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Hedging Primbs, MS&E 345 1 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 2 Hedging Hedging is about the reduction of risk. We will consider dynamic hedging in which a portfolio is dynamically traded in order to reduce risk. Simply put, a portfolio is hedged against a certain risk if the portfolio value is not sensitive to that risk. Primbs, MS&E 345 3 The Basic Idea Given a risky portfolio or asset: c Form a new portfolio by purchasing or selling other assets in the market: P c 1S1 ... n S n Choose the amounts of the other assets, 1...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 4 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 5 Example: Hedging a call option with the underlying stock Underlying stock: dS Sdt Sdz Option: dc (ct ScS 1 S cSS ) dt ScS dz 2 2 2 We are long the option and would like to hedge our risk with the stock. Portfolio: P c S Portfolio change: dP dc dS (ct ScS 1 2 S 2 cSS )dt ScS dz Sdt Sdz 2 (ct ScS 1 2 S 2 cSS S )dt (ScS S )dz 2 Primbs, MS&E 345 6 Example: Hedging a call option with the underlying stock Underlying stock: dS Sdt Sdz Option: dc (ct ScS 1 S cSS ) dt ScS dz 2 2 2 We are long the option and would like to hedge our risk with the stock. Portfolio: P c S Portfolio change: dP (ct ScS 1 2 S 2 cSS S )dt (ScS S )dz 2 To eliminate risk, choose (ScS S ) 0 cS The portfolio is hedged over the next instantaneous dt. Primbs, MS&E 345 7 Black-Scholes: Provided we can trade continuously, we have formed a riskless portfolio: dP (ct ScS 1 2 S 2 cSS S )dt (ScS S )dz 2 with cS dP (ct 1 2 S 2 cSS ) dt 2 Since this is riskless, it must earn the risk free rate: dP (ct 1 2 S 2 cSS )dt rPdt r (c cs S )dt 2 ct rScs 1 2 S 2 cSS rc 2 Black-Scholes for the final time! Primbs, MS&E 345 8 Example: Hedging an interest rate derivative Short rate: dr adt bdz Asset 1: dB1 ( Bt1 aBr 1 b 2 Brr )dt bBr dz 1 2 1 1 Asset 2: dB 2 ( Bt2 aBr2 1 b 2 Brr )dt bBr2 dz 2 2 We are long B1 and would like to hedge with B2. Portfolio: P B1 B 2 Portfolio change: dP dB1 dB 2 ( Bt1 aBr 1 b 2 Brr )dt bBr dz 1 2 1 1 [(Bt2 aBr2 1 b 2 Brr )dt bBr2 dz] 2 2 Primbs, MS&E 345 9 Example: Hedging an interest rate derivative Short rate: dr adt bdz Asset 1: dB1 ( Bt1 aBr 1 b 2 Brr )dt bBr dz 1 2 1 1 Asset 2: dB 2 ( Bt2 aBr2 1 b 2 Brr )dt bBr2 dz 2 2 We are long B1 and would like to hedge with B2. dP ( Bt1 aBr 1 b 2 Brr )dt bBr dz 1 2 1 1 [(Bt2 aBr2 1 b 2 Brr )dt bBr2 dz] 2 2 Br1 To eliminate risk, choose: Br Br2 0 1 2 Br Then the portfolio: P B1 B 2 is hedged over the next instantaneous dt. Primbs, MS&E 345 10 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 11 Hedging assets with Poisson jumps: Underlying stock: dS Sdt Sdz (Y 1)Sd Option: dc (ct ScS 1 2 S 2cSS )dt ScS dz c(YSt , t ) c(St , t )d 2 Portfolio: P c S Portfolio change: dP (ct ScS 1 2 S 2cSS S )dt (ScS S )dz 2 [[c(YSt , t ) c( St , t )] (Y 1) S ]d Primbs, MS&E 345 12 Hedging assets with Poisson jumps: Portfolio change: dP (ct ScS 1 2 S 2cSS S )dt (ScS S )dz 2 [[c(YSt , t ) c( St , t )] (Y 1) S ]d If we let A [c(YSt , t ) c( St , t )] B (Y 1) S then the variance of the portfolio change is: var(dP) [ E[ A2 ] 2E[ AB ] 2 E[ B 2 ]]dt [var( A) 2 cov( A, B ) 2 var( B )]2 dt 2 (ScS S ) 2 dt Primbs, MS&E 345 13 Hedging assets with Poisson jumps: Minimizing over alpha gives: [E[ AB ] 2 dt cov( A, B ) 2 S 2 cS ] * [E[ B 2 ] 2 dt var( B ) 2 S 2 ] and sending dt to zero leads to: [E[ AB ] 2 S 2 cS ] * [ E[ B 2 ] 2 S 