Options

This Finance`Options` subpackage contains tools to compute options using the Black-Scholes model.

This loads the Options subpackage.

In[1]:=<< Finance`Options`

Here are the basic option functions.

In[2]:=?Finance`Options`*

Delta Norm

Elasticity Option

HistoricalVolatility Rho

ImpliedVolatility Theta

Lambda

An option object.

This is a put option for an asset named XYZ that expires on December 20, 1992 with an exercise (or striking) price of 40.

In[3]:=xyzft = Option[{"Put", XYZ, 40, {12, 20, 1992}}]

Out[3]=

However, we have to remember that an option is a derivative security and we need to specify the price and volatility of the underlying asset.

In[4]:=(Price[XYZ] ^= 40;
Volatility[XYZ] ^= 3/10;)

We also have to define the riskless interest rate.

In[5]:=r = 12/100

Out[5]=

We define the settlement date to be August 20, 1992.

In[6]:=settlement = {8, 20, 1992}

Out[6]=

This is a call option for an asset named ABC. The striking price is 40 and the expiration date is December 19, 1992.

In[7]:=abcft = Option[{"Call", ABC, 40, {12, 19, 1992}}]

Out[7]=

This time we specify the volatility but not the price of the underlying asset.

In[8]:=Volatility[ABC] = 3/10

Out[8]=

A basic function.

We compute the value of xyzft given the riskless interest rate r = 12%. Mathematica gives us an exact result.

In[9]:=val1 = Value[xyzft, settlement, r]

Out[9]=

Here is a numerical approximation.

In[10]:=N[%]

Out[10]=

Another way to get an approximation is to write one of the arguments with a decimal point.

In[11]:=Value[xyzft, settlement, .12]

Out[11]=

We define an option's value as a function of two variables, the price of the underlying asset x and the time to maturity t.

In[12]:=optionvalue[x_, t_] = Value[abcft,
ToCalendar[ToJulian[settlement] + Floor[t]], r] /.
Price[ABC] -> x;

This illustrates how the graph of the option value changes as the time to maturity varies.

In[13]:=Plot[Evaluate[Table[optionvalue[x, t],
{t, 0, 120, 10}]], {x, 25, 45},
PlotLabel ->
" Call Option Value\nfor Different Maturities",
FrameLabel -> {"Stock Price", ""}];

Here is the equivalent three-dimensional graph.

In[14]:=Plot3D[optionvalue[x, t], {x, 25, 55}, {t, 0, 120},
AxesLabel->{"Stock Price", "Time", "Option \nValue "}];

Sensitivity measures functions.

Here is the analytical expression for the delta of the put option xyzft.

In[15]:=Delta[xyzft, settlement, r]

Out[15]=

This is a numerical approximation.

In[16]:=N[%]

Out[16]=

We define the delta of the option abxft as a function of two variables, x and t, the price of the underlying asset and the time to maturity, respectively.

In[17]:=optiondelta[x_, t_] = Delta[abcft,
ToCalendar[ToJulian[settlement] + Floor[t]], r] /.
Price[ABC] -> x;

This illustrates how the graph of the option delta changes as the time to maturity varies.

In[18]:=Plot[Evaluate[Table[optiondelta[x, j],
{j, 0, 120, 10}]], {x, 25, 45},
PlotLabel ->
" Call Option Delta\nfor Different Maturities",
FrameLabel -> {"Stock Price", ""}];

Here is the equivalent three-dimensional graph.

In[19]:=Plot3D[optiondelta[x, t], {x, 25, 55}, {t, 0, 120},
AxesLabel->{"Stock Price", "Time", "Option \nDelta"}];

Here is the analytical expression for the theta of the put option xyzft.

In[20]:=Theta[xyzft, settlement, r]

Out[20]=

This is a numerical approximation.

In[21]:=N[%]

Out[21]=

We define the theta of the option abcft as a function of two variables, x and t, the price of the underlying asset and the time to maturity, respectively.

In[22]:=optiontheta[x_, t_] = Theta[abcft,
ToCalendar[ToJulian[settlement] + Floor[t]], r] /.
Price[ABC] -> x;

This illustrates how the graph of the option theta changes as the time to maturity varies.

In[23]:=Plot[Evaluate[Table[optiontheta[x, j],
{j, 0, 110, 10}]], {x, 25, 45},
PlotLabel ->
" Call Option Theta\nfor Different Maturities",
FrameLabel -> {"Stock Price", ""}];

Here is the equivalent three-dimensional graph.

In[24]:=Plot3D[optiontheta[x, t], {x, 25, 55}, {t, 0, 110},
AxesLabel->{"Stock Price", "Time", "Option \nTheta"}];

Here is the analytical expression for the gamma of the put option xyzft.

In[25]:=Gamma[xyzft, settlement, r]

Out[25]=

This is a numerical approximation.

In[26]:=N[%]

Out[26]=

We define the gamma of the option abcft as a function of two variables, x and t, the price of the underlying asset and the time to maturity, respectively.

In[27]:=optiongamma[x_, t_] = Gamma[abcft,
ToCalendar[ToJulian[settlement] + Floor[t]], r] /.
Price[ABC] -> x;

This illustrates how the graph of the option gamma changes as the time to maturity varies.

