Option Pricing: Black-Scholes Made Easy—
The Movie
An introduction to how hedge funds use mathematical models to value derivatives and other financial instruments
By Jerry Marlow

The modern, sophisticated way to think of investing is this: Every financial forecast is a probability distribution. To value an investment is to evaluate its probability distribution.

This principle applies to every type of investment, not just stocks and options but also to bonds, commodities, collateralized debt obligations and other derivative securities.

Stochastic calculus and Monte Carlo simulations allow you to evaluate probability distributions and value investments.

The easiest way to understand these fundamental principles of modern financial theory is to gain an understanding of how probability distributions determine the value of stock options.

The fastest and easiest way to understand how probability distributions determine the value of stock options is with the stock option simulator.

With the stock option simulator, not only can you understand these principles, you can see how to apply them to investing in stock options.

If you've read about the credit crisis and heard of mortgage- backed securities and other derivative instruments being "marked to model" but weren't entirely clear on what the expression meant, the movie above lets you watch a model value stock options. Related models value credit derivatives and other financial contracts.

The Adobe Shockwave movie on this site and the book Option Pricing: Black-Scholes Made Easy by Jerry Marlow take you step by step through a series of easy-to- understand lessons and simulations.

In a very short time, you will learn:
How you can use call and put options to make a lot of money— if you have a good idea of whether the stock market is going to go up or down
What a probability distribution is
How potential price paths translate into probability distributions
How probability distributions translate into potential price paths
How to express your stock forecasts as probability distributions
How to use probability distributions to value stock options
What implied volatillity is and how to use it in your option trading
How to extract the market-consensus forecast from an option's price

Movie Script

In 1973 Fischer Black and Myron Scholes published the seminal paper for what has become known as Black-Scholes option- pricing theory.

Yes, Black-Scholes option-pricing theory provided a new way to value stock options, but more importantly it started a revolution in how hedge funds and other market participants think about and value financial assets.

Today, hedge fund traders and other sophisticated market participants understand every financial forecast to be a probability distribution. To value an investment is to evaluate its probability distribution.

To evaluate probability distributions, hedge funds and other market participants use geometric-Brownian-motion models, binomial models, Monte Carlo simulations and stochastic calculus.

Today one provider of asset-valuation software sells more than thirty different financial models that can be used to value over one hundred different derivative securities and other financial contracts.

The value of any investment is the probability-weighted present value of its future payoffs or cash flows.

Whatever model a market participant uses to value any instrument, the objective is the same:

1) Model the assetís potential future payoffs or cash flows and how likely different payoffs or cash flows are.

2) Calculate the probability- weighted present value of those potential payoffs or cash flows.

The easiest way to begin to understand this revolutionary way of thinking is to learn how Black-Scholes option pricing theory values stock options.

The easiest way to understand how Black-Scholes option pricing theory values stock options is with the book and simulator Option Pricing: Black-Scholes Made Easy.

This movie gives you a quick introduction to Black-Scholes option pricing theory. It shows you some of the simulations you can run with the Black-Scholes Made Easy simulator.

As the movie plays, you can use the slower and faster buttons to increase or decrease the amount of time you have to read text. The buttons affect only how long the movie pauses to let you read text. Any speed changes you make apply to all subsequent pauses.

The simulations run as fast as your computer's processing speed allows.

To pause movie, click ||.

To resume play, click >.

If, after you watch the movie, you want to learn more about stock options, option-trading strategies and this revolutionary way of thinking, use the link on the page below to buy the book Option Pricing: Black-Scholes Made Easy by Jerry Marlow from amazon.com.

The book includes the simulation software on a CD. To use the web version of the simulator, click on the link below that says Black-Scholes Made Easy simulatoró web version. The web version requires a password from the book.

Happy learning!

Just in case you don't know anything, let's take a quick look at how calls and puts work.

Let's say that today a stock is trading at $100.00.

You buy a call option on the stock for $10.02. It has a strike price of $110.00. (yellow line).

The option expires in one year (252 trading days).

If, at the end of 252 trading days, the stock price is above the strike price, you get a payoff.

If the stock price finishes above the green line, you make a profit.

Let's say you buy a put for $2.56. It has a strike price of $80.00 (yellow line).

If the stock price finishes below the strike price, you get a payoff.

If the stock price finishes below the green line, you make a profit.

In the return descriptions on the right of the screen, CC return means continuously compounded rate of return. Financial modeling almost always uses continuously compounded rates of return. In modeling, they are known as geometric rates of return.

Financial firms use simple rates of return only in their sales pitches, not in their financial modeling.

If you want to get rich, should you buy a call? Should you buy a put?

Should you sell a call? Should you sell a put?

Some combination thereof?

Depends.

Depends on your forecast for the stock. Depends on the stock forecast implied by the option price.

Depends on how your forecast compares to the forecast implied by the option price.

Huh?

