Delta Gamma Vega approximation of a Call

[fullofquestions]

1. The underlying price S = 100, volatility = 25%, and risk free rate = 5 %. The Call option is priced at 15.52 with a maturity of 40 days, delta = .98, gamma = .006 and vega = 1.55. What is the price of the option if the Price goes to 105 and volatility to 28%, i.e. what is the delta-gamma-vega approximation for the call?

delta P = .98 * .03 * 105 + .5 * .006 * .03^2 * 105 + []

1. I've always had the dangling question of wether delta (.98) is negative in the delta-gamma approximation (like in the modified duration and convexity approximation for bonds) or if the sign depends on the instrument?
2. I believe the answers were distinct enough so that factoring the Vega approximation did not help any. If you needed to factor in Vega, would it be 1/3 * Vega * .03^3 * 105?

 

[David Harper CFA FRM]

Hi FoQ,
As a truncated Taylor, I would here use:

change in P ~ delta*d_stock + 0.5(gamma)(d_stock^2) + vega*d_volatilty =
(.98 delta * $5 change in stock) + (.5 * .006 gamma * $5^2) + (1.55 vega * 0.03 change in volatility) = + $5.02
i.e., vega is a first derivative, so it does not need to be cubed, it is "on a par" with delta

1. depends on instrument. As this is a call, increase in stock corresponds to increase in call option
(another way to visualize: the option curve here goes "up and to the right", so tangent slope--i.e., delta--is positive...compared to price/yield for bond curve which goes "up and to the left" such that tangent slope--ie., dollar duration--is negative. Also, the almost 1.0 delta betrays this option is significantly "in the money". The delta of this put, however, = 0.98 - 1 = -.02

2. 1/3 * Vega * .03^3 * 105? As above, no, that would be appropriate for a third derivative (e.g., the first derivative of gamma is called "speed" and is unlikely to have impact) ... but vega is first derivative

David

append, i look up 2nd derivative with respect to volatility: called "vomma"; and 3rd deriv wrt vol is called "ultima"