Investing is decision making under conditions of uncertainty. Uncertainty is defined by probability distributions. The probability distribution with which people are most familiar is the normal, bell‑shaped curve.

Every financial forecast is a probability distribution. To value a financial asset is to value the cash flows or payoffs that the asset's forecast allows for. Mathematically speaking, to value a financial asset 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, other derivative securities and even real estate investments.

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

Black‑Scholes Options Pricing Theory assumes that a forecast for a stock's future price is a normal, bell‑shaped curve drawn on a lognormal price axis.

The Black‑Scholes value of a call option is the area of the bell‑shaped curve above the call's strike price. The Black‑Scholes value of a put option is the area of the bell‑shaped curve below the put's strike price.

The Black‑Scholes Made Easy movie, book and simulator give you an easy way to explore, learn and master these basic concepts of modern financial theory.

The movie and the book 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

Jerry Marlow

(917) 817‑8659

jerry@jerrymarlow.com.

The Black‑Scholes Made Easy Movie is an Adobe Shockwave file. Shockwave files require that your browser have the Adobe Shockwave player installed.

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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 the movie from the beginning, refresh your browser window (F5).

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

*An introduction to how hedge funds use mathematical models to value derivatives and other financial instruments*

By Jerry Marlow

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 asset’s 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 simulator— 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?

Let’s 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. It’s 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