To give you mastery of these topics, let's start with how you might express your forecast.

Then we'll look at how the forecast for the underlying determines the value of an option on the underlying.

If you have an idea of how high and how low a stock price might go over a given time horizon, the simulator can turn your idea into a forecast of expected return and volatility.

Let's say a stock currently is trading at $100.00. Over the coming year you think the stock price could go as high as $200.00 or as low as $50.00

From your forecast, the simulator can simulate as many potential price paths as you would like.

The simulator tabulates the outcome of each price path with a little square.

The simulator can simulate price-path outcomes without drawing the price path.

The little squares build a histogram.

You can interpret the histogram as a probability distribution.

Probability distributions are also known as probability density functions or PDFs.

The probability distribution is the stock forecast.

All investment forecasts are probability distributions!

Investing is decision making under conditions of uncertainty.

Probability distributions define the uncertainty.

In Black-Scholes option pricing theory, probability distributions for stocks are assumed to be bell-shaped curves drawn on an axis of geometric rates of return or drawn on a lognormal price axis.

If you're not familiar with them, here's the easiest way to orient yourself to continuously compounded or geometric rates of return:

A geometric return of 69.315% doubles the value of your investment.

A geometric return of minus 69.315% halves the value of your investment.

At the start of our one-year investment horizon, the stock is trading at $100.00.

On our return and price axes, you'll notice that a return of 69.315% is at the same height as $200.00, double our starting value.

A return of -69.315% is at the same height as $50.00, half our starting value.

You can divide your forecast into deciles.

There's one chance in ten that the stock price will finish in any one decile.

You can divide your forecast into standard deviations.

When people speak of the volatility of a stock forecast, they cite the annualized standard deviation of the forecast.

The more a stock price jumps around, the more spread out the bell-shaped curve.

The more spread out the bell-shaped curve, the higher the standard deviation of volatility.

This forecast is drawn for a one-year time horizon. It has a median return of 0% and a standard deviation of 23%.

Three standard deviations up from the median is a return of 69%.

Three standard deviations down is a return of -69%.

The tail of the probability distribution going up trails off into an infinite return and an infinite dollar amount.

The tail going down trails off to a return of negative infinity and approaches but never gets to a dollar amount of $0.

In theory, the price of a stock can never get to zero. Stock prices are like Xeno's arrow.

Xeno pointed out that, if he shot an arrow at you, to get to you the arrow would have to travel half the distance to you. From there, it would have to travel half the remaining distance. From there, it would have to travel half the remaining distance. And so on.

No matter how far the arrow travels, it still has to travel half the remaining distance.

Hence, no matter how far the arrow travels, it can never reach you.

On average, 99.7% of price paths will finish within three standard deviations of the median return.

The simulator draws outline bell-shaped curves out to three standard deviations going up and three going down.

Price paths and little squares may end up beyond three standard deviations.

The median or middle of this return forecast is 0%. The forecast average return, however, is 2.7%.

Why is the forecast average return higher than the median return?

On a lognormal scale, dollar amounts going up from a 0% return grow faster than dollar amounts going down shrink.

Two price paths that mirror one another do not average out to a 0% return.

Imagine that you have two stocks in your portfolio. Each has a market price of $100. Your portfolio has a value of $200.

Over the course of a year, one stock has a return of 69%. It doubles in value to $200.

The other stock has a return of -69%. It halves in value to $50.

What is the median return of the two stocks? What is the value of your portfolio now?

The median return of the two stocks is 0%. Yet the new value of your portfolio is $250.00.

The return on your portfolio is ln($250/$200) = 22.3%.

The average return on the two stocks is the same as the return on your portfolio: 22.3%.

When we average the price-path outcomes represented by all the little squares in the bell-shaped curve, we find this relationship:

Forecast average return

= Median return + .5 x (Standard deviation squared)

The math of this equation is easiest for a bell-shaped curve with a median return of 0% and a standard deviation of volatility of 40%.

Forecast average return

= Median ret + .5 x SD^2

= 0.00 + .5(.40^2)

= 0.00 + .5 x .16

= .08

= 8%

Hence, a forecast with a median return of 0% and a standard deviation of volatility of 40% has a forecast average return of 8%.

To refer to forecast average return, most people in finance say "expected return."

That expression, however, is misleading. You can never expect to earn the expected return.

If we start with a forecast average return of 8% and a standard deviation of volatility of 40%, then

Median return

= Fcst avrg ret - .5 x SD^2

= .08 - .5(.40^2)

= 0.08 - .5 x .16

= .08 - .08

= 0%

A forecast with an average return of 8% and a volatility of 40% has a median return of 0%.

This relationship among forecast average return, volatility and median return implies the following:

If we hold forecast average return constant and keep increasing expected volatility, then the middle of the bell-shaped curve keeps dropping lower and lower.

If the forecast average return is less than .5 x SD^2, then more potential price paths will lose value than increase in value.

