Explain Gamma. Use delta and convexity in your explanation. It is like duration
ID: 2804948 • Letter: E
Question
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.
Explanation / Answer
Am presuming this is your question and the rest is an answer. "Explain Gamma. Use delta and convexity in your explanation."
Gamma is the second derivative of the option price with respect to the price of the underlying. It measures how much the rate of change of the option price with respect to the underlying, i.e., the delta, itself changes with the price of the underlying. This is important because very often peoploe try to construct a delta neutral portfolio, having zero delta, i.e., one in which the price does not change with respect to the change in the price of the underlying. But the deltas themselves change when the underlying's price changes so a true underlying price insensitive option portfolio will need to be delta and gamma neutral.
Duration and convexity are used with reference to bonds to refer to the time in which the bond would repay its price through coupons and the face value, both discounted to the present, and the second derivative of the price with respect to the interest rate respectively. Duration can also be called the first derivative with respect to the interest rate of the bond price. Hence similar to option strategies, bond strategies may use portfolio management techniques to immunize the portfolio against changes in the interest rate through buying bonds that have opposing trends with respect to duration and convexity. Hence they can be considered similar to the delta and gamma resepctively.
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