Derivatives Inside Black Scholes by ulf16328

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									Derivatives
Inside Black Scholes


Professor André Farber
Solvay Business School
Université Libre de Bruxelles
                      Lessons from the binomial model

•     Need to model the stock price evolution
•     Binomial model:
      – discrete time, discrete variable
      – volatility captured by u and d
•     Markov process
           • Future movements in stock price depend only on where we are,
              not the history of how we got where we are
           • Consistent with weak-form market efficiency
•     Risk neutral valuation
      – The value of a derivative is its expected payoff in a risk-neutral world
          discounted at the risk-free rate
                          p  f u  (1  p )  f d          e rt  d
                     f                            with p 
                                   e rt                     ud

July 9, 2010                      Derivatives 08 Inside Black Scholes     |2
                  Black Scholes differential equation: assumptions

• S follows the geometric Brownian motion: dS = µS dt +  S dz
   – Volatility  constant
   – No dividend payment (until maturity of option)
   – Continuous market
   – Perfect capital markets
   – Short sales possible
   – No transaction costs, no taxes
   – Constant interest rate

• Consider a derivative asset with value f(S,t)
• By how much will f change if S changes by dS?
• Answer: Ito’s lemna


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                        Ito’s lemna

• Rule to calculate the differential of a variable that is a function of a
  stochastic process and of time:
• Let G(x,t) be a continuous and differentiable function
• where x follows a stochastic process dx =a(x,t) dt + b(x,t) dz

• Ito’s lemna. G follows a stochastic process:

                      G      G 1  2 G 2         G
               dG  (    a      2  b )  dt      b  dz
                      x      t 2 x              x


                                Drift                               Volatility


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                    Ito’s lemna: some intuition

• If x is a real variable, applying Taylor:
                    G      G      1 2G 2 2G                1 2G 2
               G     x     t         x       x  t         t ..
                    x      t      2 x 2      xt           2 t 2
                                                                        An approximation
                                      G      G                        dx², dt², dx dt negligeables
• In ordinary calculus:        dG       dx     dt
                                      x      t


                                            G      G      1  ²G
• In stochastic calculus:         dG          dx     dt         dx²
                                            x      t      2 x ²


• Because, if x follows an Ito process, dx² = b² dt you have to keep it



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                            Lognormal property of stock prices

• Suppose:                             dS=  S dt +  S dz
• Using Ito’s lemna:                   d ln(S) = ( - 0.5 ²) dt +  dz

• Consequence:
                                                                              ln(ST) – ln(S0) = ln(ST/S0)
                                           ²                                 Continuously compounded
      ln( S T )  ln( S 0 ) ~ N [(            )T ,  T ]
                                            2                                 return between 0 and T




                                       ²                                     ln(ST) is normally distributed
     ln( S T ) ~ N [ln(S 0 )  (          )T ,  T ]                        so that ST has a lognormal
                                       2
                                                                              distribution




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                    Derivation of PDE (partial differential equation)

• Back to the valuation of a derivative f(S,t):
• If S changes by dS, using Ito’s lemna:
                           f         f 1  2 f                   f
                     df  (    S      2   2  S 2 )  dt        S  dz
                           S         t 2 S                      S

• Note: same Wiener process for S and f
•  possibility to create an instantaneously riskless position by combining
  the underlying asset and the derivative
• Composition of riskless portfolio
        • -1              sell (short) one derivative
        • fS = ∂f /∂S     buy (long) DELTA shares

• Value of portfolio: V = - f + fS S


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                    Here comes the PDE!

• Using Ito’s lemna

                                 f 1  2 f 2 2
                        dV  (             S ) dt
                                 t 2 S  2



• This is a riskless portfolio!!!
• Its expected return should be equal to the risk free interest rate:
                                  dV = r V dt
• This leads to:

                         f      f 1  2 f 2 2
                             rS            S  rf
                         t      S 2 S  2




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                          Understanding the PDE

• Assume we are in a risk neutral world
                                                                               Expected change
                          f      f 1  f 2 2  2
                                                                               of the value of
                              rS           S  rf
                          t      S 2 S 2                                    derivative
                                                                               security
        Change of the
        value with
        respect to time     Change of the value                          Change of the
                            with respect to the                          value with
                            price of the                                 respect to
                            underlying asset                             volatility




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                    Black Scholes’ PDE and the binomial model

• We have:
       • BS PDE :        f’t + rS f’S + ½ ² f”SS = r f
       • Binomial model: p fu + (1-p) fd = ert
• Use Taylor approximation:
       • fu = f + (u-1) S f’S + ½ (u–1)² S² f”SS + f’t t
       • fd = f + (d-1) S f’S + ½ (d–1)² S² f”SS + f’t t
       • u = 1 + √t + ½ ²t
       • d = 1 – √t + ½ ²t
       • ert = 1 + rt
• Substituting in the binomial option pricing model leads to the differential
  equation derived by Black and Scholes



