Behavioral Black-Scholes

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					        Review on Option Pricing
Behavioral Model For Option Price




      Behavioral Black-Scholes

                      Nilüfer Calıskan
                              . .

                     Swiss Banking Institute
                       Zurich University


                  April/2008
         Vorlesung Behavioral Finance




                 Nilüfer Calıskan
                         . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                          Basics
              Review on Option Pricing    Pricing
      Behavioral Model For Option Price   Important Assumptions
                                          Black-Scholes Model




Options are one of the specific types of Derivative Assets
(Contracts about possible Future Transactions)
Options specicifically gives the right to buy or sell the
underlying asset at given price at a given time period.
European options has a fixed and certain time for
exercising the option to make the possible transaction.
Call options give the right to buy the underlying while the
put options give the right to sell the underlying at a fixed
price.




                       Nilüfer Calıskan
                               . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                          Basics
              Review on Option Pricing    Pricing
      Behavioral Model For Option Price   Important Assumptions
                                          Black-Scholes Model




Under Arbitrage-Free Complete Markets, one can form
unique risk neutral probabilities for each possible payoff of
the asset depending on the possible states.
Risk-Neutral Probabilities are formed under the existence
of Stochastic Discount Factor, which is ensured by Market
Completeness.




                      Figure: States Evolution

                       Nilüfer Calıskan
                               . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                          Basics
              Review on Option Pricing    Pricing
      Behavioral Model For Option Price   Important Assumptions
                                          Black-Scholes Model




In the Standard Case, from the first order condition of basic
consumtion based pricing, the arbitrage-free price of the
security must satisfy the following equality⇒


  p = E(mx) =                π(s)m(s)x(s)                                                (1)
                         s
    = π(up)m(up)x(up) + π(down)m(down)x(down)                                            (2)

where m represents the Stochastic Discount Factor and x
is the payoff of the next period in a 2−period model.



                       Nilüfer Calıskan
                               . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                           Basics
              Review on Option Pricing     Pricing
      Behavioral Model For Option Price    Important Assumptions
                                           Black-Scholes Model




The Discount factor can be seen as such a transformation
of probability measure that the price of the security
satisfies the relation
                                          1 ∗
                                  p=         E (x)                                        (3)
                                          Rf
where * shows the new measure that we call Risk Neutral
Measure and R f denotes the risk-free rate, which is in fact
the sum of all state prices.
Given that all the conditions for existence are satisfied, the
task here then for the price of option is to derive the Risk
Neutral Probabilities because what we see in the market
are the Objective probabilities.

                       Nilüfer Calıskan
                               . .         Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                          Basics
              Review on Option Pricing    Pricing
      Behavioral Model For Option Price   Important Assumptions
                                          Black-Scholes Model




The transformation method can differ among different
securities. For the options, we can use the fact that under
complete markets we can we can replicate the option
strategy by a portfolio consisting of bonds and underlying
and by making use of the arbitrage condition, one can
derive the risk neutral probabilities.

                     π ∗ = R f m(s)π(s) = R f pc(s)                                      (4)
                           m(s)
                         =        π(s),                                                  (5)
                           E(m)
                                1
                     Rf =                                                                (6)
                               pc(s)


                       Nilüfer Calıskan
                               . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                               Basics
                   Review on Option Pricing    Pricing
           Behavioral Model For Option Price   Important Assumptions
                                               Black-Scholes Model


Important Assumptions




     Efficient Markets
     No Arbitrage Opportunities
     Homogenous Beliefs




                            Nilüfer Calıskan
                                    . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                               Basics
                   Review on Option Pricing    Pricing
           Behavioral Model For Option Price   Important Assumptions
                                               Black-Scholes Model


Black-Scholes


     Asset Prices follow Lognormal Distribution→ Geometric
     Brownian Motion
     Risk Neutral Pricing→ Equivalent Martingale Measure
     Self financing strategy & complete markets
     In a complete market, one can form a portfolio that
     replicates the option price at every instant ⇒
     Black-Scholes Partial Differential Equation (PDE)
     Risk Neutral Pricing



                            Nilüfer Calıskan
                                    . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                                 Basics
                     Review on Option Pricing    Pricing
             Behavioral Model For Option Price   Important Assumptions
                                                 Black-Scholes Model


Black-Scholes Formula
  In continuous time also as a limiting case of the Binomial
  Model:
  The European Call option price with underlying security qz ,
  maturity t, and strike price K is

            CBS (qz , K , σ, t, r ) = qz N(d1 ) − Ke−rt N(d2 )                                  (7)
                                           √                  √
  where d1 = [ln(qz /K ) + (r + σ 2 /2)t]/σ t, and d2 = d1 − σ t
  and security price follows

                                  dqz
                                      = µdt + σdZ                                               (8)
                                  qz

  where Z is a Wiener process.
                              Nilüfer Calıskan
                                      . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                                Basics
                    Review on Option Pricing    Pricing
            Behavioral Model For Option Price   Important Assumptions
                                                Black-Scholes Model


Why Black-Scholes is so popular?




