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Lecture04[1]

# Lecture04[1] - STAT 2820 Chapter 4 Option Strategies by K.C...

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STAT 2820 Chapter 4 Option Strategies by K.C. Cheung 4.1 Insuring a Long Position: Floors 4.1.1 Suppose that we invest in a certain asset at time 0, for instance, one share of stock ABC, and plan to sell it in T years. If the stock price drops after T years, we can only sell for less. To reduce (hedge) this risk, put option on this stock can help. If we buy put option on one share of stock ABC with strike price K and T years to maturity, then the total payoff after T years is Payoff = S T + ( K - S T ) + = ( K, if S T < K S T , if S T K 4.1.2 The downside risk is eliminated: if the stock price after T years drops below K , we could still obtain \$ K . Of course, this protection from downside risk comes with a price: we need to pay for the put options. If we let p be the price of put option on 1 share of stock ABC, then at time 0 we have to pay p in total. If r is the annual risk-free interest rate compounded continuously, then the overall P/L is P/L = Payoff - FV(initial investment) = S T + ( K - S T ) + - ( S 0 + p ) e rT 4.1.3 Remark: (a) We may rewrite the payoff as Payoff = S T + ( K - S T ) + = K + ( S T - K ) + so the combined position (stock + put option) has the same payoff as the following strategy: ( Deposit \$ Ke - rT in a bank account at time 0 for T years Buy call options on one share of stock ABC with strike price K and T years to maturity So we obtain a general principle: Underlying asset + Put option = Bond + Call option

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STAT 2820 Chapter 4 2 4.2 Insuring a Short Position: Caps 4.2.1 Suppose that we short sell a certain asset at time 0, for instance, one share of stock ABC. If stock price rise we experience a loss. To insure the short position, we can buy call option on this stock to protect against a higher price of repurchasing the stock. If we buy call option on one share of stock ABC with strike price K and T years to maturity, then the total payoff after T years is Payoff = - S T + ( S T - K ) + = ( - S T , if S T < K - K, if S T K 4.2.2 The upside risk is eliminated: if the stock price after T years rises above K , we could still use \$ K to close out the short position, instead of using a higher price \$ S T . This protection from upside risk comes with a price: we need to pay for the call options. If we let c be the price of a call option on 1 share of stock ABC, then at time 0 we have to pay c in total. If r is the annual risk-free interest rate compounded continuously, then the overall P/L is P/L = Payoff - FV(initial investment) = - S T + ( S T - K ) + - ( c - S 0 ) e rT 4.2.3 Remarks: (a) The term “ - S 0 ” in the above expression means that when short selling one share of stock at time 0, we borrow one share from our broker then sell them to the market at the spot price S 0 . So we receive S 0 in total. (b) We may rewrite the payoff as Payoff = - S T + ( S T - K ) + = - K + ( K - S T ) + so the combined position (short stock + call option) has the same payoff as the following strategy: ( Borrow \$ Ke - rT for T years Buy put option on one share of stock ABC with strike price K and T years to maturity So we obtain a principle: ( - Underlying asset) + Call option = ( - Bond) + Put option which is the same as the one at the end of page 1.
STAT 2820 Chapter 4 3 4.3 Covered Calls and Covered Puts 4.3.1

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