European Option Prices and Greeks in 3D

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This Demonstration displays the prices of European call options, put options, or the "Greeks" associated with these options (delta, gamma, vega, theta, and rho). The display is 3D, with the stock price on the axis and the days to expiration on the axis. Sliders let you change the strike price, risk-free rate, and volatility parameters. You can see how a specific parameter realization travels across the 3D surface as the stock price changes and time to expiration approaches.

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The 3D surface clearly shows how the call (or put) price curve changes shape and approaches the intrinsic value curve as time to expiration approaches. Alternatively, the delta parameter becomes very steep as expiration approaches, and this steepness is seen as the gamma parameter spikes for near-the-money options as expiration approaches. The vega parameter surface shows the influence of (a constant) diminishing of volatility as expiration approaches or the stock price moves away from the strike price. To further aid in understanding, you can drag to rotate the 3D displays to match the two-dimensional graphs available in most financial derivatives textbooks.

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Contributed by: Joseph D. Haley (November 2012)
(St. Cloud State University)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Snapshot 1: Rotate the view slightly to look under the surface. It shows the delta curve at 90 days to expiration as a smooth, cumulative normal distribution. As time to expiration approaches, this curve becomes very steep, showing the increased difficulty of maintaining an accurate delta hedge.

Snapshot 2: The gamma parameter spikes when the option is near expiration and the stock price is near the strike price. This (relatively) large value corresponds with the steepness of the delta curve under the same circumstances.

Snapshot 3: Rotate the view to a two-dimensional display showing that the vega parameter shrinks as expiration approaches. The horizontal grid lines represent a fixed stock price.



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