# 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
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

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