     Closed Form Solution
     All sensitivities available in closed form
     Very Easy to implement and interpret
     Good Proxy




                             Nilüfer Calıskan
                                     . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                               Basics
                   Review on Option Pricing    Pricing
           Behavioral Model For Option Price   Important Assumptions
                                               Black-Scholes Model


Alternative Pricing Models




     Stochastic Volatility Models
     Jump Diffusion Models
     Information Based Models
     Behavioral Based Models




                            Nilüfer Calıskan
                                    . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                               Risk Neutral Densities and Option Pricing
                                               Option Pricing Examples
                   Review on Option Pricing
                                               Smile Patterns
           Behavioral Model For Option Price
                                               Pitfall:Beliefs Do Not Matter in Black-Scholes
                                               Summary


Behavioral Based Models




     Option Smile
     Heterogeneous Beliefs
     Sentiment measures the degree of bias in the
     representative investor’s probability density function.




                            Nilüfer Calıskan
                                    . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                                     Risk Neutral Densities and Option Pricing
                                                     Option Pricing Examples
                      Review on Option Pricing
                                                     Smile Patterns
              Behavioral Model For Option Price
                                                     Pitfall:Beliefs Do Not Matter in Black-Scholes
                                                     Summary



Theorem (Theorem 14.1 of Shefrin 2005)
Let ν be an equilibrium state price vector.
(1) ν satisfies

                             ν(xt ) = δR,t PR (xt )g(xt )−γR (xt )
                                       t
                                                                                                       (9)

where γR , δR , and PR have the structure below:

                     1/γR (xt ) =            θj (xt )(1/γj )                                          (10)
                                        j
                              t
                             δR,t =          ν(xt )ζ(xt )γR (xt )                                     (11)
                                       xt

                                      ν(xt )ζ(xt )γR (xt )
                        PR (xt ) =            t
                                                                                                      (12)
                                             δR,t
                                        J
                                             cj (x0 )(Dj (xt )/ν(xt ))1/γj
                           ζ(xt ) =                     J
                                                                              .                       (13)
                                       j=1              k =1   ck (x0 )


                               Nilüfer Calıskan
                                       . .           Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                                      Risk Neutral Densities and Option Pricing
                                                      Option Pricing Examples
                       Review on Option Pricing
                                                      Smile Patterns
               Behavioral Model For Option Price
                                                      Pitfall:Beliefs Do Not Matter in Black-Scholes
                                                      Summary


Theorem (Theorem 21.1 of Shefrin 2005)
Given that theorem 14.1 holds, the general expression for the price of a
European call option on a security Z , featuring exercise price K and
expiration date t, is determined as follows.
(1)Let S(t − 1) be the set of successor nodes xt to xt−1 . The risk-neutral
density η(xt ) associated with event xt , conditional on xt−1 , is defined by:

                                                       ν(xt )
                                   η(xt ) =                     ν(yt )
                                                   yt ∈S(t−1)



(2)Let AE denote the event qZ ≥ K , in which the call option is exercised, and
Pν AE be its probability under the risk neutral density Pν . The product of a
sigle period interest rates defines the cumulative return
 t
ic (xt ) = i1 (x0 )i1 (x1 )...i1 (xt−1 ) to holding the short term risk free security with
reinvestment, from date 0 to date t. Then the x0 price of the call option is
given by:

                                             t
              qc (x0 ) = Eν (qz (xt ) − K )/ic (xt ) | AE , x0 P {AE | x0 }

                                Nilüfer Calıskan
                                        . .           Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                                  Risk Neutral Densities and Option Pricing
                                                  Option Pricing Examples
                      Review on Option Pricing
                                                  Smile Patterns
              Behavioral Model For Option Price
                                                  Pitfall:Beliefs Do Not Matter in Black-Scholes
                                                  Summary




Theorem (Theorem 21.2 of Shefrin 2005)
(1) Let PR {AE } be the exercise in the money probability under the
representative investor’s probability distribution. Then the price of the option
defined in the previous theorem satisfies:

           qc (x0 ) = δR,t ER (qz (xt ) − K )g(xt )−γR (xt ) | AE PR {AE }
                       t
                                                                                                   (14)

(2)Define the t−step probability distribution Φ(xt ) over date t events xt ,
conditional on x0 , as follows:
                                                   ν(xt )
                                   Φ(xt |x0 ) =                                                    (15)
                                                   yt ν(yt )

