Prices of State Contingent Claims implicit in Option Prices

1. 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 MengTian

2. 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’

3. 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)]

4. Pricing Elementary Contingent Claims • P(M,T)=(1/∆M){[c(M- ∆M,T)-c(M,T)]-[c(M,T)-c(M+ ∆M,T)]}

5. 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

6. 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

7. Example using Black-Scholes

8. 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):

9. 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:

10. Properties of Elementary Contingent-Claim Prices (Cont.) The elasticity of P(M,T) with respect to the dividend rate is:

11. 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:

12. 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

13. 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

14. 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