MFE 4 Delta and Gamma Hedging

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Market Making – Delta Hedging

Market Making – Delta and Gamma Hedging


- Market making can control risk by computing the option delta and taking an offsetting position in shares.


Risk without hedging

(For a writer of a call option)

- Delta is negative → risk for the writer when the stock price rise

- If stock price does not change the writer benefits from time decay (the effect of the increase of the option value by the shorter time to expiration) ( effect)


Profit calculation

- Our borrowing capacity is equal to the market value of our securities. We pay interest on the borrowed amount


Net cash flow=change in borrowing capacity-cash used to purchase additional shares – interest=

Overnight gain on shares


Overnight gain on the option





A hedged portfolio that does not require additional cash investments to remain hedged is self-financing (when a stock moves one standard deviation).


Delta - gamma approximation

- Delta is the linear approximation to change in the value of the stock. The approximation is good for small changes; is less accurate for larger changes, since delta changes when the price of the stock changes

- Gamma gives the quadratic approximation.

-         The new predicted call price is not perfect since gamma change when the stock price changes.

Effect of time

- The call price decreases as time passes (time decay)

- Formula taking into account the time decay:

Market-Maker’s profit

- When the stock price changes over a period of time h and the position is delta-hedged, the profit is:


- For a call:

- Gamma and interest work against the market-maker’s profit

- Theta works in favor of market-maker’s profit

- It’s the magnitude (not the direction) of the price change that determines the profit. There’s profit when the price change is small, and loss when the change is large.



- Black and Scholes argued that money invested in a hedged position should earn risk-free rate since the income stream is risk free.

Text Box: Assumptions:
- Asset pays no div
- Option pays no div
- r and  are constant
- Stock moves   over a small time interval

This equation is valid for American options when early exercise is not optimal.


- Delta hedging in practice:

  1. To be gamma-neutral the market maker must buy or sell options to offset gamma of the position
  2. Static option replication: uses options to hedge options (using put-call parity)
  3. Market-makers can but out-of-the money options (but is almost impossible to get the required amount on the market)
  4. Create a new product by selling the hedging error.

- Gamma hedging in practice.

  1. We can’t gamma hedge by buying stock but we can by buying options.
  2. Market-makers are reluctant to buy options since they have to pay bid-ask spread and because the aggregate market won’t have enough options (the market cannot have a gamma neutral position).





SOA 2009 Spring QA - #9


B-S framework

MM delta hedges and sells a call option.

r=10%;;; h=1(day); Profit (h=1)=0 T=3 months K=50



Profit=0 when S moves 1


From the Normal dist:



We can use the data from this problem to graph some of the Greeks. I used an excel worksheet (you can find it at the end of the page).


SOA Spring 2007 # 10



You sold 1000xCall(I)

Delta and Gamma hedge your position by buying or selling options II.




To gamma-hedge the position:

X= 873


Is the position delta-hedged? Yes since

The market-maker has to sell 96 stocks (B)


SOA Spring 2007 # 19



B-S framework






Delta-gamma approximation



CAS Spring 2007 # 32


MM sold 100C(100 calls), delta-hedged by purchasing the stock

r=9% T=12m N()=0.5793 N()=0.5





MM bought 10,000x0.54=5,400 to delta-hedge the position


Profit= -100(56.08)+5,400(1)=-208 (C)



CAS Fall 2007 # 24


AA owns 100 P(40) and 5 S

AA can write P(35)

How do we delta and gamma neutralize the position?







Delta position (buying X put options and Y shares) is now:



We have to sell 5 shares