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Macaulay Duration
 denominator is price P(i)

Modified duration
 represents sensitivity of price change in the interest rate


Duration of a portfolio
 = present value weighted average of the durations of the individual components.
 i.e., Durations are weighted by price



Conditions for full immunization with one cash outflow
 1.
 2.
 3. There is one cash inflow before and one after the outflow

payoff and profit with purchased call option
payoff
 profit
 K is the strike price
 S_T is the spot price out T years
 C is the premium for the option

payoff and profit with purchased put option
 payoff
 profit
 K is the strike price
 S_T is the spot price out T years
 C is the premium for the option


Protective put
 Long position in the asset + purchased put
 Profit equivalent to purchased call

Insuring a short position
 Short position in the asset + purchased call
 Profit equivalent to purchased put

Covered call
 Long position in the asset + written call
 Profit equivalent to written put

Covered put
 Short position in the asset + written put
 Profit equivalent to written call

Putcall parity
 Call(K,T)  Put(K,T) = PV(F_{0,T}  K)
 Recall how this relates to synthetic forwards vs long forwards.
 Note PV(F) = S_{0}

Cash and carry
 Short forward + purchased stock (with borrowed money)
 Can be used for arbitrage and marketmaking

_{}Four ways to buy a stock
 Outright purchase:
 Receive stock at time 0
 Pay S_{0} at time 0
 Borrow to pay for stock:
 Receive stock at time 0
 Pay S_{0}e^{rT} at time T
 Prepaid forward:
 Receive stock at time T
 Pay at time 0
 = S_{0}  PV(dividends)
 Forward contract: