delta hedge

Black-Scholes assumptions envision a risk-neutral world

Black-Scholes sets expected return equal to risk-free rate

For strike prices at extremes of wide distributions, use more intervals

Black-Scholes value of a put

If option has time value, don't exercise it early

Out of the money options have only time value

As put goes deep into the money, may be advantageous to exercise early.

Option's value may be its early-exercise value&151;Black's approximation

When to exercise deep-in-money put if underlying pays lumpy dividends?

May be optimal to exercise on last ex‑dividend date

Option value depends on location of little squares relative to strike price

What if put goes deep into money and underlying pays dividend yield?

What if call goes deep into money and underlying pays lumpy dividends?

Maybe exercise on last day before underlying goes ex‑dividend for last time

What if call goes deep into money and underlying pays dividend yield?

Depends on yield, time value, volatility, expected return, risk-free rate

If call on underlying that pays no dividends, never exercise early

Deeper into money, less sensitive option value is to changes

Vega&151;If volatility increases, value of call goes up

Delta&151;When spot price increases, value of call goes up

Theta&151;If underlying pays no dividends, call value goes down over time

Rho&151;Increase in risk-free rate increases median return. Call value goes up.

Vega&151;If volatility increases, distribution spreads and drops. Put value goes up.

Delta&151;When spot price increases, value of put goes down

Theta&151;As time passes, put's value goes down. Usually!

Rho&151;Increase in risk-free rate increases median return. Put value goes down.

From Black-Scholes value, extract stock's implied volatility

Draw risk-neutralized, market-equilibrium forecast for stock

If agree, then stock and option have same expected return

If disagree, then use option to leverage expected return

Bid and ask prices give different implied volatilities

Different strike prices give us volatility smile

Different expiration dates give us term structure of volatility

Theoreticians keep building alternative models

Market-equilibrium forecasts

Calculate your forecast without dividends for a call's underlying

Simulate potential price paths of a call's underlying

Enter dividend schedule for a call's underlying

Calculate calls' probabilities of profit and expected returns

Simulate call's potential investment outcomes

If you think somebody's bubble is about to burst, buy puts

Calculate your forecast without dividends for a put's underlying

Enter dividend schedule for a put's underlying

Calculate puts' probabilities of profit and expected returns

Simulate put's potential investment outcomes

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Your value at risk

An investment strategy that allows you to express your views and have your portfolio's value never go down

Invest risk free an amount that interest will grow back to original portfolio value

Translate your beliefs into a forecast

Invest foregone interest in options

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