"So what?," being impertinent, you might ask. "What have probability distributions got to do with options?"

Everything!

Let's say, your forecast for a stock is a standard deviation of 20% and a forecast average return of 5%.

For a one-year time horizon, that forecast would look like this.

You can buy a call option on the stock.

It expires in one year.

It has a strike price of $120.00.

How much is the call worth? What is its value?

How do you figure that out?

As a fundamental strategy of investing, you want to buy undervalued assets and sell overvalued assets.

To buy undervalued options and sell overvalued options, you need a way to value them. How do you value this option?

If you'd figured that out before 1973, you would've gotten a Nobel Prize.

Too late! Fischer Black, Myron Scholes and Robert Merton beat you to it.

The Black-Scholes option- pricing formula revolutionized financial thinking.

Want to see it?

Piece of cake, huh?

Letís look at an easy way to understand the calculations that the formula makes.

You've seen how call options produce payoffs if the stock price finishes above the option's strike price.

Let's find the probability- weighted present value of the payoffs from all the little squares in this probability distribution.

With this forecast over this time horizon, the highest stock price we might expect after one year would be $206.72.

With a strike price of $120.00, the option's payoff would be $86.72.

To fill up this bell-shaped curve with little squares, we need 2,000 squares. Hence the probability of this outcome is 1/2,000 or 0.0005.

The probability- weighted value of this payoff a year from now is $86.72 x 0.0005 which is a little more than 4 cents.

To find the probability- weighted value of this payoff as of today, we discount it (reduce it) by the forecast average return.

The probability- weighted value of this payoff as of today is $0.04124451.

We want to add up the probability- weighted present values of the payoffs of all the little squares.

We save this one in the box labeled cumulative probability- weighted present value.

What's the second highest stock price this forecast might give us after one year?

$194.45.

This stock price produces a payoff of $74.45. It adds $0.03540814 to our cumulative probability- weighted present value.

Thus we sweep through the probability distribution. We keep adding the probability- weighted present value of each payoff to the cumulative value.

As the little squares get closer and closer to the strike price, the payoffs add less and less to the cumulative probability- weighted present value.

Once the little squares reach the strike price, the payoffs are zero. They add nothing to the cumulative probability- weighted present value.

The cumulative probability- weighted present value for all the potential payoffs of this option based on this stock forecast rounds to $3.25.

$3.25 is the probability- weighted present value of this option's potential future cash flows.

The value of ANY investment is the probability- weighted present value of its potential future cash flows.

Hence, $3.25 is the value of this option.

In principle, these are the same calculations that the Black- Scholes formula makes.

The formula is more elegant and accurate. It is a stochastic calculus formula.

Stochastic is from the Greek stokhastikos which means able to guess or conjecturing. Stochastic calculus is the mathematics of guessing or evaluating uncertainty.

Stochastic calculus evaluates probability distributions. Itís formulas are difficult and complex.

In our easier to understand graphic representation, the value of a call option depends on its potential payoffs.

Potential payoffs depend almost entirely on two things:

How many little squares are above the strike price.

How far the little squares are above the strike price.

The greater the number of little squares above the strike price; the more potential payoffs. The more potential payoffs; the greater the value of the call.

The higher the little squares above the strike price; the greater the potential payoffs. The greater the potential payoffs; the greater the value of the call.

Where the little squares fall relative to the strike price depends on the factors in the Black-Scholes equation:

Option's strike price

Current market price of the underlying

Forecast's volatility

Forecast's average return

Option's time to expiration

Dividends, if any, that the underlying pays

We look in turn at how changes in each of these factors affect:

Where the little squares fall relative to the strike price

The number of potential payoffs

The size of potential payoffs

The value of the option

Increase in Strike Price of Call

The effect of a change in the option's strike price is the simplest variation to draw and understand.

We increase the strike price from $120.00 to $130.00.

The yellow line moves from $120.00 up to $130.00

The bell-shaped curve stays where it is.

How does a higher strike price affect where the little squares fall relative to the strike price?

Will the option have more potential payoffs or fewer?

Will the highest potential payoffs be larger or smaller?

Will the call have more value or less?

With a higher strike price, fewer little squares are above the strike price. The option has fewer potential payoffs.

The squares farthest above the strike price are nearer the strike price. Potential payoffs are smaller.

A higher strike price lowers the value of a call option.

Increasing the strike price from $120.00 to $130.00 reduces the value of the call from $3.25 to $1.64.

Decrease in Strike Price of Call

Let's lower the strike price of the call to $90.00.

A strike price of $90.00 gives us an "in-the-money" call option. That is, the strike price is less than the current market price of the underlying.

What do you think the cumulative probability- weighted present value of this option's potential payoffs will be?

With a lower strike price, more little squares are above the strike price. The option has more potential payoffs.

The little squares farthest above the strike price are farther from the strike price. Potential payoffs are greater.

A lower strike price increases the value of a call.

