As I wrote the title of this post, I couldn’t help but think of the classic Elton John song, Bennie and the Jets (cue music…) Bennie.. Bennie.. Bennie and the Je..e.ets. Elton John’s hit song was recorded in May of 1973. The Black-Scholes model was first published in a 1973 paper titled “The Pricing of Options and Corporate Liabilities”. Elton John went on to become an international pop superstar while Myron Scholes and another key contributor to the development of the Black-Scholes equation, Robert Merton, went on to win the Nobel Prize in Economics. Maybe some musically inclined option quants can come up with a silly parody something like “Black-Scholes.. Black-Scholes.. Black-Scholes and the Gre..ee..eeks”.

My awful attempt at humor aside, it’s beyond dispute that the Black-Scholes equation revolutionized derivatives pricing and established Fischer Black, Myron Scholes and Robert Merton as the ‘rock stars’ of modern finance.

### The Black-Scholes Equation

So let’s take a closer look at the Black-Scholes equation. First, it is important to understand the basic assumptions of the Black-Scholes model, which I’ve taken from the excellent book, “Options, Futures and Other Derivatives”, by John Hull:

- The underlying asset price follows a ‘random walk’ in accordance with a process known as Geometric Brownian Motion
- Shorts sales are permitted
- No transaction costs are incurred
- No dividends are paid prior to expiration of the option
- Security trading is continuous, with no jumps in the underlying asset price
- The risk-free interest rate, r, is constant

The first assumption leads to a model of stock price behavior that I briefly touched on in my last post in which the underlying asset’s return is assumed to be constant and normally distributed, the asset’s price is lognormally distributed and its volatility is constant. The discrete time version of the lognormal model of stock prices is described by the equation below, which is also the starting point for the derivation of the Black-Scholes differential equation, where delta S is the change in stock price over an instantaneous period of time, t, mu represents the underlying return, sigma its volatility and delta z is a normally distributed random (stochastic) variable:

The next step is to apply Ito’s Lemma, which describes how a function, f, of the underlying S and time t is related to changes in S and t. Then a ‘riskless’ portfolio is formed consisting of a long position in the derivative and a short position in ‘delta’ units of the underlying (I’ll get to what delta means in a moment). This effectively cancels out the randomness of the underlying stochastic process and create an instantaneously riskless position. The position is riskless only for an instant because the relationship between f, S and t is constantly changing as the market moves. I won’t go into all the math, which is beyond the scope of this post, but the end result is the Black-Scholes differential equation:

Lastly, to solve for the value of a call and put, the appropriate boundary conditions must be used, which should be familiar from my last post as the *payoff* functions for a call and put respectively.

For a call, the boundary condition is:

For a put, the boundary condition is:

This results in the following equation for the value of a call:

where d1 and d2 are defined as follows:

and N is the normal cumulative probability density function. As such, N(x) is the probability that a normally distributed random variable is less than x.

To illustrate how these equations are used in practice to calculate the value of a call and put, I’ve uploaded a spread sheet to my Box account at https://app.box.com/s/iqekbg6phhf9m23tuzki . A screen shot is below:

Note that N(x) is implemented in LibreOffice (and Excel) as the function *normsdist.
*Also you can see that the call and put values agree with those from my previous post.

### The Greeks

So let’s move on to the second part of this post, in which I’ll introduce and show how to calculate the *Greeks*. In the context of option valuation, the Greeks refer to a standard set of option sensitivity measures. They are:

- Delta (df/dS) – the change in the value of an option (f) given a one point change in the price of the underlying (S). Delta is defined as the first derivative of f with respect to S.
- Gamma (d2f/d2S) – the change in the delta of an option given a one point change in the price of the underlying (S). Gamma is defined as the second derivative of f with respect to S.
- Theta (df/dt) – the change in the value of an option (f) given a 1 day decrease in the option’s time to maturity (t). Theta is often referred to as the decay rate of an option. It is defined as the first derivative of f with respect to t.
- Vega (df/dsigma) – the change in the value of an option (f) given a one point change in the volatility (sigma). Vega is defined as the first derivative of f with respect to sigma.
- Rho (df/dr) – the change in the value of an option (f) given a one basis point change in the risk-free rate (r). Rho is defined as the first derivative of f with respect to r.

In options trading, the Greeks are critical to managing the risk of an options portfolio. For example, vega can be used to measure the PnL impact of a change in market volatility.

Additionally, as most option traders seek to limit their exposure to movements in the underlying, the delta of an option is used as a hedge ratio. To achieve a delta neutral position, a trader must offset his delta by buying (selling) delta units of the underlying if the trader’s option delta is negative (positive).

Also, by virtue of the fact that delta and gamma are the first and second derivatives of the option valuation function (f) with respect to S, the new value of an option can be estimated by means of a Taylor series expansion. As such, delta and gamma are similar in many respects to duration and convexity in the fixed income domain (see my March 2013 post, Introducing QuantLib: Duration and Convexity).

