Delta and time decay.xls
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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)
- 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
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:
- 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.
This equation is valid for American options when early exercise is not optimal.
- Delta hedging in practice:
- Gamma hedging in practice.
SOA 2009 Spring QA - #9
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:
Is the position delta-hedged? Yes since
The market-maker has to sell 96 stocks (B)
SOA Spring 2007 # 19
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