The qc satisfies:

            qc (x0 ) = EΦ {(qz (xt ) − K ) | AE , x0 } PΦ {AE |x0 } /itt (x0 )                     (16)



                               Nilüfer Calıskan
                                       . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                             Risk Neutral Densities and Option Pricing
                                             Option Pricing Examples
                 Review on Option Pricing
                                             Smile Patterns
         Behavioral Model For Option Price
                                             Pitfall:Beliefs Do Not Matter in Black-Scholes
                                             Summary




Theorems say:
   Risk-Neutral Density pricing equations as given above tend
   to obscure the properties of the representative invsetor’s
   beliefs affect asset prices.
   Alternatively, one can have the second option pricing
   formula which shows the direct impact of the
   representative invsetor’s beliefs on call option prices.




                          Nilüfer Calıskan
                                  . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                           Risk Neutral Densities and Option Pricing
                                           Option Pricing Examples
               Review on Option Pricing
                                           Smile Patterns
       Behavioral Model For Option Price
                                           Pitfall:Beliefs Do Not Matter in Black-Scholes
                                           Summary




Heterogeneity can cause the interest rates and volatility to
be stochastic.
Stochastic interest rate and volatility directly affects the
option price.
Heterogeneity introduces smile effects into equilibrium
option prices, which leads to different implied volatilities for
put and call options.




                        Nilüfer Calıskan
                                . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                                Risk Neutral Densities and Option Pricing
                                                Option Pricing Examples
                    Review on Option Pricing
                                                Smile Patterns
            Behavioral Model For Option Price
                                                Pitfall:Beliefs Do Not Matter in Black-Scholes
                                                Summary


Discrete Time Example

  Binomial Model:
      Single Asset: Market Prortfolio, two investors with
      Logarithmic Utility holding the one half of the Market
      Portfolio.
      u = 1.05, d = 1/u and Πu = 0.7 and 1 − Πu are objective
      branching probabilities(δ = 1).
      Different beliefs about branching probabilities P1,u and P2,u
      so the representative investor belief

                               PR,u = P1,u + P2,u /2                                             (17)



                             Nilüfer Calıskan
                                     . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                                Risk Neutral Densities and Option Pricing
                                                Option Pricing Examples
                    Review on Option Pricing
                                                Smile Patterns
            Behavioral Model For Option Price
                                                Pitfall:Beliefs Do Not Matter in Black-Scholes
                                                Summary


Discrete Time Example Continues



     The Equilibrium State Price satisfies in this setting:

                    νu = PR,u /u = P1,u /u + P2,u /u /2                                          (18)

     In the log-utility setting, the equilibrium interest rate:

                                              1
                            i=                                                                   (19)
                                   PR,u /u + (1 − PR,u /d)




                             Nilüfer Calıskan
                                     . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                    Risk Neutral Densities and Option Pricing
                                    Option Pricing Examples
        Review on Option Pricing
                                    Smile Patterns
Behavioral Model For Option Price
                                    Pitfall:Beliefs Do Not Matter in Black-Scholes
                                    Summary




                 Nilüfer Calıskan
                         . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                          Risk Neutral Densities and Option Pricing
                                          Option Pricing Examples
              Review on Option Pricing
                                          Smile Patterns
      Behavioral Model For Option Price
                                          Pitfall:Beliefs Do Not Matter in Black-Scholes
                                          Summary


For 2.9, the one period discount factor 1/1.029 = 0.9719
For 0.87, the one periond discount factor
1/1.0087 = 0.9914
Weighted Average Property ⇒ the wealth weighted convex
combination of them.
The wealth weight are 50% so one period equilibrium
interest rate is 1.87.
However, since they disagree on probability, they will bet
on eacgh other so the wealth weights will be changed at
the end of period.
Thus in the next period both the interest rate and
representative probability will change because the wealth
levels change.
This introduces the stochastic volatility and stochastic
interest rates.
                       Nilüfer Calıskan
                               . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                    Risk Neutral Densities and Option Pricing
                                    Option Pricing Examples
        Review on Option Pricing
                                    Smile Patterns
Behavioral Model For Option Price
                                    Pitfall:Beliefs Do Not Matter in Black-Scholes
                                    Summary




                 Nilüfer Calıskan
                         . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                                 Risk Neutral Densities and Option Pricing
                                                 Option Pricing Examples
                     Review on Option Pricing
                                                 Smile Patterns
             Behavioral Model For Option Price
                                                 Pitfall:Beliefs Do Not Matter in Black-Scholes
                                                 Summary