2 ] This hedge doesn’t really work so well. With big jumps in assets values, we can’t exactly say that we have eliminated risk. Primbs, MS&E 345 14 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 15 Complete versus Incomplete markets: Broadly speaking, a complete market is one in which you can replicate any desired payoff by trading assets in the market. A market is incomplete when this is not possible. Completeness relates to hedging in that if I consider a completely hedged portfolio: P c 1S1 ... n S n Then this will look like a bond: c 1S1 ... n S n B By rearranging, this is equivalent to replicating c: c B (1S1 ... n S n ) Primbs, MS&E 345 16 Incompleteness generally comes from two main sources: 1) There are not enough assets in the market to “span” the uncertainty. (An example would be standard stochastic volatility.) 2) Trading strategies are limited or not ideal 1) discrete trading 2) transaction costs, short selling constraints, etc. When we cannot replicate a payoff perfectly, we cannot argue for a unique price determined by a replicating portfolio. Any price will implicitly depend on risk preferences. This is why we saw the market price of risk emerging in certain problems. The market was incomplete. Primbs, MS&E 345 17 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 18 Example: Hedging a call option with the underlying stock Here is a quicker route to obtaining hedging parameters as long as asset prices move continuously: Portfolio: P c S The risk comes from S. So let’s set the partial derivative of the portfolio with respect to S equal to zero. PS cS 0 cS Primbs, MS&E 345 19 Example: Hedging an interest rate derivative 1 We have bonds B ( r , t ) and B 2 (r , t ) where r is the short rate Form a hedged portfolio: P(r , t ) B1 (r , t ) B 2 (r , t ) and r is where the risk comes from: Set the partial derivative of the portfolio with respect to r equal to zero: Br1 Pr 0 Br Br2 0 1 2 Br Primbs, MS&E 345 20 More generally, we can look at the sensitivity of a portfolio over time t through a Taylor expansion: We have a derivative which is a function of a factor. f (S , t ) A Taylor Expansion: f f t t f S S 1 f tt (t ) 2 1 f SS (S ) 2 f tS tS ... 2 2 Approach: eliminate as many “random” terms as possible Primbs, MS&E 345 21 Delta hedged call option: We have a derivative which is a function of a factor: c( S , t ) A Taylor Expansion: c ct t cS S 1 ctt (t ) 2 1 cSS (S ) 2 ctS tS ... 2 2 Now consider the hedged portfolio: P c S P c S ct t (cS )S 1 ctt (t ) 2 1 cSS (S ) 2 ctS tS ... 2 2 cS Eliminates the risk of S. Still left with higher order risk... Primbs, MS&E 345 22 Delta hedge in pictures: Current Price: S = 10, Risk Free Rate: r = 0.05 Delta Hedge 6 call delta 4 2 Call Option Price 0 -2 Slope = -4 Position in Bonds -6 0 5 10 15 Stock Price Hedged Call Option Parameters: (K=10, T=0.2, =0.3) Primbs, MS&E 345 23 A delta-gamma hedge: We have a derivative which is a function of a factor: c( S , t ) We will hedge the stock and another derivative on it: S and c ( 2) (S , t ) A Taylor Expansion: c ct t cS S 1 ctt (t ) 2 1 cSS (S ) 2 ctS tS ... 2 2 c ( 2 ) ct( 2 ) t cS2 ) S 1 ctt2 ) (t ) 2 1 cSS) (S ) 2 ctS2 ) tS ... ( 2 ( 2 (2 ( The hedged portfolio: P c 1 S 2 c ( 2 ) P c 1S 2 c ( 2 ) (ct 2ct( 2) )t (cS 1 2 cS2) )S 1 (ctt 2ctt2) )(t ) 2 ( 2 ( 1 (cSS 2cSS) )(S ) 2 (ctS 2ctS2) )tS ... 