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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 ]  2E[ 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 tS  ...
                                   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 tS  ...
                                  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 tS  ...
                                            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 tS  ...
                                   2               2

         c ( 2 )  ct( 2 ) t  cS2 ) S  1 ctt2 ) (t ) 2  1 cSS) (S ) 2  ctS2 ) tS  ...
                                  (
                                            2
                                               (
                                                               2
                                                                  (2             (



    The hedged portfolio:         P  c  1 S   2 c ( 2 )
    P  c  1S   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) )tS  ...
                                  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) )tS  ...
                            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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