Stock Option Tutorials and Seminars
by Jerry Marlow

In a tutorial or seminar, you will gain a sophisticated understanding of stock options quickly and easily because I build my teaching around the dynamic and highly visual simulation software that I have created.

Screen captures with captions poorly represent the smoothness and clarity of dynamic simulations accompanied by lecture and discussion. Nonetheless, the screen captures here may give you some idea of the topics I cover and how I cover them.

To step through screen captures of simulations, click on the right arrow above.

In most browsers, to give yourself more viewing room, hit the F11 key on your keyboard.

To see a screen capture of the simulation used to explain a particular concept, scroll down on this page and click on the concept.

If you want to look at the simulations for a particular topic, click

Black-Scholes Option Pricing Theory

Binomial Option Pricing Theory

Volatility Smiles and the Term Structure of Volatility
If you are interested in
An in-person tutorial
A telephone tutorial
Organizing a seminar and having me run it
Having me speak to your group
contact:
Jerry Marlow, MBA
(917) 817-8659
jerrymarlow@jerrymarlow.com
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Black-Scholes Option Pricing Theory

Most of the Black-Scholes screen captures below are from the option simulator and book: Option Pricing—Black-Scholes Made Easy by Jery Marlow

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

Lose all your money, rate of return is negative infinity

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?

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

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

Simulation 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

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— 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— If volatility increases, value of call goes up

Delta— When spot price increases, value of call goes up

Theta— If underlying pays no dividends, call value goes down over time

Rho— Increase in risk-free rate increases median return. Call value goes up.

Vega— If volatility increases, distribution spreads and drops. Put value goes up.

Delta— When spot price increases, value of put goes down

Theta— As time passes, put's value goes down. Usually!

Rho— 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

Conversation with simulations

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

 

Binomial Option Pricing Theory

Selected creen captures from my simulation software that I use in seminars and tutorials on binomial option pricing theory.

Modeling the evolution of a stock price on a non-recombining binomial tree

We arbitrarily change the down continuously compounded rate of return to -0.287682

Up and down probabilities at each node are both 0.5

We arbitrarily change the probability of going up at each node to 0.6

With probability bars, we can represent graphically the probability of the stock price going to the different terminal nodes.

A recombining tree with the same characteristics as the non-recombining one we drew initially

At each node, the probability of the stock price going up is equal to the probability of it going down. The probabilities of the stock price getting to each terminal value form a binomial distribution.

Valuing a European-style call option that has a strike price of $40.00. The long yellow horizontal line represents the strike price.

Calculate the probability-weighted values of the payoffs and sum them.

Calculate the option payoffs at all the final nodes. Backward induction: Walking values back down the tree

At each previous node, calculate the probability-weighted, point-in-time value of the subsequent values.

Using backward induction to walk the values back down the tree gives us a probability-weighted present value of $21.66. This is the value of this European-style call option on a stock that pays no dividends valued on this tree. It is the same value we got when we simply calculated the probabilities of the payoffs and found their probability-weighted present value. Using backward induction to value a European-style put option

Using this model, this number of time steps, and this forecast for the underlying stock, the value of this European-style put option with a strike price of $10.00 is $0.37.

Using backward induction to value an American-style put option

We add another step

We compare the point-in-time value of the subsequent payoffs with the payoff at the present node. We retain the larger value and continue our backward induction with it.

Binomial models translate forecasts of expected return and volatility into up and down rates of return and probabilities.The Equal Probabilities Model sets the up probability and the down probability equal to.5.

The Cox, Ross, and Rubinstein Model makes the up and down jumps equal.

The General Additive Model makes the up and down jumps equal.

Risk-neutral models evaluate probability distributions against strike prices.

Different binomial Ns give different option values: they generate different probability distributions relative to the strike price

When an underlying pays a dividend yield, the price-path evolution is depressed below what it otherwise would be.

When an underlying pays no dividends, the right of early exercise adds no value to a call option.

Payment of quarterly or other lumpy dividends depresses price paths at the time of the payments.The right of early exercise may add value to a call option on an underlying that pays dividends.

Probability distributions and simulations show the financial forecast and expectations on which the value is based.

 

Volatility Smiles and the Term Structure of Volatility

Selected screen captures from my simulation software that I use when I discuss how volatility smiles differ for call and put bid and ask prices and with options' times to expiration.

In seminars, we look at and discuss how historical or realized volatilities differ from implied volatilties. We look at term structures of volatility for puts and calls, bid and ask prices. We discuss how the term structure of volatility may evolve over time.

When an option price is less than the theoretical value would be if the underlying had an expected volatility of 0%, Marlow negative implied volatility preserves comparative information.

Compare volatility smiles from bid and ask prices for calls and puts with 10 days to expiration. Compare historical volatility for a reach-back period of 10 days.

Compare volatility smiles from bid and ask prices for calls and puts with 29 days to expiration. Compare historical volatility for a reach-back period of 29 days.

Compare volatility smiles from bid and ask prices for calls and puts with 54 days to expiration. Compare historical volatility for a reach-back period of 54 days.

Compare volatility smiles from bid and ask prices for calls and puts with 93 days to expiration. Compare historical volatility for a reach-back period of 93 days.

Compare volatility smiles from bid and ask prices for calls and puts with 154 days to expiration. Compare historical volatility for a reach-back period of 154 days.

Compare volatility smiles from bid and ask prices for calls and puts with 217 days to expiration. Compare historical volatility for a reach-back period of 217 days.

Compare volatility smiles from bid and ask prices for calls and puts with 349 days to expiration. Compare historical volatility for a reach-back period of 349 days.

 
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Stock Option Simulator and Option Market Simulator
Stock option tutorial with free simulator
Master quickly and easily the ideas that revolutionized financial thinking.
Order Option Pricing: Black Scholes Made Easy from amazon.com.
Option market simulator: Option Market Sim (Work in progress)
Think you can beat the option market?
First, make sure you can beat the option market simulator.
Option Market Sim picks up where Option Pricing—Black-Scholes Made Easy leaves off. This option market simulator lets you buy and sell multiple calls and puts with different times to expiration and simulate their potential payoffs under different market conditions.
Stock Option Tutorials and Seminars
Tutorials and seminars on investing in stock options
Seminar: How to value stock options in divorce proceedings
Download seminar notes: How to value stock options in divorce proceedings

Valuation of Stock Options in Divorce Proceedings
Getting divorced? Don't let the court value your spouse's stock options at their so-called "intrinsic value"!
Valuation of stock options in divorce proceedings
   
Jerry Marlow, MBA
(917) 817-8659
jerrymarlow@jerrymarlow.com
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Stock options -- The Movie