In[28]:=Plot[Evaluate[Table[optiongamma[x, j],
{j, 0, 110, 10}]], {x, 25, 45},
PlotLabel ->
" Call Option Gamma\nfor Different Maturities",
FrameLabel -> {"Stock Price", ""}];

Here is the equivalent three-dimensional graph.

In[29]:=Plot3D[optiongamma[x, t], {x, 25, 55}, {t, 0, 110},
AxesLabel->{"Stock Price", "Time", "Option Gamma"}];

Here is the analytical expression for the rho of the put option xyzft.

In[30]:=Rho[xyzft, settlement, r]

Out[30]=

This is a numerical approximation.

In[31]:=N[%]

Out[31]=

We define the rho of the option abcft as a function of two variables, x and t, the price of the underlying asset and the time to maturity, respectively.

In[32]:=optionrho[x_, t_] = Rho[abcft,
ToCalendar[ToJulian[settlement] + Floor[t]], r] /.
Price[ABC] -> x;

This illustrates how the graph of the option rho changes as the time to maturity varies.

In[33]:=Plot[Evaluate[Table[optionrho[x, j],
{j, 0, 120, 10}]], {x, 25, 45},
PlotLabel ->
" Call Option Rho\nfor Different Maturities",
FrameLabel -> {"Stock Price", ""}];

Here is the equivalent three-dimensional graph.

In[34]:=Plot3D[optionrho[x, t], {x, 25, 55}, {t, 0, 110},
AxesLabel->{"Stock Price", "Time", "Option \nRho"}];

Here is the analytical expression for the lambda of the put option xyzft. Kappa, vega, and epsilon are sometimes used instead of lambda.

In[35]:=Lambda[xyzft, settlement, r]

Out[35]=

This is a numerical approximation.

In[36]:=N[%]

Out[36]=

We define the lambda of the option abcft as a function of two variables, x and t, the price of the underlying asset and the time to maturity, respectively.

In[37]:=optionlambda[x_, t_] = Lambda[abcft,
ToCalendar[ToJulian[settlement] + Floor[t]], r] /.
Price[ABC] -> x;

This illustrates how the graph of the option lambda changes as the time to maturity varies.

In[38]:=Plot[Evaluate[Table[optionlambda[x, t],
{t, 0, 110, 10}]], {x, 25, 45},
PlotLabel ->
" Call Option Lambda\nfor Different maturities",
FrameLabel -> {"Stock Price", ""}];

Here is the equivalent three-dimensional graph.

In[39]:=Plot3D[optionlambda[x, t], {x, 25, 55}, {t, 0, 110},
AxesLabel->{"Stock Price", "Time", "Option \nLambda"}];

Here is the analytical expression for the elasticity of the put option xyzft.

In[40]:=Elasticity[xyzft, settlement, r]

Out[40]=

This is a numerical approximation.

In[41]:=N[%]

Out[41]=

We define the elasticity of the option abcft as a function of two variables, x and t, the price of the underlying asset and the time to maturity, respectively.

In[42]:=optionelasticity[x_, t_] = Elasticity[abcft,
ToCalendar[ToJulian[settlement] + Floor[t]], r] /.
Price[ABC] -> x;

This illustrates how the graph of the option elasticity changes as the time to maturity varies.

In[43]:=Plot[Evaluate[Table[optionelasticity[x, t],
{t, 0, 100, 10}]], {x, 25, 45},
PlotLabel -> "Call Option Elasticity\n"<>
"for Different Maturities",
FrameLabel -> {"Stock Price", ""}];

Here is the equivalent three-dimensional graph.

In[44]:=Plot3D[optionelasticity[x, t], {x, 25, 55}, {t, 0, 100},
AxesLabel->{"Stock Price", "Time",
"Option \nElasticity"}];

Implied volatility function.

This calculates the implied volatility.

In[45]:=ImpliedVolatility[xyzft, settlement, r, val1]

Out[45]=

Auxiliary functions.

To use the Black-Scholes model we need to know the underlying asset's volatility. The function HistoricalVolatility computes the maximum likelihood estimate of the asset's volatility given a list of past prices.

We read in the prices for one year of IBM stock from the file IBM-90.prc as pairs {date, price}.

In[46]:=prices[IBM] ^=
Last[Transpose[ReadList[
ToFileName[{"Finance", "Data"}, "IBM-90.prc"],
{Number, Number}]]];

Here is the graph of IBM's prices.

In[47]:=ListPlot[prices[IBM], PlotJoined -> True];

Now we compute the maximum likelihood estimate for IBM's stock volatility. This is the daily volatility since we used daily stock prices.

In[48]:=HistoricalVolatility[prices[IBM]]

Out[48]=

To get an estimate of the yearly volatility we multiply by the square root of the number of days in a year. Usually we only take the number of trading days, which is about 250.

In[49]:=% Sqrt[250.]

Out[49]=

The cumulative normal distribution is often used in modern finance and especially in the context of option pricing. Here is the graph of the cumulative standard normal distribution function.

In[50]:=Plot[Norm[x], {x, -5, 5}];

Its derivative with respect to x is indeed the standard normal distribution.

In[51]:=D[Norm[x], x]

Out[51]=

Here is the graph of the standard normal distribution.

In[52]:=Plot[Evaluate[D[Norm[x], x]], {x, -5, 5}];