13-Hedging4

# 13-Hedging4 - Hedging Primbs MS&E 345 1 Basic Idea Hedging...

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Primbs, MS&E 345 1 Hedging

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Primbs, MS&E 345 2 Basic Idea Hedging under Ito Processes Hedging Poisson Jumps Complete vs. Incomplete markets Delta and Delta-Gamma hedges Greeks and Taylor expansions Complications with Hedging
Primbs, MS&E 345 3 Hedging Hedging is about the reduction of risk. Simply put, a portfolio is hedged against a certain risk if the portfolio value is not sensitive to that risk. We will consider dynamic hedging in which a portfolio is dynamically traded in order to reduce risk.

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Primbs, MS&E 345 4 The Basic Idea c Given a risky portfolio or asset: Form a new portfolio by purchasing or selling other assets in the market: n n S S c P α + + + = ... 1 1 Choose the amounts of the other assets, 1 ...α 2 , in order to “eliminate the risk” in the portfolio. Ito’s lemma will tell us how much risk the portfolio has over the next instantaneous dt .
Primbs, MS&E 345 5 Basic Idea Hedging under Ito Processes Hedging Poisson Jumps Complete vs. Incomplete markets Delta and Delta-Gamma hedges Greeks and Taylor expansions Complications with Hedging

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Primbs, MS&E 345 6 Example: Hedging a call option with the underlying stock Underlying stock: Sdz Sdt dS σ μ + = Option: dz Sc dt c S Sc c dc S SS S t + + + = ) ( 2 2 2 1 We are long the option and would like to hedge our risk with the stock. [ ] [ ] Sdz Sdt dz Sc dt c S Sc c S SS S t α + + + + + = ) ( 2 2 2 1 dz S Sc dt S c S Sc c S SS S t ) ( ) ( 2 2 2 1 ασ αμ + + + + + = S c P + = Portfolio: dS dc dP + = Portfolio change:
Primbs, MS&E 345 7 Example: Hedging a call option with the underlying stock Underlying stock: Sdz Sdt dS σ μ + = Option: dz Sc dt c S Sc c dc S SS S t + + + = ) ( 2 2 2 1 We are long the option and would like to hedge our risk with the stock. S c P α + = dz S Sc dt S c S Sc c dP S SS S t ) ( ) ( 2 2 2 1 ασ αμ + + + + + = To eliminate risk, choose 0 ) ( = + S Sc S S c - = The portfolio is hedged over the next instantaneous dt . Portfolio: Portfolio change:

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Primbs, MS&E 345 8 Black-Scholes: Provided we can trade continuously, we have formed a riskless portfolio: dz S Sc dt S c S Sc c dP S SS S t ) ( ) ( 2 2 2 1 ασ σ αμ μ + + + + + = S c - = α with dt c S c dP SS t ) ( 2 2 2 1 + = Since this is riskless, it must earn the risk free rate: dt S c c r rPdt dt c S c dP s SS t ) ( ) ( 2 2 2 1 - = = + = rc c S rSc c SS s t = + + 2 2 2 1 Black-Scholes for the final time!
Primbs, MS&E 345 9 Example: Hedging an interest rate derivative Short rate: bdz adt dr + = Asset 1: dz bB dt B b aB B dB r rr r t 1 1 2 2 1 1 1 1 ) ( + + + = We are long B 1 and would like to hedge with B 2 . 2 1 dB dB dP α + = 2 1 B B P + = Portfolio: Portfolio change: Asset 2: dz bB dt B b aB B dB r rr r t 2 2 2 2 1 2 2 2 ) ( + + + = ] ) [( ) ( 2 2 2 2 1 2 2 1 1 2 2 1 1 1 dz bB dt B b aB B dz bB dt B b aB B r rr r t r rr r t + + + + + + + =

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Primbs, MS&E 345 10 Example: Hedging an interest rate derivative Short rate: bdz adt dr + = Asset 1: dz bB dt B b aB B dB r rr r t 1 1 2 2 1 1 1 1 ) ( + + + = We are long B 1 and would like to hedge with B 2 .
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## This note was uploaded on 06/17/2010 for the course MS&E 345 taught by Professor Jimprimbs during the Winter '10 term at Stanford.

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13-Hedging4 - Hedging Primbs MS&E 345 1 Basic Idea Hedging...

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