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                             And now, the Black Scholes formulas

• Closed form solutions for European options on non dividend paying stocks
  assuming:
       • Constant volatility
       • Constant risk-free interest rate

               Call option:
                                                    C  S 0  N (d1 )  Ke  rT  N (d 2 )

               Put option:                           P  Ke  rT N (  d 2 )  S 0  N (  d 1 )


                             ln( S 0 / Ke  rT )
                      d1                           0.5 T                    d 2  d1   T
                                   T
           N(x) = cumulative probability distribution function for a standardized normal variable

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                        Understanding Black Scholes

• Remember the call valuation formula derived in the binomial model:
                               C =  S0 – B
• Compare with the BS formula for a call option:
                           C  S 0  N (d1 )  Ke  rT  N (d 2 )

• Same structure:
• N(d1) is the delta of the option
        • # shares to buy to create a synthetic call
        • The rate of change of the option price with respect to the price of
          the underlying asset (the partial derivative CS)
• K e-rT N(d2) is the amount to borrow to create a synthetic call
          N(d2) = risk-neutral probability that the option will be exercised at
          maturity
July 9, 2010                         Derivatives 08 Inside Black Scholes      |12
                              A closer look at d1 and d2



                 ln( S 0 / Ke  rT )                              d 2  d1   T
          d1                           0.5 T
                       T
                 2 elements determine d1 and d2

                                                      A measure of the “moneyness” of the
                 S0 /   Ke-rt                         option.
                                                      The distance between the exercise price
                                                      and the stock price



                T
                                                        Time adjusted volatility.
                                                        The volatility of the return on
                                                        the underlying asset between
                                                        now and maturity.


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                            Example

           Stock price S0 = 100
           Exercise price K = 100 (at the money option)
           Maturity T = 1 year
           Interest rate (continuous) r = 5%
           Volatility  = 0.15

           ln(S0 / K e-rT) = ln(1.0513) = 0.05
           √T = 0.15
           d1 = (0.05)/(0.15) + (0.5)(0.15) = 0.4083
               N(d1) = 0.6585
                                                           European call :
           d2 = 0.4083 – 0.15 = 0.2583
                                                           100  0.6585 - 100  0.95123  0.6019
               N(d2) = 0.6019                              = 8.60


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               Relationship between call value and spot price




                             For call option,
                             time value > 0




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                      European put option

• European call option: C = S0 N(d1) – PV(K) N(d2)

                                  Delta of call option          Risk-neutral probability of exercising
                                                                the option = Proba(ST>X)
• Put-Call Parity: P = C – S0 + PV(K)

• European put option: P = S0 [N(d1)-1] + PV(K)[1-N(d2)]
                                 Delta of put option               Risk-neutral probability of exercising
                                                                   the option = Proba(ST<X)

•                           P = - S0 N(-d1) +PV(K) N(-d2)

          (Remember: N(x) – 1 = N(-x)




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                            Example


 •    Stock price S0 = 100
 •    Exercise price K = 100 (at the money option)
 •    Maturity T = 1 year
 •    Interest rate (continuous) r = 5%
 •    Volatility  = 0.15

           N(-d1) = 1 – N(d1) = 1 – 0.6585 = 0.3415

               N(-d2) = 1 – N(d2) = 1 – 0.6019 = 0.3981


               European put option
               - 100 x 0.3415 + 95.123 x 0.3981 = 3.72


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               Relationship between Put Value and Spot Price




                                 For put option, time
                                 value >0 or <0




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                    Dividend paying stock

• If the underlying asset pays a dividend, substract the present value of future
  dividends from the stock price before using Black Scholes.

• If stock pays a continuous dividend yield q, replace stock price S0 by S0e-qT.
   – Three important applications:
        • Options on stock indices (q is the continuous dividend yield)
        • Currency options (q is the foreign risk-free interest rate)
        • Options on futures contracts (q is the risk-free interest rate)




July 9, 2010                    Derivatives 08 Inside Black Scholes     |19
                            Dividend paying stock: binomial model

      t = 1 u = 1.25, d = 0.80
      r = 5% q = 3%                       uS0 eqt with dividends reinvested
      Derivative: Call K = 100            128.81
                                          uS0            ex dividend                       fu
                                          125                                              25
                 S0
                                      dS0 eqt with dividends reinvested
                 100
                                      82.44
                                                                                          fd
                                     dS0             ex dividend                          0
         f =  S0 + M                80
                                                               f = [ p fu + (1-p) fd] e-rt = 11.64
          Replicating portfolio:
            uS0 eqt + M ert = fu
                128.81 + M 1.0513 = 25                        p = (e(r-q)t – d) / (u – d) = 0.489
                dS0 eqt + M ert = fd
                82.44 + M 1.0513 = 0
                                                               = (fu – fd) / (u – d )S0eqt = 0.539