Decreasing the strike price from $120.00 to $90.00 increases the value of the call from $3.25 to $16.70.

Increase in Price of Underlying

If the current market price of the underlying were higher, where would the little squares fall relative to the strike price?

Would we have more of fewer potential payoffs?

Would potential payoffs by larger or smaller?

Would the value of the call be higher or lower?

If the current market price of the underlying is $110.00, the bell-shaped curve relative to the strike price looks like this.

If the current price of the underlying is higher, more little squares fall above the strike price.

The option has more potential payoffs.

Some squares are farther above the strike price. Potential payoffs are greater.

If the current market price of the underlying is higher, then the value of this call is higher: $7.00 instead of $3.25.

Decrease in Price of Underlying

Where do the little squares fall relative to the strike price if the current market price of the underlying is lower; not $100.00 or $110.00 but $90.00?

Is the value of the call higher or lower?

Why?

If the current market price of the underlying is $90.00, then the bell-shaped curve relative to the strike price looks like this.

If the current market price of the underlying is lower, fewer little squares fall above the strike price.

The option has fewer potential payoffs.

The highest squares are not as far above the strike price. Potential payoffs are smaller.

Three different current asset prices give three different call values:

For $ 90, $1.16

For $100.00, $3.25

For $110.00, $7.00

Increase in Forecast Volatility

How does an increase in forecast volatility affect the number and size of an option's potential payoffs?

To work with a higher volatility, we first need to rescale the vertical axes. Currently our vertical axes change in increments of 10%. We rescale both the return and dollar axes to increments of 20%.

Our baseline forecast has a standard deviation of volatility of 20%. At this scale, that volatility looks like this.

The forecast's bell-shaped curve looks like this.

We double the forecast's standard deviation of volatility to 40%. That amount of volatility looks like this.

The forecast's bell-shaped curve looks like this.

We compare the higher- volatility bell-shaped curve to the lower-volatility one.

Because the two forecasts have the same forecast average return, the middle of the higher-volatility bell-shaped curve sits slightly lower than the middle of the lower-volatility one.

Where do the little squares fall now relative to the strike price?

How does an increase in expected volatility affect the value of the call?

Why?

Increasing the standard deviation of volatility from 20% to 40% increases the value of the call from $3.25 to $10.80.

Even though the middle of the higher-volatility forecast sits lower than the middle of the lower-volatility forecast, because of the greater spread of the higher-volatility forecast, more of its little squares are above the strike price.

Decrease in Expected Volatility

How does a decrease in expected volatility affect where the little squares fall relative to the strike price?

At this scale, a standard deviation of volatility of 10% looks like this.

The forecast's bell-shaped curve looks like this.

What do you think the value of the call will be?

With a decrease in expected volatility, fewer little squares fall above the strike price.

The highest squares are nearer the strike price.

Decreasing the standard deviation of volatility from 20% to 10% decreases the value of the call from $3.25 to $0.46.

Increase in Forecast Average Return

Our baseline forecast has a current asset price of $100.00 and a forecast average return of 5%.

Forecast average return is the average return on the underlying for all the little squares in the forecast.

Changing the forecast average return changes how high the forecast sits on the return and price axes.

Increasing the forecast average return from 5% to 15% makes the bell-shaped curve sit higher on the axes.

How does that increase affect where the little squares fall relative to the strike price?

How does a higher forecast average return affect the value of the call?

With a higher forecast average return, more little squares fall above the strike price.

The highest squares are farther above the strike price.

When we increase the forecast average return, we also increase the rate at which we discount the probability- weighted future values back to the present.

The higher discount rate does not offset the greater returns of the little squares because the greater return pushes additional little squares above the strike price.

Increasing the forecast average return from 5% to 10% increases the value of the call from $3.25 to $6.56.

Decrease in Forecast Average Return

Where do the little squares fall relative to the strike price when we lower the forecast average return?

Our baseline forecast has an average return of 5%.

Decreasing the forecast average return from 5% to -10% makes the bell-shaped curve sit lower on the axes.

How does that decrease affect where the little squares fall relative to the strike price?

With a lower forecast average return, fewer potential outcomes fall above the strike price.

The highest squares are nearer the strike price.

Using a negative rate of return to discount future values back to the present is kind of wacky. But so is a forecast with a negative forecast average return.

Decreasing the forecast average return from 5% to -10% reduces the value of the option from $3.25 to $0.82.

The Black-Scholes formula uses the risk-free rate of return r as the forecast average return.

Valuations that use the risk-free rate as the forecast average returns are known as risk-neutral valuations.

Risk-neutral valuation methodologies are at the heart of non-arbitrage pricing theory.

Increase in Option's Time to Expiration

We've been working with forecasts over a one-year investment horizon. We've seen where potential price paths and little squares end up.

For a given level of volatility and forecast average return, over 252 trading days, the market price of a stock is likely to get only so far from where it is today.

