Recently, I had to get to grips with the Black Scholes algorithm and its implementation in a few different languages. I decided to go with Java first as that’s what I’m most proficient at. The objective here was to code the algorithm from scratch rather than use a third party library and remain in ignorance.

### A critique of existing Black Scholes implementations in Java

Like any good programmer I first searched to see what was out there. I found numerous such questions on the web as well as only three implementations in Java. The first was Espen Haug’s authoritative webpage implementing Black Scholes in a number of different languages. Espen Haug is known for his two books: The Complete Guide to Option Pricing Formulas and Derivatives: Models on Models; the second – a teaching resource implementation by Robert Sedgewick and Kevin Wayne and third, an implementation by Joshua Davis on koders.com.

To be quite frank I wasn’t happy with any of these implementations. The third implementation was identical to the first one. The second implementation appeared only to support call option prices. And they all exhibited, in my opinion, classic signs of bad code. Let me be more specific by taking Espen Haug’s implementation as an example though many of my reservations would apply to the others too.

Espen Haug’s program, first of all, was written exactly as you would write a C or C++ program; more like C actually. I don’t think it could have been any further from idiomatic Java. First of all it was syntactically incomplete. It was composed only of two functions not residing within a class. The casing of the function and variable naming was arbitrary and not like that of Java at all. The lack of whitespacing and indenting made it all the more hard to read and the constant declarations within the second method invocation was just sheer laziness. I couldn’t help but get the impression that if it had been possible to put the whole thing on one line the author would have.

So far, you could argue, though I would refute it, that my criticisms have been synthetic but if you were to do so you would have spoken too soon. My biggest objection, by far, is still to come and it is I believe a fundamental problem that makes these implementations unacceptable to me. It is the fact that none of the implementations give the reader any idea of how they’ve been composed. To put it another way – although the implementations may be correct and the authors far more intelligent than I – the programs have not been decomposed into their constituent parts for the benefit of the reader’s comprehension. And for this reason there’s insufficient reuse of certain code. For example, if there was a function for the standard normal probability function (which is bundled into the CND() code) then it would be reused by the CND() function as well as the calculation of the Greeks.

The existing implementations may be suitable if I was just looking to copy and paste some code and not care about what it did or its maintainability and readability. However, in this case, I was looking to learn about the algorithm itself from these sources and wanted to know specifically how the implementations had been composed from the formulas and to which areas of code each constituent formula corresponded. In the ideal case I should have been able to look at the formulas and the implementation side by side and be able to relate one to the other by merely identifying blocks of code. I knew I could do better so I did.

### A decomposed implementation

Here I present the constituent parts in the implementation of the Black Scholes algorithm and formulas that correspond to each part – an end result I refer to as a decomposed implementation. This is purely the derivation of an implementation from the mathematical formulas. It is NOT a discussion about the mathematical or financial aspects of the algorithm – this I’m simply not qualified to write about. Please see expert authors if you’re looking for a semantic treatment.

#### Black Scholes Formula

The Black Scholes formula prices European call or put options and consists, at the top level, of two formulas – one that calculates the price of a call option (c) and another calculates the price of a put option (p).

where:

The meanings of the variables used above are as below.

- s = price of stock
- x = strike price
- r = risk free interest rate
- t = time in years until option expiration
- σ (sigma) – implied volatility of stock
- Φ (phi) – standard normal cumulative distribution function

In order to implement these formulas let’s take our first step in decomposing it. So far the formula above consists almost entirely of scalar primitive variables with the exception of one: Φ (phi). This is the standard normal cumulative distribution function (CDF). As it is a prerequisite to the formulas it must be implemented first. Let’s look at the CDF formula.

The CDF is not used directly in Black Scholes being an infinite integral. Instead a very accurate numerical approximation method is used. There are a number of different ways CDF can be approximated depending on the level of accuracy desired. In Black Scholes generally the Abramowitz & Stegun (1964) numerical approximation is used which is given below.

