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A State Contingent Claim Approach To Asset Valuation Kate Barraclough

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Overview Aim is to empirically test the application of Arrow-Debreu state preference theory State preference approach is applied to pricing both to stocks and options Model values under state pricing are compared to other asset pricing models State preference approach is found to provide an overall improvement on the other models for both stocks and options

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Background Basic form of any asset pricing equation: where P t is the price at time t, E t is a conditional expectations operator, X t+1 is the asset’s payoff and M t+1 is the stochastic discount factor The stochastic discount factor represents investors’ marginal rates of substitution between consumption in the current period, and consumption in time t, state s.

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State Contingent Claims Stochastic discount factor may be characterised by the set of state contingent claim prices A state contingent claim will have a positive payoff in state s and zero elsewhere Ross (1976) - state contingent claims will be implicit in the price of traded securities and in a complete market investors may form a portfolio with a positive payoff in state s and zero elsewhere

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State Preference Approach State price is the price today of one unit of consumption at time t, state s The price of a risky asset may be determined as the payoff in time t state s multiplied by the state price and summed over all possible S states:

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State Price Computation Breeden and Litzenberger (1978) – state contingent claim may be modelled as the second derivative of a call option price Construct a butterfly spread with a unit payoff – buy one call with strike M-ΔM, one call with strike M+ΔM, and sell two calls with strike M M-ΔMM M+ΔM ΔMΔM

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State Price Computation If the value of the underlying asset is M next period, then the payoff on the option portfolio will be ΔM and zero otherwise. Normalising for a unit payoff: Taking the limit as ΔM tends to zero then the price of a portfolio paying one unit will be given by: Evaluating the portfolio at X=M provides

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State Price Computation Black-Scholes option pricing formula provides a closed form solution: where But cannot compute in continuous increments for each strike

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Delta Securities Delta security – unit payoff if the price of the underlying asset is greater than or equal to some level Y: State price - cost of a security with a unit payoff if the level of the underlying asset is between some levels is between some levels Y 1 and Y 2 :

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Advantages Incorporate market microstructure features –Minimum tick size in the underlying instrument –Price limits, circuit breakers etc Limit range of distribution with empirical maximum and minimum returns –Not price extreme observations –Reduce computational burden

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S&P 500 Index Options First test of the paper is to apply the state preference approach to pricing S&P 500 index options Compare model values to the Black-Scholes option pricing formula and Stutzer’s canonical valuation approach

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State Preference Approach Historical maximum and minimum T-day returns are determined from the empirical distribution of returns on the S&P 500 index Possible future index levels are calculated in increments of 5c The state price corresponding to each index level are determined by the delta security method

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State Preference Approach The option price is calculated as the sum of the expected payoff at each index level multiplied by the respective state price Call option prices will be given by: And put options:

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Canonical Valuation Stutzer (1996) – determine risk-neutral probabilities from the empirical distribution of returns on the underlying instrument Solving the unconstrained minimisation problem: provides the probability distribution:

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Canonical Valuation Option prices are determined with reference to historical index returns Calculated as the sum of the expected payoff at each index level multiplied by the respective risk-neutral probability:

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Options Data S&P 500 index options Weekly observations January 1990 through December 1993 Two proxies for expected stock market volatility: –CBOE Market Volatility Index (VIX) –40 day historical volatility Daily observations on the S&P 500 index used to determine historical distribution of index returns

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Results – Table 1

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Results – Table 2

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Results – Table 3

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Results – Table 4

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Results – Table 5

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Results – Table 6

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Results – Figure 1

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Results – Figure 2

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Results – Figure 3

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Results – Figure 4

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Results – Figure 5

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Stock Valuation The second test is to apply the state preference approach to stock valuation Comparison is made between the state preference approach and Ohlson’s (1995) residual income model Stutzer’s canonical valuation approach is also applied

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State Preference Approach Linear projection to determine stock movements in reference to the market: Payoffs are given by: Providing the valuation expression:

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Canonical Valuation Determine risk-neutral probabilities from the historical distribution of index returns as described previously Apply to the payoff function from the previous slide to provide the valuation expression:

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Residual Income Model The residual income model specifies a relationship between market value, book value, and contemporaneous and future earnings Based on the dividend discount model: And a clean surplus relation:

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Residual Income Model Combining the dividend discount model and the clean surplus relation: Test empirically using cross sectional regression estimates:

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Stock Data Sample covers all companies in the COMPUSTAT database from 1993 through 2004 Linear projection – based on monthly observations over the previous 5 years Stock and index returns are from the CRSP database Consensus earnings from I/B/E/S proxy for the market’s expectation of future earnings

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Results – Table 7

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Results – Figure 6

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Summary S&P 500 index options: –state preference approach provides an overall improvement compared to Black-Scholes and canonical valuation Stocks –state preference approach provides a significant improvement on the residual income model for stock valuation

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Future Directions Extend canonical valuation approach –Additional constraint of previous day’s call option price – similarity to using previous day’s implied volatilities for Black-Scholes –Include market microstructure considerations - 5c/10c price increments rather than empirical movement Implications for investor risk preferences –VIX index vs realised volatility

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Future Directions Compare results for stock valuation over different maturities –1 week –1 month –Quarterly May be improvements available for residual income model

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