If the price of a stock has more time to jump around, it can get farther away from where it is today.

Let's see where potential price paths end up if the price has 352 trading days in which to jump around.

For comparison, we redraw the forecast for a 252-trading-day horizon.

With a longer investment horizon, the forecast's bell-shaped curve is more spread out.

With the forecast average return used here, the bell-shaped curve also sits higher for a 352-day forecast than it does for a 252-day forecast.

More little squares are above the strike price. The highest squares are farther above the strike price.

How does the value of a call option that expires in 352 trading days differ from the value of one that expires in 252 trading days?

Increasing the option's time to expiration from 252 trading days to 352 days increases the value of the option from $3.25 to $5.11.

Decrease in Option's Time to Expiration

Let's look at how a shorter time to expiration affects where the little squares fall relative to the option's strike price.

First, for comparison, we re-draw the bell-shaped curve for the 252-day forecast.

The annualized standard deviation of volatility in our forecast is 20%. When we draw the forecast for a one-year investment horizon, one standard deviation going up is marked off at 23%. One standard deviation going down is marked off at minus 17%.

Let's see where potential price paths end up for an investment horizon of one fourth year, 63 trading days.

Our time to expiration is one quarter of a year. Our bell- shaped curve is less spread out.

To be more precise, when our investment period was one year, our standard deviation of volatility was 20%.

When we reduce our investment period to 1/4 year, our standard deviation of volatility falls by 1/2 to 10%.

Coincidence?

I think not.

When you want to bore people at cocktail parties to tears, you'll want to say over and over again, "The standard deviation of volatility varies with the square root of time."

The square root of 1/4 is 1/2.

1/2 x 20% = 10%.

Can you handle the math?

If yes, then you can become a Wall Street options trader.

Remember the Black-Scholes formula?

Sigma represents the annualized standard deviation of volatility.

In the denominators (bottoms) of the d1 and d2 equations, by what do you see sigma multiplied?

That's right!

Square root of T.

Say it! Loud! "The standard deviation of volatility varies with the square root of time T."

How does a shorter time to expiration affect the option's potential payoffs?

With a shorter time to expiration, potential payoffs are fewer. They are smaller.

Reducing the option's time to expiration from 252 to 63 trading days reduces the value of the option from $3.25 to $0.20.

The Effect of Lumpy Dividends on the Value of a Call Option

In all the forecasts and price-path simulations with which we've been working, the underlying stock paid no dividends.

How are a stock's potential price paths and a call option's potential payoffs different if the stock pays lumpy dividends?

To illustrate the effect of lumpy dividends on a stock's price path, let's pretend we've found a stock that has a forecast average return of 10% and a forecast volatility of zero. The stock pays no dividends.

No volatility would mean no uncertainty.

For that imaginary stock, every single potential price path would look like this.

Now let's introduce dividends. Let's say the stock pays dividends of $2.00 every quarter.

Where does the money come from? What does the price path look like now?

If our zero-volatility stock pays quarterly dividends, its potential price path looks like this.

The dividends come out of the stock price. Every time the stock pays a dividend, the stock price drops by the amount of the dividend from where it would have been otherwise.

The same principle applies to volatile stocks.

If we introduce quarterly two-dollar dividends to the volatile price path drawn here, we get a potential price path that looks like this.

We simulate a couple of thousand potential price paths with quarterly dividends of $2.00.

We get probability distributions that looks like these.

The probability distribution on the price axis sits lower than the probability distribution on the return axis.

Forecast average return includes both the stock's potential price appreciation and expected dividends.

How does the stock's payment of dividends affect the value of the call?

If the underlying pays quarterly dividends of $2.00, the value of our baseline call option falls from $3.25 to $1.49.

The Effect of Dividend Yield on the Value of a Call Option

Some market indices on which you can buy options contain stocks that pay dividends at different times of year.

To model potential price paths of these indices, we model the dividend payments as a continuous dividend yield.

First we simulate a potential price path with no dividend yield.

We introduce a dividend yield of 8%.

We show what the same price path would look like with the continuous dividend yield.

You can think of the dividend yield as a continuous leakage of money from the stock price.

What effect does a continuous dividend yield have on the price forecast for the underlying?

With no dividend yield, the return and price forecasts would look like this. They sit at the same height.

We introduce a dividend yield of 8%. The return and price forecasts look like this.

The price forecast sits lower than the return forecast.

How does the lower-sitting bell-shaped curve affect the call's potential payoffs and the value of the option?

If the underlying pays a continuous dividend yield of 8%, the value of our baseline call option falls from $3.25 to $1.51.

Whether you use these graphic methods or the Black-Scholes formula to value call options, you are evaluating probability distributions.

Every financial forecast is a probability distribution. To value an asset is to evaluate its probability distribution.

Once we have a value for an option, we can calculate potential returns of investing in the option.

