prices of state contingent claims implicit in option prices douglas t breeden & robert h...
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Prices of State Contingent Claims
implicit in Option PricesDouglas T Breeden & Robert H Litzenberger, The Journal of Business, 1978
By Aditya M Kashikar and Meng Tian
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Central Idea
• Time-state preference model, multiperiod economy
• Deriving prices of primitive securities from prices of call options
• Main motivation: to bridge the divide between theory and empirical
• The simplest state contingent claim is an ‘elementary claim’
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Pricing Elementary Contingent Claims• ‘Elementary Claim’ on a security or portfolio pays 1$ in T periods if
the value of the security or portfolio is M at that time• P(M,T) derived using Call option prices c(X,T)• P(1,T)=[c(0,T)-c(1,T)]-[(c(1,T)-c(2,T)]
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Pricing Elementary Contingent Claims• P(M,T)=(1/∆M){[c(M- ∆M,T)-c(M,T)]-[c(M,T)-c(M+ ∆M,T)]}
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Pricing Contingent Claims (Derivatives)• If a security has payoffs over time that are known functions of a
portfolio (underlying asset), then it can be priced as below
• Hence, any derivative claims can be priced using a portfolio of calls• Assumptions used: Perfect markets, c(X,T) is twice differentiable for (3)• No assumptions on stochastic process on underlying’s price or option
price. No assumptions on individual preferences
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Example using Black-Scholes
• Where d1=d2+ σT√T, r T = - {ln[B(T)]}/T• European Call has B(T)=exp(-rT), δ=0 and σT
2 = σ2
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Example using Black-Scholes
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Properties of Elementary Contingent-Claim PricesThe price of $1.00 contingent upon aggregate wealth being M in T periods:
The increased probabilities of HIGH levels of Mt and the decreased probabilities of LOW levels of Mt increase and decrease, respectively, their contingent-claim prices given by P(M,T):
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Properties of Elementary Contingent-Claim Prices (Cont.)The elasticity of the pricing function with respect to the instantaneous standard deviation of the rate of growth of aggregate wealth is:
The elasticity of P(M,T) with respect to the price of a T-period riskless discount bond, B(T), is:
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Properties of Elementary Contingent-Claim Prices (Cont.)
The elasticity of P(M,T) with respect to the dividend rate is:
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Properties of Elementary Contingent-Claim Prices (Cont.)
Market value structure:The elasticity of the claim price with respect to the level of the market that it is contingent upon (its exercise price) is:
Maturity structure:The effects of a changing probability of the given level of M at T as T changes:
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Extensions: Pricing any Security
• Time-additive, State-independent utility over lifetime• Homogenous expectations conditional on aggregate consumption• One-to-one mapping between aggregate wealth and aggregate
consumption
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Extensions: CAPM
• If cash flows are jointly log-normally distributed with aggregate consumption
• Under earlier assumptions and Pareto-optimal Capital Markets, Black-Scholes prices options correctly under CRRA
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Summary
• State-Contingent claim prices are implicit in Option Prices
• Due to put-call parity, the same analysis can be carried for puts
• Under certain assumptions, any security can be priced using option prices through elementary claims