So now that we’ve covered all the key background concepts related to the valuation of options with Black-Scholes and the measurement of option price sensitivity, I’ll show how easy it is to value an option in QuantLib using the BlackScholesCalculator class. This class, in keeping with the Black-Scholes assumptions above, takes a constant volatility (sigma) and rate (r) as input along with the underlying’s price (S), the option’s strike (K) and the option’s time to maturity (t).

Unlike the original, classic Black-Scholes model, the QuantLib BlackScholesCalculator also supports an optional dividend yield. Also, a subtle quirk of the BlackScholesCalculator implementation to watch out for is that the constructor expects sigma to be multiplied by the square root of time.

```
#include <iostream>
#include <cstdlib>
#define BOOST_AUTO_TEST_MAIN
#include <boost/test/unit_test.hpp>
#include <boost/detail/lightweight_test.hpp>
#include <ql/quantlib.hpp>
#include <boost/format.hpp>
namespace {
using namespace QuantLib;
BOOST_AUTO_TEST_CASE(testBlackScholes) {
Real strike = 110.0;
Real timeToMaturity = .5; //years
Real spot = 100.0;
Rate riskFree = .03;
Rate dividendYield = 0.0;
Volatility sigma = .20;
//QuantLib requires sigma * sqrt(T) rather than just sigma/volatility
Real vol = sigma * std::sqrt(timeToMaturity);
//calculate dividend discount factor assuming continuous compounding (e^-rt)
DiscountFactor growth = std::exp(-dividendYield * timeToMaturity);
//calculate payoff discount factor assuming continuous compounding
DiscountFactor discount = std::exp(-riskFree * timeToMaturity);
//instantiate payoff function for a call
boost::shared_ptr<PlainVanillaPayoff> vanillaCallPayoff =
boost::shared_ptr<PlainVanillaPayoff>(new PlainVanillaPayoff(Option::Type::Call, strike));
BlackScholesCalculator bsCalculator(vanillaCallPayoff, spot, growth, vol, discount);
std::cout << boost::format("Value of 110.0 call is %.4f") % bsCalculator.value() << std::endl;
std::cout << boost::format("Delta of 110.0 call is %.4f") % bsCalculator.delta() << std::endl;
std::cout << boost::format("Gamma of 110.0 call is %.4f") % bsCalculator.gamma() << std::endl;
std::cout << boost::format("Vega of 110.0 call is %.4f") % bsCalculator.vega(timeToMaturity)/100 << std::endl;
std::cout << boost::format("Theta of 110.0 call is %.4f") % (bsCalculator.thetaPerDay(timeToMaturity)) << std::endl;
Real changeInSpot = 1.0;
BlackScholesCalculator bsCalculatorSpotUpOneDollar(Option::Type::Call, strike, spot + changeInSpot, growth, vol, discount);
std::cout << boost::format("Value of 110.0 call (spot up $%d) is %.4f") % changeInSpot % bsCalculatorSpotUpOneDollar.value() << std::endl;
std::cout << boost::format("Value of 110.0 call (spot up $%d) estimated from delta is %.4f") % changeInSpot % (bsCalculator.value() + bsCalculator.delta() * changeInSpot) << std::endl;
//use a Taylor series expansion to estimate the new price of a call given delta and gamma
std::cout << boost::format("Value of 110.0 call (spot up $%d) estimated from delta and gamma is %.4f") % changeInSpot % (bsCalculator.value() + (bsCalculator.delta() * changeInSpot) + (.5 * bsCalculator.gamma() * changeInSpot)) << std::endl;
//calculate new price of a call given a one point change in volatility
Real changeInSigma = .01;
BlackScholesCalculator bsCalculatorSigmaUpOnePoint(Option::Type::Call, strike, spot, growth, (sigma + changeInSigma) * std::sqrt(timeToMaturity) , discount);
std::cout << boost::format("Value of 110.0 call (sigma up %.2f) is %.4f") % changeInSigma % bsCalculatorSigmaUpOnePoint.value() << std::endl;
//estimate new price of call given one point change in volatility using vega
std::cout << boost::format("Value of 110.0 call (sigma up %.2f) estimated from vega) is %.4f") % changeInSigma % (bsCalculator.value() + (bsCalculator.vega(timeToMaturity)/100)) << std::endl;
}}
```

When run, the code produces the following output:

Value of 110.0 call is 2.6119

Delta of 110.0 call is 0.3095

Gamma of 110.0 call is 0.0249

Vega of 110.0 call is 0.2493

Theta of 110.0 call is -0.0160

Value of 110.0 call (spot up $1) is 2.9340

Value of 110.0 call (spot up $1) estimated from delta is 2.9214

Value of 110.0 call (spot up $1) estimated from delta and gamma is 2.9339

Value of 110.0 call (sigma up 0.01) is 2.8631

Value of 110.0 call (sigma up 0.01 estimated from vega) is 2.8612

As you can see, for small changes in the Black-Scholes input parameters, the Greeks can accurately estimate the new price of the option. For larger changes in parameter values or when needing to consider the combined effect of multiple parameter changes (often called a *scenario*), it is necessary to revalue all of the options in a portfolio. This can be expensive and time consuming depending on the performance of the option pricing code, the number of options and the complexity of the scenario.