The yellow line represents the strike price. The distance from the yellow line to the green line represents the cost of the option.

If the payoff of an option is less than what the option cost, then you have a negative return.

If a call finishes out of the money, then you lose all your money. You have a continuously compounded return of negative infinity which is a simple return of -100%.

We can show potential option returns graphically.

To accommodate the higher potential option returns, we rescale our vertical axes.

If we tabulate enough return outcomes, we build a histogram of potential option returns.

Given a stock forecast, an option strike price and an option price, we can draw a probability distribution for the option.

In essence, the probability distribution for the option is the probability for the stock filtered through the option's strike price.

If the option has a strike price of $120.00, its probability distribution looks radically different from that of the stock.

Let's see what happens if we keep lowering the strike price, recalculating the value of the option and redrawing the probability distributions.

Strike price: $110.00

Option price: $6.04

Strike price: $100.00

Option price: $10.45

Strike price: $90.00

Option price: $16.70

Strike price: $80.00

Option price: $24.59

Strike price: $50.00

Option price: $52.44

Strike price: $25.00

Option price: $76.22

Strike price: $0.01

Option price: $99.99

If the option has a strike price of $0.01, then the probability distribution of the option is indistinguishable from the probability distribution of the underlying stock.

Let's go back to the probability distribution of our baseline call option which has a strike price of $120.00 and an option cost of $3.25.

The forecast average return for the stock is 5%. What is the forecast average return for the option?

If we sweep through the forecasts, as we go we can calculate a running average return for the option.

5%. Even though the two forecasts are radically different, they have the same average return.

If a stock forecast is the same as the forecast used to value the option, then the option forecast has the same average return as the stock forecast.

In risk-neutral valuation, forecast average returns are always set equal to the risk-free rate. Hence, stock forecasts and option forecasts have the same forecast average return.

If we keep this call option that has a strike price of $120 and a cost of $3.25 and run against it a stock forecast with an average return of 10%, what forecast average return will we get for the option?

47.1% versus 10% for the underlying stock. The forecast average return for the call option is higher.

If a stock forecast's average return is higher than the one used to value a call option, then the call leverages the stock forecast's average return.

To express the same idea another way, if a stock forecast's average return includes a risk premium, then the call option leverages the stock forecast's risk premium.

I hope that these simulations have given you insights into how potential price-path evolutions translate into probability distributions and how probability distributions translate into asset values. If you grasp these concepts, then you are well on your way to understanding how hedge funds and other market participants use financial models to value derivatives and other financial contracts.

To gain a deeper understanding, buy the book and simulator Option Pricing: Black-Scholes Made Easy.

If you would like to run this movie on your web site, let me know. I can customize it however you would like and license it to you.

If your company sells asset- valuation software and you are looking for a marketing writer who can get financial firms excited about your models, I would like to write for you.

If you manage assets and you are looking for a writer who can get investors excited about your firm's asset-management strategies, I would like to write for you.

Give me a call or send me an e-mail.

Jerry Marlow
(917) 817-8659
jerrymarlow@jerrymarlow.com
www.jerrymarlow.com

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www.jerrymarlow.com


Client Raves and Praise from the Financial Press

Extremely easy to follow. I can't praise Black-Scholes Made Easy highly enough. Any student or trainee having to study this topic really should buy this tutorial.

Having a degree in Mathematics and a professional accountancy qualification did not prepare me for the explanations of Black Scholes to be found in most text books. They may have got a Nobel prize for their option pricing model but Black and Scholes were never going to get an award for clarity of explanation.

Having grappled with this area for a few months, I decided I needed a little more innovative help; hence my purchase of Jerry Marlow's interactive tutorial.

Two days later and I feel I could go for the next Nobel prize myself! So many things click into place so quickly, it's marvelous. Jerry gives his email address which I had to resort to for one query. He answered most helpfully within a couple of hours.
Peter Stewart

The simulation is a good one. A deep insight into the price process pops out clearly. In my next semester course "Black-Scholes Made Easy" will be a useful tool for giving the necessary intuitive view of the matter.
Lucio Geronazzo, Professor
Mathematical Models for Financial Markets
Il Dipartimento di Matematica per le Decisioni
Università degli Studi di Firenze
Florence, Italy

Your simulations are simply the best, making every part of Black-Scholes so understandable. I knew Black-Scholes' option pricing model was important and used it every day in my trading, but never knew why or how it worked. Your book and CD-ROM provided me clear understanding of the model, and best of all, all I need to understand the model was basic mathematical background and some common sense.

Thank you very much.
Dong-Wook Kim
Futures & Options Trader
Equity & Derivatives Team
Hanwha Securities. Co., Ltd
Seoul, Republic of Korea

Just a brief thank you for the two-day course on valuing options (with an emphasis on divorce valuation) that you presented to my group last week. The group consisted of a number of my peers/ competitors, all of us moderately to heavily experienced in forensic accounting and business valuations. Thus, it was not an easy audience to satisfy. However, you did it. We all recognize how dry the options valuation topic is, but nonetheless, my personal reaction and those of my fellow attendees was most positive. All of us felt that we got considerable value out of the program, and insight into how to best approach the issue of the valuation of options.

