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# Delta hedging

#### nikogeorgiev

##### Member
Hi David

This is supposed to be an easy one but I am struggling with one Schweser question relating to delta hedging.

Suppose that a call option on Stock Y with a strike price of $50 trades at$3 and that the delta on this option is equal to 0.5. Derivatives trader Ralph currently owns 10,000 shares of stock Y and delta hedges his position with 200 call option contracts.

After the hedge is initiated, the price of Stock Y increases which increases the delta of the call option to 0.8. In order to main his delta hedge, Ralph should buy/sell how many shares?

Kind regards
N

I am lost here, why is he hedging with only 200 contracts given that the delta is 0.5? And from their the question doesn't quite make sense to me. What am I missing here?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @nikogeorgiev Keep in mind that each call option contract is for 100 options. Ralph starts with 10,000 owned shares, each with delta = 1.0, so the shares have position delta = +10,000 * 1.0 = +10,000. The key formula is Quantity * %Greek = Position Greek; e.g., if you write 300 call options each with (percentage) delta of 0.5, then your position delta = -300* 0.5 = -150. If you buy 400 option with per-option gamma of 0.030, then your position gamma = +400 * 0.030 = 12.0.

So to hedge his +10,000 position delta, he needs a trade that neutralizes with -10,000 position delta, where if the per option delta = 0.50, then his quantity is given by -10,000 / 0.50 = -20,000, so he needs to write -20,000 options which is 200 contracts. So the option trade contributes position delta = quantity * delta = -(200 * 100) * 0.5 = -10,000, such that his delta is neutralized: (10,000 shares * 1.0 per) + [-(200 * 100) * 0.5] = zero.

If the call option delta increases to 0.80, then his written options have a new position delta = -(200 * 100) * 0.80 = -16,000. To neutralize with shares (each share has delta of 1.0), he should own 16,000 shares. So he needs to buy 6,000 shares. I hope that's helpful!

#### nikogeorgiev

##### Member
Absolutely amazing explanation.

Why clarifying question, if I may?

Are option contracts always for 100 options or we are making an assumption here based on what is common in the market place?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @nikogeorgiev Glad to help! Re: Are option contracts always for 100 options or we are making an assumption here based on what is common in the market place? We do seem to assume that. Hull says "In the United States, an option contract is a contract to buy or sell 100 shares." (Chapter 1) and then:
"Stock Options Most trading in stock options is on exchanges. In the United States, the exchanges include the Chicago Board Options Exchange (www.cboe.com), NYSE Euronext (www.euronext.com), which acquired the American Stock Exchange in 2008, the International Securities Exchange (www.ise.com), and the Boston Options Exchange (www. bostonoptions.com). Options trade on several thousand different stocks. One contract gives the holder the right to buy or sell 100 shares at the specified strike price. This contract size is convenient because the shares themselves are usually traded in lots of 100." -- Hull, John C.. Options, Futures, and Other Derivatives (Page 214). Pearson Education. Kindle Edition.

I think this has been challenged in the forum. But no exceptions have been cited. I don't trade options. Thanks!

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