The Abramowitz & Stegun numerical approximation above uses six constant values in its formula. However it also relies on another function in turn – the standard normal probability density function (PDF) denoted above by Z(x). We will return to implement the CDF later but we must drop down another step in our decomposition to implement the PDF which is our lowest level prerequisite.

#### Standard normal probability density function

The standard normal probability density function (PDF) is as below.

As you can see the PDF is defined purely in terms of variables and there are no further functions involved so let’s begin our Java implementation with this formula.

public static double standardNormalDistribution(double x) { double top = exp(-0.5 * pow(x, 2)); double bottom = sqrt(2 * PI); return top / bottom; }

Note that even the implementation of this function is decomposed into its top half and its bottom half allowing it be easily related to the formula above visually by the reader. This is how I prefer formulas to be coded. Now that we’ve implemented the PDF let us return to the next function up which required the PDF: the CDF.

#### Standard normal cumulative distribution function

As we discussed previously the CDF implementation uses the Abramowitz & Stegun (1964) numerical approximation formula consisting of six constant values as below.

Since we now have a PDF implementation above we can now use this in our CDF Java implementation below. This implements the CDF with one little difference: on line 9 and 17 it handles negative values appropriately. On line 16 we use the PDF function defined earlier on.

private static final double P = 0.2316419; private static final double B1 = 0.319381530; private static final double B2 = -0.356563782; private static final double B3 = 1.781477937; private static final double B4 = -1.821255978; private static final double B5 = 1.330274429; public static double cumulativeDistribution(double x) { double t = 1 / (1 + P * abs(x)); double t1 = B1 * pow(t, 1); double t2 = B2 * pow(t, 2); double t3 = B3 * pow(t, 3); double t4 = B4 * pow(t, 4); double t5 = B5 * pow(t, 5); double b = t1 + t2 + t3 + t4 + t5; double cd = 1 - standardNormalDistribution(x) * b; return x < 0 ? 1 - cd : cd; }

Once again, note the nature of the code above. The function and variables are named appropriately with the variable naming being the same as in the formula, each step of the formula is broken down into a separate line and constants are declared separately and not inline. And the code is again easy to relate to the formula. Let us now move to the next function implementation required by the Black Scholes formula. This time, it is not one, but two: d1 and d2. Let’s look at d1 first. Now, bear in mind, although d1 and d2 do not require the CDF directly the Black Scholes formula does so that’s why we looked at CDF first. Though – we could also have looked at d1 and d2 first. It would have been an equally valid decomposition.

#### d1 – A sub-formula of Black Scholes

The d1 formula is as follows which I reproduce here again for convenience.

Before I present the d1 Java implementation here is the legend of the variable names used which are mostly substitutions of the greek letters used in the maths formula.

- s = Spot price of underlying stock/asset
- k = Strike price
- r = Risk free annual interest rate continuously compounded
- t = Time in years until option expiration (maturity)
- v = Implied volatility of returns of underlying stock/asset

Here is the Java implementation of d1 using the names above. Once again note the formula is broken down into regions which makes it easy to relate to the formula.

private static double d1(double s, double k, double r, double t, double v) { double top = log(s / k) + (r + pow(v, 2) / 2) * t; double bottom = v * sqrt(t); return top / bottom; }

Now let’s move onto d2 which is significantly simpler.

#### d2 – A sub-formula of Black Scholes

The d2 formula is as below.

Its implementation using the same variable names is as below.

private static double d2(double s, double k, double r, double t, double v) { return d1(s, k, r, t, v) - v * sqrt(t); }

Once you have prerequisite function implementations it’s so easy to compose higher level functions. Here d2 uses d1 from earlier on. Now let us return to the parent functions – the call and put formulas themselves.

#### Black Scholes formula

Now that we have prerequisite functions it should be simple to code up the top level formulas – the call and put price calculations.