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                            Black Scholes Merton with constant dividend
                            yield
      The partial differential               f             f 1  2 f 2 2
      equation:                                  (r  q) S            S  rf
      (See Hull 5th ed. Appendix 13A)
                                             t             S 2 S  2

                                                                 Expected growth rate of stock


         Call option
                                            C  S 0 e  qT  N (d 1 )  Ke  rT  N (d 2 )

          Put option                        P  Ke  rT N ( d 2 )  S 0 e  qT  N (d1 )



                      ln( S 0 e  qT / Ke  rT )                              d 2  d1   T
               d1                                  0.5 T
                               T

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                                Options on stock indices

• Option contracts are on a multiple times the index ($100 in US)
• The most popular underlying US indices are
       –       the Dow Jones Industrial (European) DJX
       –       the S&P 100 (American) OEX
       –       the S&P 500 (European) SPX
• Contracts are settled in cash

•    Example: July 2, 2002 S&P 500 = 968.65
•    SPX September
•    Strike       Call     Put
•    900           -       15.60
     1,005        30       53.50
     1,025        21.40 59.80
•    Source: Wall Street Journal

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                   Options on futures

• A call option on a futures contract.
• Payoff at maturity:
       • A long position on the underlying futures contract
       • A cash amount = Futures price – Strike price

• Example: a 1-month call option on a 3-month gold futures contract
      • Strike price = $310 / troy ounce
      • Size of contract = 100 troy ounces
      • Suppose futures price = $320 at options maturity
      • Exercise call option
               » Long one futures
               » + 100 (320 – 310) = $1,000 in cash


July 9, 2010                   Derivatives 08 Inside Black Scholes    |23
                           Option on futures: binomial model

                                                uF0 → fu

           Futures price F0
                                                dF0 →fd

               Replicating portfolio:  futures + cash                                 fu  fd
                                                                                 
                                                                                      uF0  dF0
                (uF0 – F0) + M ert = fu
                                                                                      pf u  (1  p) f d
                (dF0 – F0) + M   ert   = fd                                   f 
                                                                                             e rt

                      f=M                                                                1 d
                                                                                  p
                                                                                         ud


July 9, 2010                              Derivatives 08 Inside Black Scholes                      |24
                       Options on futures versus options on dividend
                       paying stock
        Compare now the formulas obtained for the option on futures and for an
        option on a dividend paying stock:

          Futures                                           Dividend paying stock


            pf u  (1  p) f d                                 pf u  (1  p) f d
        f                                                 f 
                   e rt                                              e rt

          1 d                                                 e ( r  q ) t  d
       p                                                   p
          ud                                                        ud

               Futures prices behave in the same way as a stock paying a
               continuous dividend yield at the risk-free interest rate r

July 9, 2010                        Derivatives 08 Inside Black Scholes             |25
                            Black’s model


         Assumption: futures price has lognormal distribution



Ce         rT
                  [ F0 N (d1 )  KN (d 2 )]
                                                       P  e  rT [ KN (d 2 )  F0 N (d1 )]


            F0                                                 F0
               )
               ln(                                                )
                                                                ln(
     d1    X  0.5 T                                  d2    X  0.5 T  d   T
                                                              T
                                                                             1
           T




July 9, 2010                            Derivatives 08 Inside Black Scholes        |26
               Implied volatility – Call option




July 9, 2010             Derivatives 08 Inside Black Scholes   |27
               Implied volatility – Put option




July 9, 2010             Derivatives 08 Inside Black Scholes   |28
                                      Smile
               SPX Option on S&P 500            Spot index            968.25
               September 2002 Contract          DivYield                 2%
                                                IntRate               1.86%

               July 2, 2002
               Maturity                   90 days

               Strike          Call                                            Put
                                      OpenInt          Price          ImpVol         OpenInt       Price   ImpVol

                         700                                                           3801         1.5    34.19%
                         750                                                           1581         2.9    31.59%
                         800                                                          31675           4    26.84%
                         900                                                          21723        15.6    22.17%
                         925                                                           7799          19    19.54%
                         950                                                          17419          28    19.16%
                         975                                                          16603          33    15.32%
                         980            3599              42          24.89%           4994        40.3    17.68%
                         990            3228              40          26.04%           3193          41    14.86%
                         995           11806            34.5          24.17%          23345          46    15.84%
                        1005            5404              30          23.73%           5209        53.5    16.29%
                        1025            9232            21.4          22.47%          15242        59.8     9.95%
                        1040            2286            15.1          20.97%
                        1050           11145            13.1          21.07%
                        1075            8726             7.5          19.97%
                        1100           23170             4.6          19.82%
                        1125            7556             2.4          19.16%
                        1150           18173             1.6          19.67%
                        1200            7513            0.45          19.33%
July 9, 2010                                                 Derivatives 08 Inside Black Scholes                    |29

								
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