We certainly all appreciate your offer to stay in touch, and to bounce questions off you, and to in general continue to exchange information for our mutual benefits.

Once again, thanks and best wishes.
Kalman A. Barson, CPA/ABV, CFE
The Barson Group

I have tried Jerry Marlow's Black-Scholes Made Easy and plan to make it a requirement for my investments classes. Its animations provide intuitively clear visual explanations of the fundamental nature of stock market risk. It is an ideal complement to textbooks such as my own (Bodie, Kane, and Marcus, Investments). I have been using my own Excel simulations to illustrate this material in the past, but Marlow's animations blow mine away. Simply wonderful stuff!!
Zvi Bodie, Professor of Finance
Boston University School of Management

Thank you for presenting the Black-Scholes Made Easy and option-valuation seminar. It was interesting and informative, and as the title of your book suggests, your presentation skills and graphics, did make it easy to follow and understand.

My dozen CPA colleagues attending the seminar from other firms, all came away with the same impression: you were able to take a subject that is, at best, difficult to understand and make it intelligible.

I look forward to future seminars, and wish you much success with the option-valuation program.
Marshall A. Morris

Futures Magazine calls the Black-Scholes animation "A joy to use" and of "enormous value as a teaching tool for students, novice traders and those who suddenly find themselves in need of a reference for option behavior."

I am new to options and will be using the book and software to learn about them. I have started to read the first chapters and using the software and I am pleased to report that it has made understanding options so much easier. Thank you.
Robert Lim

Financial Engineering News says,
If you're well versed in the stochastic mathematics underlying Black-Scholes-Merton option pricing theory, this book will interest you as a way of visually demonstrating how the value of options vary with the key input parameters such as time to maturity, strike price, dividends paid, stock price volatility, etc. And if you're relatively new to the field (or a student), this book offers a unique tool for learning about option pricing in a way that blends theory (albeit with minimal math) with an easy to use, graphics rich program which runs on Windows based PCs.

Marlow's visual template replaces the traditional 'hockey stick' payoff diagrams with probability functions— both theoretical and numerically generated (Monte Carlo simulations) to help readers visualize how an option's initial and subsequent values are best measured by the calculated likelihood of a payoff and profit...

I especially liked how the software allows users to provide 3-sigma estimates of a stock price's upper and lower boundaries within a specified timeframe, then translates these inputs into statistical parameters such as expected return, median return, volatility, and probability distributions of future prices and returns during the timeframe selected...

In the final part of the book, Marlow takes the concepts he's introduced in the previous sections to guide readers through an explanation of how one can use these concepts to assess option opportunities. It's very practical advice...

I'd also recommend the book and software to anyone (like me) who's had a strong dose of option pricing theory but wants to enhance his or her intuitive grasp of the topic....
Jim Finnegan, Editor

Look Inside >> 
February 2002

I am now studying in Harvard Summer School and taking a course Risk Management and Derivatives. My professor Bulent Aybar introduced your book Option Pricing: Black-Scholes Made Easy to me. I have just finished my undergraduate sophomore year in Tsinghua University in China and I really have a wonderful experience here in Boston during this summer. Your book is friendly and easy to understand. I like your writing style. You express complex ideas in easy words.
Ji Tuo

Your book Option Pricing: Black-Scholes Made Easy with the Black-Scholes animation is wonderful! It truly is a visual way to understand this subject.
Savio Braganza

I've read several books on options and have traded options on a regular basis. I felt like I had a pretty good handle on the concepts and mechanics. Black-Scholes Made Easy took these abstract ideas and provided a concrete visual experience to solidify them in my head. To see the probabilities and assumptions expressed graphically really enhanced my understanding of options as well as stocks. I believe anyone wanting to get a firm grasp on options, whatever their current level of knowledge, would benefit from Black-Scholes Made Easy. I wish that this program had been available when I first starting learning options.
Kirk Guillebeau
Financial Analyst

I thoroughly enjoyed the presentation. I am looking forward to receiving all the "goodies" which you promised.
I. Alan Hirschfeld CPA/ABV MBA
RosenfarbWinters, LLC
Forensic Accountants & Business Consultants

Thanks for a grand tutorial on Friday, and for making a huge contribution to the investment community. You are truly an "explainer extraordinaire."
Dennis A. Webb, MAI, ASA
Primus Valuations
Marina Del Rey, CA

Jerry Marlow has rendered a major service in mapping the results of using the Black-Scholes option pricing formula... If you want to see the formula at work, then get this book... This is the formula without all the math. Marlow has made a valuable contribution to the understanding of contemporary, mathematical finance.
Andrew Allentuck
globeinvestor.com Review

Black-Scholes is an essential tool for understanding option pricing. No buyer or seller of options should be without it. However, it is something that is given perfunctory treatment at best in most finance texts, and few MBA programs cover it in sufficient detail to make it useful as an analytical tool.