The Java implementation is as below using the same variable naming as in the legend given above. There is an additional boolean variable which is set to true if the input is a call option and false otherwise.

public static double calculate(boolean callOption, double s, double k, double r, double t, double v) { if (callOption) { double cd1 = cumulativeDistribution(d1(s, k, r, t, v)); double cd2 = cumulativeDistribution(d2(s, k, r, t, v)); return s * cd1 - k * exp(-r * t) * cd2; } else { double cd1 = cumulativeDistribution(-d1(s, k, r, t, v)); double cd2 = cumulativeDistribution(-d2(s, k, r, t, v)); return k * exp(-r * t) * cd2 - s * cd1; } }

#### Black Scholes in Java: The complete implementation

The complete Black Scholes Java implementation is given below to see at a glance.

package name.dhruba.black.scholes.formula; import static java.lang.Math.PI; import static java.lang.Math.abs; import static java.lang.Math.exp; import static java.lang.Math.log; import static java.lang.Math.pow; import static java.lang.Math.sqrt; public enum BlackScholesFormula { _; private static final double P = 0.2316419; private static final double B1 = 0.319381530; private static final double B2 = -0.356563782; private static final double B3 = 1.781477937; private static final double B4 = -1.821255978; private static final double B5 = 1.330274429; public static double calculate(boolean callOption, double s, double k, double r, double t, double v) { if (callOption) { double cd1 = cumulativeDistribution(d1(s, k, r, t, v)); double cd2 = cumulativeDistribution(d2(s, k, r, t, v)); return s * cd1 - k * exp(-r * t) * cd2; } else { double cd1 = cumulativeDistribution(-d1(s, k, r, t, v)); double cd2 = cumulativeDistribution(-d2(s, k, r, t, v)); return k * exp(-r * t) * cd2 - s * cd1; } } private static double d1(double s, double k, double r, double t, double v) { double top = log(s / k) + (r + pow(v, 2) / 2) * t; double bottom = v * sqrt(t); return top / bottom; } private static double d2(double s, double k, double r, double t, double v) { return d1(s, k, r, t, v) - v * sqrt(t); } public static double cumulativeDistribution(double x) { double t = 1 / (1 + P * abs(x)); double t1 = B1 * pow(t, 1); double t2 = B2 * pow(t, 2); double t3 = B3 * pow(t, 3); double t4 = B4 * pow(t, 4); double t5 = B5 * pow(t, 5); double b = t1 + t2 + t3 + t4 + t5; double cd = 1 - standardNormalDistribution(x) * b; return x < 0 ? 1 - cd : cd; } public static double standardNormalDistribution(double x) { double top = exp(-0.5 * pow(x, 2)); double bottom = sqrt(2 * PI); return top / bottom; } }

Compare this implementation to the alternative implementations linked to at the beginning of the article. Do you see the differences? Financial algorithms are sufficiently complex without programmers obfuscating their implementations even further. The function by function decomposition technique coupled with region based decomposition of individual functions makes such complex code maintainable and readable which should be our primary objectives when developing such algorithms. This is the guide I wished I had when I started looking into Black Scholes. Bear in mind I have omitted the javadoc on these methods for brevity which would normally contain online links to formulas etc.

#### Example outputs

A couple of examples follow of the Black Scholes calculation – one call and one put.

public static void main(String[] args) { boolean call; double s, k, r, t, v, price; // call option example call = true; s = 56.25; k = 55; r = 0.0285; t = 0.34; v = 0.28; price = BlackScholesFormula.calculate(call, s, k, r, t, v); System.out.println(price); // 4.56 // put option example call = false; s = 49; k = 50; r = 0.001; t = 0.25; v = 0.20; price = BlackScholesFormula.calculate(call, s, k, r, t, v); System.out.println(price); // 2.51 }

Wolfram Alpha has an excellent online calculator that will allow you to check the result of the call and the put.

In my next and final article in the Black Scholes series I alter the implementation above to incorporate the Greeks (delta, gamma, vega, theta, rho) into it. Stay tuned.

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Patrick AUsually, I have the price of an option from my broker. A formula (in Java) for calculating volatility would be more useful since I don’t really trust my broker’s (IB, TDA or any other) vol quote. I suspect it is based on mid-price but that would be distorted by a large bid-Ask spread. Any thoughts?