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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

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black-scholes model, option pricing, behavioral finance, black scholes, implied volatility, black-scholes option pricing model, myron scholes, stock price, black-scholes formula, of the black, fischer black, risk management, prospect theory, fair value, asset pricing

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posted: | 7/9/2010 |

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