2 (2 ( Primbs, MS&E 345 24 A delta-gamma hedge: We have a derivative which is a function of a factor: c( S , t ) We will hedge the stock and another derivative on it: S and c ( 2) (S , t ) P (ct 2ct( 2) )t (cS 1 2 cS2) )S 1 (ctt 2 ctt2) )(t ) 2 ( 2 ( 1 (cSS 2 cSS) )(S ) 2 (ctS 2ctS2) )tS ... 2 (2 ( Eliminate S and (S)2 terms: cSS ( 2 ) cS 1 c ( 2) 2 S 0 1 ( 2 ) cS cS cSS cSS 2cSS) 0 (2 cSS 2 ( 2) cSS Primbs, MS&E 345 25 Delta-Gamma hedge in pictures: Current Price: S = 10, Risk Free Rate: r = 0.05 Delta vs. Delta-Gamma Hedge 6 call delta delta-gamma 4 2 Call Option Price 0 -2 -4 -6 0 5 10 15 Stock Price Hedged Call Option Parameters: (K=10, T=0.2, =0.3) 2nd Call Option Parameters: (K=8, T=0.4, =0.25) Primbs, MS&E 345 26 Delta-Gamma hedge in pictures: Current Price: S = 10, Risk Free Rate: r = 0.05 Delta vs. Delta-Gamma Hedge 3 call delta 2.5 delta-gamma 2 Call Option Price 1.5 1 0.5 0 -0.5 -1 8 8.5 9 9.5 10 10.5 11 11.5 12 Stock Price Hedged Call Option Parameters: (K=10, T=0.2, =0.3) 2nd Call Option Parameters: (K=8, T=0.4, =0.25) Primbs, MS&E 345 27 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 28 The Greeks A call option depends on many parameters: c( S , , r , t ) A Taylor Expansion: c ct t cS S c cr r 1 cSS (S ) 2 ... 2 theta delta vega rho gamma ct cS c cr cSS Primbs, MS&E 345 29 European Call Option Price 2.5 2 1.5 Price 1 0.5 0 8 8.5 9 9.5 10 10.5 11 11.5 12 S (S=10, K=10, T=0.2, r=0.05, =0.2) Primbs, MS&E 345 30 European Call Option Delta 1 0.9 0.8 0.7 0.6 Delta 0.5 0.4 0.3 0.2 0.1 0 8 8.5 9 9.5 10 10.5 11 11.5 12 S (S=10, K=10, T=0.2, r=0.05, =0.2) Primbs, MS&E 345 31 European Call Option Gamma 0.5 0.45 0.4 0.35 0.3 Gamma 0.25 0.2 0.15 0.1 0.05 0 8 8.5 9 9.5 10 10.5 11 11.5 12 S (S=10, K=10, T=0.2, r=0.05, =0.2) Primbs, MS&E 345 32 European Call Option Theta 0 -0.2 -0.4 -0.6 Theta -0.8 -1 -1.2 -1.4 8 8.5 9 9.5 10 10.5 11 11.5 12 S (S=10, K=10, T=0.2, r=0.05, =0.2) Primbs, MS&E 345 33 European Call Option Rho 2 1.8 1.6 1.4 1.2 Rho 1 0.8 0.6 0.4 0.2 0 8 8.5 9 9.5 10 10.5 11 11.5 12 S (S=10, K=10, T=0.2, r=0.05, =0.2) Primbs, MS&E 345 34 European Call Option Vega 1.8 1.6 1.4 1.2 1 Vega 0.8 0.6 0.4 0.2 0 8 8.5 9 9.5 10 10.5 11 11.5 12 S (S=10, K=10, T=0.2, r=0.05, =0.2) Primbs, MS&E 345 35 Just as we set up Delta and Delta-Gamma hedges, we can hedge against changes in other parameters. However, normally it is difficult and expensive to try to hedge all these parameters. Instead, you might delta hedge and monitor the other Greeks, only taking action when needed. In general, the Greeks provide an important summary of the sensitivity of a portfolio to underlying uncertainties. Primbs, MS&E 345 36 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 37 Options with discontinuous payoffs tend to be very difficult to hedge, especially close to the discontinuities. The problem is that the holdings in the hedged portfolio are extremely sensitive to small changes in the underlying asset. A digital option provides a good example of this... Primbs, MS&E 345 38 Digital Option Price 1 0.9 0.8 T=0.01 0.7 T=0.05 0.6 Option Price 0.5 0.4 0.3 0.2 0.1 0 7 8 9 10 11 12 13 S (S=10, K=10, r=0.05, =0.25) Primbs, MS&E 345 39 Digital Option Delta 1.6 1.4 T=0.01 1.2 1 T=0.05 Delta 0.8 0.6 0.4 0.2 0 7 8 9 10 11 12 13 S (S=10, K=10, r=0.05, =0.25) Primbs, MS&E 345 40 Digital Option Gamma 4 3 T=0.01 2 1 T=0.05 Gamma 0 -1 -2 -3 -4 7 8 9 10 11 12 13 S (S=10, K=10, r=0.05, =0.25) Primbs, MS&E 345 41 Primbs, MS&E 345 42

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posted: | 11/4/2011 |

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