## суббота, 12 июня 2010 г.

### Example

A call option expiring in 99 days on 100 shares of XYZ stock is struck at $50, with XYZ currently trading at$48. With future realized volatility over the life of the option estimated at 25%, the theoretical value of the option is $1.89. The hedge parameters Δ, Γ, κ, θ are (0.439, 0.0631, 9.6, and -0.022), respectively. Assume that on the following day, XYZ stock rises to$48.5 and volatility falls to 23.5%. We can calculate the estimated value of the call option by applying the hedge parameters to the new model inputs as:

$dC = (0.439 \cdot 0.5) + \left(0.0631 \cdot \frac{0.5^2}{2} \right) + (9.6 \cdot -0.015) + (-0.022 \cdot 1) = 0.0614$

Under this scenario, the value of the option increases by $0.0614 to$1.9514, realizing a profit of $6.14. Note that for a delta neutral portfolio, where by the trader had also sold 44 shares of XYZ stock as a hedge, the net loss under the same scenario would be ($15.86).