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Call Option Sensitivity Measures (single option)

Financial Toolbox™ Graphics Example 2

This example creates a three-dimensional plot showing how gamma changes relative to price for a Black-Scholes option. Recall that gamma is the second derivative of the option price relative to the underlying security price. The plot shows a three-dimensional surface whose z-value is the gamma of an option as price (x-axis) and time (y-axis) vary. It adds yet a fourth dimension by showing option delta (the first derivative of option price to security price) as the color of the surface.

Set Up

```if ~exist('normpdf') msgbox('The Statistics and Machine Learning Toolbox is required to run this example.', ... 'Product dependency') return end ```

Set Up Option Parameters

```% Range of stock prices range = 10:70; span = length(range); j = 1:0.5:12; % Years until option expiration newj = j(ones(span,1),:)'/12; jspan = ones(length(j),1); newrange = range(jspan,:); pad = ones(size(newj)); ```

Calculate Some "Greeks", e.g., Delta and Gamma

```zval = blsgamma(newrange, 40*pad, 0.1*pad, newj, 0.35*pad); color = blsdelta(newrange, 40*pad, 0.1*pad, newj, 0.35*pad); ```

Display "Greeks" as a Function of Price and Time

• Height is gamma (second derivative of option price with respect to stock price)

• Color is delta (first derivative of option price with respect to stock price)

```% Plotting Sensitivities of an Option figure('NumberTitle', 'off', ... 'Name', 'Call Option Price Sensitivity'); mesh(range, j, zval, color); xlabel('Stock Price (\$)'); ylabel('Time (months)'); zlabel('Gamma'); title('Call Option Price Sensitivity'); axis([10 70 1 12 -inf inf]); view(-40,50) cbx = colorbar('horiz'); ax = gca; apos = ax.Position; cpos = cbx.Position; cbx.Position = [cpos(1) .075 cpos(3) cpos(4)]; ax.Position = [apos(1) .25 apos(3) .68]; box on ```

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