A Beginner’s Guide to the Black-Scholes Option Pricing Formula (Part 1)

Black-Scholes Call on European Stock
Black-Scholes d1 d2
Preface

Firstly let me apologize to the .NET developers perusing this blog, as this article is a little off-topic. However, my interests range over both .NET and derivatives, and I will be posting on both topics in the future.

Introduction

The Black-Scholes model for pricing stock options was developed by Fischer Black, Myron Scholes and Robert Merton in the early 1970’s. It is arguably the most important result in financial engineering, and is certainly a rich source of interview questions in the financial services industry.

I was recently asked by a friend if I could provide a written explanation in layman’s terms of how the Black-Scholes options pricing formula works. This isn’t necessarily all that easy as the formula involves some relatively complex mathematics. However, I think it is possible to get an intuitive understanding of what the various parts of the formula mean. This article is an attempt to explain that.

Note that to keep things simple here I am only going to discuss European call options on non-dividend paying stock. It doesn’t matter for the purposes of this article if you don’t know what that means.

What is an Option?

Firstly a reminder of what a European call option is. If I ask this in an interview I usually get the textbook answer: ‘the right but not the obligation to buy an asset at a predetermined price at a predetermined date’. My next question is always ‘what does that actually mean?’

Consider a European call option on a Microsoft share (the ‘asset’), with a strike of 30 (the ‘predetermined price’) and maturity of one year from today (the ‘predetermined date’). If I pay to enter into this contract I have the right but not the obligation to buy one share at 30 in a year’s time. Whether I actually exercise my right clearly depends on the share price in the market at that date:

– If the share price is above 30, say at 35, I can buy the share in the contract at 30 and sell it immediately at 35, making a profit of 5. Similarly if the share price is 40 I make a profit of 10.
– If the share price is below 30, say at 25, the fact that I have the right to buy at 30 is worthless: I can buy more cheaply in the open market.

Thus we get the classic ‘hockey stick’ payoff diagram as below. This shows how the amount of money I make on my contract varies with the value of Microsoft stock at the end of the year.

Call Option Payoff Diagram

So if I enter into this contract I make money if the stock price finishes above 30, but don’t lose anything if it finishes below 30. Because I can’t lose, I have to pay to enter into the contract (this is the price of the contract, or the ‘premium’). This premium is usually paid upfront at the start of the contract. The question is how much this premium is going to be?

The Basic Idea

So we’re trying to find the value today of a contract whose ultimate value depends on the value of the Microsoft stock price in one year’s time. Furthermore, the contract has different values depending on whether that stock price goes up or goes down: the payoff curve above is not symmetrical.

So intuitively we are going to need some measure, or measures, of the probabilities of the stock price ending up at various values after one year.

If we have that it may be possible to apply an expected value calculation to get to a price for the contract. This is explained further in part 2.

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26 thoughts on “A Beginner’s Guide to the Black-Scholes Option Pricing Formula (Part 1)

  1. Hi,

    I don’t know how but you really cover a lot that is needed by me, be it Black-Scholes or CAB/SCSF. Hats off and thanks a lot.

  2. Wow !! Cool … Good Explanation. Can you also write about other option pricings as Black Scholes have many limitations and cant be used in real world

  3. The chart is wrong, at the start of your contract your loss = the option premium, which you pay up front.

    So, the chart should start at -5 where -5 is the price you paid to the other party to enter into the option contract.

    Bill.

  4. Bill

    The chart is intended to show the variation of money made or lost at maturity ignoring the premium payment. It shows that we make money at maturity if the underlying stock price is above 30, but don’t lose anything if it finishes below 30. The text then goes on to say that because of this we have to pay a premium up front to enter the contract.

    So it’s not wrong, although I accept that normally in a payoff diagram you would include the premium.

    Richard

  5. Dear Rich,

    Could you you explain what Interest Rate Swap is? I noticed you have this amazing ability to explain complex issue for layman’s understanding.

    Thanks!

  6. Dear Rich,

    Could you you explain what Interest Rate Swap is? I noticed you have this amazing ability to explain complex issue for layman’s understanding.

    Thanks!

  7. Can anybody explain what is “N” exactly ?? how do we arrive at it & what does it actually represents ? Please reply in layman terms.. I know its cumulative normal standard distribution, but what exactly it means ?

  8. Kam – the normal distribution is one of the basic building blocks of statistics. You may know it as the ‘bell curve’. The easiest way to think of it is as the result of a series of experiments with random outcomes.

    For example, if you tossed a coin 100 times you would get around 50 heads (and 50 tails). If you did it again you’d still get around 50 heads, but the exact number might be slightly different. If you repeated the experiment very many times you would have a distribution of outcomes: most times the number of heads would be around 50, but you’d have some where you got no heads, some where you got all heads. The actual distribution if you drew a graph of number of heads versus number of times they appeared would resemble the normal distribution: the bell curve. If you then increased the number of coin tosses to, say, 200 and again performed the experiment multiple times you would get a smoother ‘curve’ (the x-axis would go from 0 to 200 instead of 0 to 100). There’s a thing called the central limit theorem that says roughly that if you theoretically increased the number of coin tosses to infinity then the distribution of results would be the bell curve/normal distribution exactly (with appropriate scaling).

    It turns out that this normal distribution is very useful for all kinds of statistical analysis, appears quite often in the real world, and has some mathematical properties that make it easy to work with.

    The cumulative normal distribution is just a version of the same thing. In our example it is naively a graph got by replacing the actual number of heads on the y-axis of our graph by the sum of all the heads up to that point. That is the value for 2 heads on the x-axis of the cumulative distribution is the sum of the values for 0, 1 and 2 heads on the regular distribution. The real cumulative normal is again continuous and scaled appropriately: it climbs from 0 on the left to 1 on the right. Again this version of the normal distribution is useful in statistics.

    See wikipedia for the actual graphs: I don’t think I can post them in a comment.

    https://en.wikipedia.org/wiki/Normal_distribution

  9. Could you share some other concepts usually asked in interviews for finance-related jobs? what are other essential should-know models and equations?

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