Delta and time decay.xls Size : 55 Kb Type : xls 

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 capacitycash used to purchase additional shares – interest=_{}




A hedged portfolio that does not require additional cash investments to remain hedged is selffinancing (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:
_{}
MarketMaker’s profit
 When the stock price changes _{} over a period of time h and the position is deltahedged, the profit is:
_{}
 For a call:
 Gamma and interest work against the marketmaker’s profit
 Theta works in favor of marketmaker’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.
Implications
 Black and Scholes argued that money invested in a hedged position should earn riskfree rate since the income stream is risk free.
_{}
This equation is valid for American options when early exercise is not optimal.
 Delta hedging in practice:
 Gamma hedging in practice.
Exercises
SOA 2009 Spring QA  #9
Info
BS framework
MM delta hedges and sells a call option.
r=10%;_{};_{}; h=1(day); Profit (h=1)=0 T=3 months K=50
_{}
Sol
Profit=0 when S moves 1 _{}
_{}
0.6179=(1)(N(_{})
From the Normal dist:
_{}
_{}
_{}
_{} (B)
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
Info
You sold 1000xCall(I)
Delta and Gamma hedge your position by buying or selling options II.
Sol
To gammahedge the position:
_{}
X= 873
Is the position deltahedged? Yes since
_{}
The marketmaker has to sell 96 stocks (B)
SOA Spring 2007 # 19
Info:
BS framework
_{} _{} P()=4.00 _{}_{}
P(31.50)=?
Sol
Deltagamma approximation
_{}
_{}
_{} (D)
CAS Spring 2007 # 32
Info
MM sold 100C(100 calls), deltahedged by purchasing the stock
_{}r=9% _{}T=12m N(_{})=0.5793 N(_{})=0.5
_{} _{}
Profit?
Sol
t_{}
_{}
MM bought 10,000x0.54=5,400 to deltahedge the position
Profit= 100(56.08)+5,400(1)=208 (C)
CAS Fall 2007 # 24
Info
AA owns 100 P(40) and 5 S
AA can write P(35)
How do we delta and gamma neutralize the position?
Sol
+100(0.25)X(0.50)=0
X=50
Delta position (buying X put options and Y shares) is now:
100(0.05)+5(1)50(0.10)+Y(1)=0
Y=5
We have to sell 5 shares
(E)