## Black-Scholes Option Pricing in MATLAB using the NAG Toolbox

A MATLAB user at Manchester University contacted me recently asking about Black-Scholes option pricing. The MATLAB Financial Toolbox has a range of functions that can calculate Black-Scholes put and call option prices along with several of the sensitivities (or ‘greeks‘) such as blsprice, blsdelta and so on.

The user’s problem is that we don’t have any site-wide licenses for the Financial Toolbox. We do, however, have a full site license for the NAG Toolbox for MATLAB which has a nice set of option pricing routines. Even though they calculate the same things, NAG Toolbox option pricing functions look very different to the Financial Toolbox ones and so I felt that a Rosetta Stone type article might be useful.

For Black-Scholes option pricing, there are three main differences between the two systems:

- The Financial Toolbox has separate functions for calculating the option price and each greek (e.g. blsprice, blsgamma, blsdelta etc) whereas NAG calculates the price and all greeks simultaneously with a single function call.
- Where appropriate, The MATLAB functions calculate Put and Call values with one function call whereas with NAG you need to explicitly specify Call or Put.
- NAG calculates more greeks than MATLAB.

The following code example pretty much says it all. Any variable calculated with the NAG Toolbox is prefixed NAG_ whereas anything calculated with the financial toolbox is prefixed MW_. When I developed this, I was using MATLAB 2012a with NAG Toolbox Mark 22.

%Input parameters for both NAG and MATLAB. Price=50; Strike=40; Rate=0.1; Time=0.25; Volatility=0.3; Yield=0; %calculate all greeks for a put using NAG [NAG_Put, NAG_PutDelta, NAG_Gamma, NAG_Vega, NAG_PutTheta, NAG_PutRho, NAG_PutCrho, NAG_PutVanna,... NAG_PutCharm, NAG_PutSpeed, NAG_PutColour, NAG_PutZomma,NAG_PutVomma, ifail] =... s30ab('p', Strike, Price, Time, Volatility, Rate, Yield); %calculate all greeks for a Call using NAG [NAG_Call, NAG_CallDelta, NAG_Gamma, NAG_Vega, NAG_CallTheta, NAG_CallRho, NAG_CallCrho, NAG_CallVanna,... NAG_CallCharm, NAG_CallSpeed, NAG_CallColour,NAG_CallZomma, NAG_CallVomma, ifail] = ... s30ab('c', Strike, Price, Time, Volatility, Rate, Yield); %Calculate the Elasticity (Lambda) NAG_CallLambda = Price/NAG_Call*NAG_CallDelta; NAG_PutLambda = Price/NAG_Put*NAG_PutDelta; %Calculate the same set of prices and greeks using the MATLAB Finance Toolbox [MW_Call, MW_Put] = blsprice(Price, Strike, Rate, Time, Volatility, Yield); [MW_CallDelta, MW_PutDelta] = blsdelta(Price, Strike, Rate, Time, Volatility, Yield); MW_Gamma = blsgamma(Price, Strike, Rate, Time, Volatility, Yield); MW_Vega = blsvega(Price, Strike, Rate, Time, Volatility, Yield); [MW_CallTheta, MW_PutTheta] = blstheta(Price, Strike, Rate, Time,Volatility, Yield); [MW_CallRho, MW_PutRho]= blsrho(Price, Strike, Rate, Time, Volatility,Yield); [MW_CallLambda,MW_PutLambda]=blslambda(Price, Strike, Rate, Time, Volatility,Yield);

Note that NAG doesn’t output the elasticity (Lambda) directly but it is trivial to obtain it using values that it does output. Also note that as far as I can tell, NAG outputs more Greeks than the Financial Toolbox does.

I’m not going to show the entire output of the above program because there are a lot of numbers. However, here are the Put values as calculated by NAG shown to 4 decimal places. I have checked and they agree with the Financial Toolbox to within numerical noise.

NAG_Put =0.1350 NAG_PutDelta =-0.0419 NAG_PutLambda =-15.5066 NAG_Gamma =0.0119 NAG_Vega =2.2361 NAG_PutTheta =-1.1187 NAG_PutRho =-0.5572 NAG_PutCrho = -0.5235 NAG_PutVanna =-0.4709 NAG_PutCharm =0.2229 NAG_PutSpeed =-0.0030 NAG_PutColour =-0.0275 NAG_PutZomma =0.0688 NAG_PutVomma =20.3560