Explain Gamma. Use delta and convexity in your explanation.
It is like duration of a bond.
Like duration it is primary metric of sensitivity of the option with regard to a change in the stock price.
It is the linear slope, calculated change in rise over run.
If the change in the call option is $1 and the change in the stock is $5, the delta is 1/5 = 0.2.
Call option deltas are positive, indicating a positive slope, indicating a direct relationship: when the stock rises, the option rises (maybe).
For the option it is similar but in reverse, if the option falls by $2 when the stock rises by $5, the put delta is -.2 (notice the minus), which indicates an inverse relationship. In this regard a put delta is similar to a bond’s duration.
A call delta range is 0 to 1.0 and a put delta range is 0 to -1.0.
As a call delta approaches 0 it is Out of the money and as it approaches 1.0 it is In the money. The reverse is true for puts.
To calculate the change in the (call) option based on a linear estimate, multiply the call delta times the change in the stock price. .3 delta X $3 change in stock price = $0.90 increase in call option on a linear estimated basis.
Like with duration, this is most accurate with small changes in change in stock price. Like with duration there is a correction methodology. The convexity of options is the gamma.
A low gamma option suggests that the delta should not change too much as the stock changes and a higher gamma option indicates that the delta should change more, thus creating a greater inaccuracy in predicting valuation.
The absolute value of the put delta and the absolute value of the call delta must sum to 1.0. Therefore, if the put delta is -.30, the call delta must be 0.70.
The delta is the first calculus derivative and the gamma is the 2nd.
The sum of the deltas in a position is the position delta. For example, Given a straddle with a put delta of -0.30, the call delta must be 0.70. The position delta must be 0.70 -0.30 = 0.40. If the stock rises by $1.00, the straddle should rise by $0.40 (remember X $100).
Other variables (time, volatility, strike, risk free interest rate) (eg theta, vega, NA, rho) affect the option value.