Prices of State Contingent
Claims implicit in Option Prices
Douglas T Breeden & Robert H Litzenberger, The Journal of Business, 1978
By Aditya M Kashikar and Meng Tian
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’
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)]
Pricing Elementary Contingent Claims
• P(M,T)=(1/∆M){[c(M- ∆M,T)-c(M,T)]-[c(M,T)-c(M+ ∆M,T)]}
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
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 σT2 = σ2
Example using Black-Scholes
Properties of Elementary Contingent-Claim
Prices
The 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):
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:
Properties of Elementary Contingent-Claim
Prices (Cont.)
The elasticity of P(M,T) with respect to the dividend rate is:
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:
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
Extensions: CAPM
• If cash flows are jointly log-normally distributed with aggregate
consumption
• Under earlier assumptions and Pareto-optimal Capital Markets, BlackScholes prices options correctly under CRRA
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