A market-maker sells assets or contracts to buyers and buys them from sellers. He is an intermediary between the buyers and sellers. A market-maker's function is in contrast toproprietary trading, which is "trading to express an investment strategy" (McDonald, p. 414).
A delta-hedged position is a position designed to earn the risk-free rate of interest and is used to offset the risk of an option position.
The delta-gamma approximation is used to estimate option price movements if the underlying stock price changes.
The delta-gamma approximation for call options can be expressed via the following formula:
C(St+h) = C(St) + є∆(St) + (1/2)є2Γ(St)
For a put option, the same formula holds, but delta is now negative - so the put price will decrease if the stock price increases.
Meaning of variables:
St = stock price at time t.
St+h = stock price at time t+h.
C = call option price.
є = stock price change from time t to time t + h.
∆ = option delta.
Γ = option gamma.
Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 13, pp. 413-425.
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