Jerry Marlow's simulation makes the model understandable to anyone with even a basic mathematical background. It brings the world of options trading from the world of black magic into the light of day. I strongly recommend his software to anyone who would use this tool.
William Tolin Gay
Securities Lawyer and Investor
Mission Viejo, California, USA

Bought your book on Black-Scholes Made Easy... It helps me navigate the jungle of financial mathematics and move on to other books on options pricing. Hope you are working on Binomial Pricing model. Also, what about other models like those on interest rates which are even more complex and difficult to understand? Can't wait to see your next book. Keep the good stuff coming. A Visual Guide to Binomial Pricing Model and more please please please.
Siew Gan
Senior Audit Manager
(International financial institution)
Singapore

I have read Mr. Marlow's materials, and I have used him to walk through a description of options usage for a group of 60 plus business appraisers from across the United States. I strongly urge you to consider using Mr. Marlow for your efforts; he has a good understanding of a complex subject and he is able to convey it to people without getting tangled in complex mathematics and theory.
Randall Schostag, CFA, MCBA, ASA, AVA, BVAL

I intend to use the simulation software to help visualize the Black-Scholes equation. I have a Ph.D. in chemistry and have done some work in finance. I have worked with environmental and kinetics models as a chemist. I like the idea of having a model to increase my understanding of Black-Scholes.
Dennis Nicks

I have been an equity analyst for nearly ten years and have a solid grasp on Black-Scholes, but this book makes it far easier to understand and master Black-Scholes and probability.
Chris Adams

I am a 64 year old retired woman who is mathematically illiterate. What am I doing with Jerry Marlow's Black Scholes simulator?

Well, I decided after seeing my stocks seriously decline after 2000 to do something about it. My investment is too important to me not to know what I am doing and not be in control. I decided to learn everything I could short of ever thinking I could top Alan Greenspan.
All my research led me down lots of paths, including looking at options. Believe me, if one is interested in taking control of one's investment or is interested in starting an investment club one would never want to miss Jerry Marlow's book.

Now, I do not trade in options but use his brilliant animations to arrive at my conclusions when I buy and sell stocks. For this, I find his last chapter the most useful.

If I can appreciate and understand Black Scholes, anyone can. Now I feel that I am in the world of Einstein. What a find in Jerry's book!
Rochelle Auletta

 

Movie above requires Adobe Shockwave player

To run the Adobe Shockwave movie on this page, your browser must have the Adobe Shockwave player installed. To go to Adobe and install the Shockwave player (free), click here.

In Firefox, to allow the simulator to run, you may need to allow popup. In Internet Explorer, you may need to allow blocked content.

To restart demo above from beginning, refresh your browser window (F5).

To give yourself more room to view simulator, set browser view to full screen (F11).


Learn interactively with the simulator and book Option Pricing: Black- Scholes Made Easy or with author and developer Jerry Marlow

Whether you want to trade options, learn how stock options are valued, learn about stochastic modeling, see how stochastic calculus works or are studying quantitative finance, the stock option simulator makes it easy for you to learn what you want to learn and much, much more.

The Shockwave movie on this page gives you a quick introduction to stock options and to option-pricing theory. It shows you the kinds of simulations you can run interactively with the simulator.

The book explains principles and shows you how to use the simulator to illustrate those principles.

In the simulator, change the stock's volatility forecast, the stock price, the stock's expected return, the option's strike price, the option's time to expiration, the stock's dividends or the risk-free rate. Click some more buttons. See how potential option payoffs change. See how and why the option's value changes.

Use the simulator to calculate your stock forecast and draw its probability distribution. For a given call or put option, see what probability of profit your stock forecast gives you.

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Screen Captures from the Book

To see simulator screen captures used in the book Option Pricing: Black-Scholes Made Easy, click any of the links below.

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

Buy simulator and book and start learning interactively. (Windows only.)

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Access web version of simulator
To use the web version of the simulator, click Black-Scholes Made Easy simulator— web version. The web version requires a password from the book.


Telephone tutorial or consultation with Jerry Marlow at $2/minute
Want a quick consultation or tutorial via telephone with the creator of Option Pricing: Black-Scholes Made Easy?

Call (917) 817-8659.

Jerry Marlow will explore your thinking and questions with you for a fee of $2 per minute. No minimum. No maximum.

To pay for the consultation, charge it to your credit card via PayPal.

Click the Pay Now button. On the PayPal screen, type the number of minutes that we talked into the quantity field.


Jerry Marlow, MBA
(917) 817-8659
jerrymarlow@jerrymarlow.com
www.jerrymarlow.com
www.marlowfinancialmodeling.com


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