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# L1.T4.7. Dynamic delta hedging

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
David's ProTip: I learned from Carol Alexander a useful semantic distinction (not in Hull). Consider a position in 100 call options with per-option delta of 0.6:
• The Percentage Delta is 0.6; this is the unitless first partial derivative, dc/dS
• The Position Delta is 60 because Position Delta = Quantity * Percentage Delta.
• If we are long, we use (+) quantity: Position Delta (long 100 calls) = +100 * 0.6 = +60;
• If we are short , we use (-) quantity: Position Delta (short 100 calls) = -100 * 0.6 = -60
• To neutralize is to get the position Greek to zero
This is robust, for example:
• Selling puts increases position delta because -QTY * -% delta = +position delta; i.e., % delta of puts always negative; % delta of calls is always positive
• Selling calls or puts decreases position gamma because -QTY * +% gamma = - position gamma; i.e., % gamma is always positive for both calls & puts
Just as we use dollar duration (not modified duration) to neutralize duration in the portfolio, we neutralize an option Greek by summing Position Greeks to zero. I often see candidates trying to neutralize with percentage delta directly, but you can't, you need to sum the "Position" Greeks. I hope that's useful! David

AIMs: Discuss the dynamic aspects of delta hedging. Define the delta of a portfolio.

Questions:

7.1. Yesterday, a market maker sold (wrote) 100 at-the-money (ATM) call options when the percentage delta was 0.57. The market maker immediately started a daily dynamic delta hedge by purchasing the underlying shares to achieve a a position delta of zero (i.e., to neutralize delta). Today, the share price dropped such that the call option percentage delta reduced to 0.54. What is today's dynamic delta hedge trade?
a. Buy 3.0 shares
b. Sell 3.0 shares
c. Buy 54.0 shares
d. Sell 54.0 shares

7.2. Today, a market maker takes a short position in 100 at-the-money (ATM) put options (i.e., writes or sells puts) when the percentage delta was -0.48. The market maker immediately starts a dynamic delta hedge by trading the underlying shares to neutralize delta. Tomorrow, if the stock price drops and the percentage delta drops to -0.53, what will be tomorrow's dynamic delta hedge trade?
a. Sell 5.0 shares
b. Buy 5.0 shares
c. Sell 53.0 shares
d. Buy 53.0 shares

7.3. A market maker today writes 100 at-the-money (ATM) call option contracts (i.e., short 10,000 options) and immediately starts a dynamic delta hedge by purchasing the underlying non-dividend-paying shares, but due to transaction costs will only re-balance weekly. Next week the underlying share price, volatility and riskfree rate are unchanged. What is the next week's dynamic delta hedge trade?
a. Sell some amount of shares (reduced long position in shares)
b. No transaction (maintain long position in shares)
c. Buy some amount of shares (increase long position in shares)
d. Not enough information (we need the option delta)

7.4. A market maker is trading the following three (3) positions in call and put options which are identical with respect to their underlying stock price, the strike price and the maturities: long 100 ATM call options with a percentage delta of 0.6; short 60 ATM call options; and long 50 ATM put options. Which trade will neutralize the market maker's delta?
a. Buy 6.0 shares
b. Sell 6.0 shares
c. Buy 4.0 shares
d. Sell 4.0 shares

7.5. A market maker writes 100 at-the-money call option contracts and delta hedges dynamically by daily rebalancing of a long position in the underlying shares. The delta hedge is based on an implied volatility assumption of approximately 10% per annum. However, at the end of the month, the realized (actual subsequent) volatility of the stock was over 20%. However, the stock fluctuated both up and down roughly evenly. If borrowing occurs at the constant riskfree rate, and transaction costs are ignored, what is the net profit (loss) to the market maker at the end of the month?
a. Net loss due to gamma exposure
b. Net loss due to theta (time decay)
c. Approximately break-even due to the almost continuous delta hedge and roughly even up/down movements
d. Net gain due to the gamma exposure

Answers:

Last edited:

#### Addy

##### New Member
Hi.
i Cant see the ansers. Can you please put them here..

here is i y feedback :
7.1 B
7.3 B
7.4 B

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Azarfar78, the answers to these questions are protected for paid members (we need to eat and buy xmas toys for really cute nephews and other stuff, i hope you understand). Thanks, David

#### Addy

##### New Member
Hehe.. No worries...i was new to forums so had no idea. I just hope y answers are right.. !!
Yes i hope you get some thing good for them this X mas .

#### jdlcp

##### New Member
Subscriber
David,
I believe the delta position formula and questions are off by a factor of 100. In the equation by Carol Alexander for the position delta she states that a position in 100 call options with a delta of 0.6 results in a position delta of +60. This implies that I would delta hedge my position by selling 60 shares of stock. Each call option represents 100 shares of stock so the actual position delta is 100 call options * 0.6 delta * 100 shares / option = +6000 delta. So I would hedge the position by selling 6000 shares. A simpler example would be if I bought 1 call option with a 1.0 delta. This would be equivalent to owning 100 shares of stock or equivalent to a +100 delta exposure. So 1 call option with a 0.6 delta would have a delta exposure of +60 delta and 100 call options with a 0.6 delta would have a delta exposure of +6000 delta.

Jary

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @jdlcp

The keyword is "contract." True, we assume that an option contract gives the holder the right (but not the obligation) to buy or sell 100 underlying shares. I don't see where Carol confuses "contract" with "option." I used Carol to make the point, but I tend to follow Hull: if we say "one option" we mean an option to buy or sell one (1) share; if we say "one option contract" we mean an option to buy/sell one hundred shares (100). Above, the first two question do not mention "contracts," however 7.3 and 7.5 do refer to "contracts" and so they imply 100 shares per contract. Let me know if you still disagree, but I think my calculations are okay and consistent with Hull (i.e., Hull writes many questions that refer to only a single option and not a contract). Thanks!

#### caston

##### New Member
7.1. b (to reduce the delta of the stocks you need to sell some stock)
7.2. a (a put has a negative delta so a short put gives positive delta. therefore delta of the portfolio increases from (48 to 53) which requires a short position of 5 stocks to neutralise delta)
7.3. a (delta did not change therefore the is no need to rebalance)
7.4. d (delta of a put=N(d1)-1=0.6-1=-0.4; solving 100*0.6-60*0.6+50*(-0.4)=4 therefore short 4 stocks to have zero delta)
7.5. a (rC=theta+rS*delta+0.5*sigma^2*S^2*gamma: since sigma increases to 0.2 (0.04) from 0.1 (0.01) the gamma portion increases and since delta and gamma of a call are positive this increases the value of the option, however, since the position is a short position, the increase due to gamma increases the loss to the option writer i.e the short call increases in price)