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

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 σT

2 = σ2

Example using Black-Scholes

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):

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, Black-Scholes 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