Continuous Time Example

  Black-Scholes model having two investors with equal initial
  wealth:
      Situation 1: All investors agree with investor 1 and the
      equilibrium compounded interest rate r1 , value of Z is qZ ,
      and volatility σ.
      Situation 2: All investors agree with investor 1 and the
      equilibrium compounded int erest rate r2 , value of Z is qZ ,
      and volatility σ.
      Weigthed-average property implies

         Ceq = [CBS (qZ , K , σ, t, r1 ) + CBS (qZ , K , σ, t, r2 )] /2                           (20)


                              Nilüfer Calıskan
                                      . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                               Risk Neutral Densities and Option Pricing
                                               Option Pricing Examples
                   Review on Option Pricing
                                               Smile Patterns
           Behavioral Model For Option Price
                                               Pitfall:Beliefs Do Not Matter in Black-Scholes
                                               Summary




Different interest rate and volatility arguments lead to the
equilibrium call option pricing equation as a wealth-weigthed
convex combination:
                             J
                Ceq =             wj CBS (qz , K , sigmaj , t, rj )                             (21)
                            j=1




                            Nilüfer Calıskan
                                    . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                    Risk Neutral Densities and Option Pricing
                                    Option Pricing Examples
        Review on Option Pricing
                                    Smile Patterns
Behavioral Model For Option Price
                                    Pitfall:Beliefs Do Not Matter in Black-Scholes
                                    Summary




                 Nilüfer Calıskan
                         . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                    Risk Neutral Densities and Option Pricing
                                    Option Pricing Examples
        Review on Option Pricing
                                    Smile Patterns
Behavioral Model For Option Price
                                    Pitfall:Beliefs Do Not Matter in Black-Scholes
                                    Summary




                 Nilüfer Calıskan
                         . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                              Risk Neutral Densities and Option Pricing
                                              Option Pricing Examples
                  Review on Option Pricing
                                              Smile Patterns
          Behavioral Model For Option Price
                                              Pitfall:Beliefs Do Not Matter in Black-Scholes
                                              Summary


Smile Patterns




                           Nilüfer Calıskan
                                   . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                    Risk Neutral Densities and Option Pricing
                                    Option Pricing Examples
        Review on Option Pricing
                                    Smile Patterns
Behavioral Model For Option Price
                                    Pitfall:Beliefs Do Not Matter in Black-Scholes
                                    Summary




                 Nilüfer Calıskan
                         . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                    Risk Neutral Densities and Option Pricing
                                    Option Pricing Examples
        Review on Option Pricing
                                    Smile Patterns
Behavioral Model For Option Price
                                    Pitfall:Beliefs Do Not Matter in Black-Scholes
                                    Summary




                 Nilüfer Calıskan
                         . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                              Risk Neutral Densities and Option Pricing
                                              Option Pricing Examples
                  Review on Option Pricing
                                              Smile Patterns
          Behavioral Model For Option Price
                                              Pitfall:Beliefs Do Not Matter in Black-Scholes
                                              Summary




The Equilibrium Call Option Pricing Equation is the wealth
weighted convex combination of Black-Scholes Prices of each
investor:
                                J
                  Ceq =             wj CBS qZ , K , σj , t, rj                                 (22)
                              j=1




                           Nilüfer Calıskan
                                   . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                    Risk Neutral Densities and Option Pricing
                                    Option Pricing Examples
        Review on Option Pricing
                                    Smile Patterns
Behavioral Model For Option Price
                                    Pitfall:Beliefs Do Not Matter in Black-Scholes
                                    Summary




                 Nilüfer Calıskan
                         . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                                Risk Neutral Densities and Option Pricing
                                                Option Pricing Examples
                    Review on Option Pricing
                                                Smile Patterns
            Behavioral Model For Option Price
                                                Pitfall:Beliefs Do Not Matter in Black-Scholes
                                                Summary


Pitfall:Beliefs Do Not Matter in Black-Scholes



                    dS
      Investor 1:   S    = µ1 dt + σdZ
                    dS
      Investor 2:   S    = µ1 dt + σdZ
      The expected return on the underlying security is not in the
      pricing equation.
      Thus, heterogeneity in this setting will not impact the
      option prices and does not lead to volatility smile.




                             Nilüfer Calıskan
                                     . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin
                                               Risk Neutral Densities and Option Pricing
                                               Option Pricing Examples
                   Review on Option Pricing
                                               Smile Patterns
           Behavioral Model For Option Price
                                               Pitfall:Beliefs Do Not Matter in Black-Scholes
                                               Summary


Summary



    Option Smiles can be explained as a feature of behavioral
    framework!
    Heterogeneity generally can lead the Black-Scholes price
    insufficient
    By introducing stochasticity in volatility and interest rates.




                            Nilüfer Calıskan
                                    . .        Chapter 21: Behavioral Black-Scholes, Hersh Shefrin