What Everybody Ought to Know about Options II

Last week we learned about the basics of Options. We've looked at the put call parity, moneyness of an option and calculated some simple profit and loss examples. This week we'll take a look at option greeks and see how they're used to measure market risk variables. For this post, we'll delve deeper into delta and gamma.

Delta: Δ
We use delta to describe our market exposure to the movement of the underlying.

The change in option price for a small change in the underlying stock price is denoted by the Greek letter delta Δ

Think about it this way - Before expiration, a call option's delta exposure represents some measure of all possible outcomes.

For example, an OTM call has a small probability so the delta is small (but not zero). For ITM options, these probabilities are large (but not 100). The further an option strike is from the spot price, the closer the delta is to either 0 or 100. For an ATM option, the delta is close to 50.

At expiration, option delta converges to either 100 or 0 and represents stock exposure (ITM) or zero economic interest (OTM). The ITM call would be +100Δ and the put has short exposure to the underlying and is therefore -100Δ.

Since we learned that the market exposure of a combo is identical to that of the underlying, we can also describe it using the equation:

ComboΔ = CallΔ - PutΔ = 100Δ

In this equation, delta represents percent exposure to the underlying. For example, 100Δ=100%

From this we can see that:

CallΔ - PutΔ = 100Δ

HWQ1) Based on the above, write out an equation for the following example, SPY 120C = 60Δ and the SPY 120P = -40Δ

HWQ2) Since delta represents the change in option price for a $1 change in stock price, calculate the new price of the call when S moves from $106 to$108
*Remember that change in option price = change in spot * avg. delta
Spot(S1) = $106, S2 = $108, Strike(K)=$105, Δ=0.55 Call(C)=$4.2 Put(P)=$2.15

So what about delta neutral ? All else equal and disregarding volatility, being long 100 deltas means that if the underlying stock moves up by $1, the position gains $100 and if the stock moves down by $1, the position loses $100. A position that is long 50 deltas means that if the underlying stock moves up by $1, the position gains $50 and if the stock moves down by $1, the position loses $50. Being delta neutral or 0 delta, means that the position value neither goes up nor down with the underlying stock.

Delta hedging may be accomplished by adjusting the amount bought or sold on new positions. This way, the portfolio delta can be made to sum to zero, and the portfolio is then delta neutral.

Gamma Γ:

Now that we have learned that the delta of an option is not constant and that it can increase or decrease with changes in the spot price, how do we measure this change?

The rate at which the delta changes is gamma. Gamma measures the change in the risk profile of a portfolio. (you can think about Delta and Gamma like duration and convexity in bonds, with gamma being the second derivative of the value function with respect to the underlying price.)

The change in delta of an option for a unit change in spot is represented by the Greek letter gamma (Γ).

Let's say the stock increases by $10 and this results in a $5.40 increase in the price of the 100 strike call option. This increase is attributed to delta. (0.54 in this case) However, what if the actual price of the option increased by more than $5.40? It increased by $6.30 - the additional $0.9 increase is explained by gamma.

The delta of the option changes due to gamma.
It increased as the underlying price increased. This increasing delta created additional profit as the stock appreciated.

In Summary: Δ

• Delta of a call option is always positive
• Delta of a call varies between 0 (deep OTM) to 100 (deep ITM)
• Delta of a put option is always negative
• Delta of a put varies between -100 (deep ITM) and 0(deep OTM)
• CallΔ - PutΔ = 100Δ

Summary: Γ

• Option buyers (of calls or puts) are long gamma
• Options sellers (of calls or puts) are short gamma
• Gamma is the same for calls and puts at the same strike price since their deltas are directly related.
• Their changes in delta with regard to spot are identical.

HW Q3) You are long 100 SPY 120C (with 100 shares per contract multiplier), Assume SPY=120 (ATM)
How many shares should be sold short against these calls to be delta neutral?

HW Q4) If the gamma of the option is 2 and SPY goes up to 126, how many shares should be sold to return the portfolio to delta neutral?

Comfortable with Delta and Gamma basics? Try out Q1-4 and clarify your doubts in the comments below!

//www.wallstreetoasis.com/blog/what-everybody-ought-to-know-about-trading-options