At this point, I think we’ve covered all the essentials of computing option values and sensitivities with QuantLib’s BlackScholesCalculator class. I hope you enjoyed my latest ‘Introducing QuantLib’ post. Until next time, have fun with QuantLib!

On Firefox 17.07 the equations are not displaying properly.

Ian,

Thanks for bringing this issue to my attention. I believe I’ve fixed the problem.

Regards, Mick

Hello Mick,

thanks for the post it gave a great insight into Black Scholes. I wish to run this against historical data. How would you calculate the price using Black Scholes with a given data dump similar to the following: http://optiondata.net/collections/frontpage/products/historical-options-data-2013-q1-and-q2-with-greeks-and-implied-volatility. Since there are many options with the same underlying and your example only covers one option.

Thanks for the question Asif. In order to reproduce the option prices in the table to which your link refers, you will need all of the relevant option pricing inputs, most of which are provided in the table itself (e.g. strike, underlying price, call/put, etc). In addition, in order to select the right pricing model, you will need to know whether the option contract is subject to European or American exercise. Also, important is the asset class of the underlying. A stock option is priced somewhat differently than a future option, for example. You should use the implied volatility value as the volatility of the option. In addition you will need to calculate the option’s time to maturity (as a year fraction) based on the time the option data was collected. Lastly, you will need to obtain the interest rate for whatever benchmark rate was used to calculate the implied volatility.

I recommend you look at my latest post ‘Implied Volatility’ to get a better feel for how to work with these inputs and how to process all of the options in an option chain.

I hope this helps. Mick

Hello Mick,

thanks for the reply, I have additional information and was hoping you can assist in ascertaining whether or not i am on the right track. 1) they are American exercise not European 2) stock options 3) interest rates are provided by US treasury department which is currently 0.08. Can i use Black Scholes model under these circumstances?

Asif,

Black-Scholes is intended for pricing European options without dividends. Binomial trees or an analytic engine such as QuantLib’s BaroneAdesiWhaleyApproximationEngine should be used to price American stock options with dividends. I suggest you take a look at the equityoption.cpp exammple on the QuantLib Web site for more information. In an upcoming post, I will be demonstrating how to price American options with dividends.

Good questions and, again, thanks for reading my blog. Mick

How would you determine whether to buy a put or call?

It depends on what trading strategy you are pursuing. A put can be used to provide downside protection on a long underlying position. In this case, you are not buying the put to make money on it but rather as a form of insurance – a hedge. Alternatively, you might want to buy ‘cheap’ puts with an eye towards making a profit later.

A call option on stock, on the other hand, can be used to establish a long position in the stock for a fraction of the cost of buying shares in the stock itself, which may be profitable if you are anticipating a significant upward move in the stock. As such, a call provides a form of leverage.

There are a lot of option strategies that involve buying and selling calls and/or put options, with various risk-reward profiles. I suggest you check out the many resources on the Web for more information.

Thanks for reading my blog. Mick

Hi Mick,

Great post by the way. I really appreciate your pragmatism and I am really delightful about your posts since they are very informative. I would like to ask you if you know a method to calculate standard Greeks (delta, gamma, theta, vega, rho) for all the pricing models included in the QuantLib library. It would be really nice of you if you can explain an example about how to calculate missing Greeks for binomial pricing method. At the moment only delta, gamma and theta are provided. It might be interesting to sort out an effective way to calculate also rho and vega, perhaps by means of finite difference method. Thanks in advance.

Antonio,

Thanks for the positive feedback. I really appreciate it. I unfortunately do not have an answer for you regarding a method to calculate Vega and Rho when using QuantLib’s binomial option pricing models. I’m fairly certain the QuantLib implementation itself needs to be extended to implement these Greeks.

Thanks for reading my blog.

Regards, Mick

Your code for “Value of 110.0 call (spot up $1) estimated from delta and gamma” is wrong as the gamma should be multipled by changeInSpot to the square. Howerver for changeInSpot=1 it does not change the numerical result.

I am learning quantlib, so this is very helpful to me.

Just noticing that on line 44 of the code there the following statement

std::court << boost::format("Vega of 110.0 call is %.4f") % bsCalculator.vega(timeToMaturity)/100 << std::endl;

On my compiler visual studio desktop express C++ I need to include

bsCalculator.vega(timeToMaturity)/100

in parentheses as follows;

std::court << boost::format("Vega of 110.0 call is %.4f") % (bsCalculator.vega(timeToMaturity)/100 )) << std::endl;

Otherwise I get an error.

Thanks!