One-Period Valuation Theory

Transcription

1 One-Period Valuation Theory Part 1: Basic Framework Chris Telmer March, 2013 Develop a simple framework for understanding what the pricing kernel is and how it s related to the economics of risk, return and decision-making under uncertainty. where q asset price q = E(md) m pricing kernel or stochastic discount factor d asset payoff 1 / 22 2 / 22 Basic Idea: State Prices Probability Space and Financial Markets Valuation of riskless cash flows: {c 1, c 2, c 3 } q 1 = 1 1+i 1 c 1 = d 1 c 1 Valuation of risky cash flows: { c 1, c 2, c 3 } S states of nature, s {1,2,...,S} Objective probabilities: P s = Prob(state =s) Securities market: q h 1 = ψ h 1c h 1 q l 1 = ψ l 1c l 1 q R N D (R N R S ) q 1 = q h 1 +q l 1 q asset prices D asset payoffs 3 / 22 4 / 22

2 and Payoffs Portfolio: θ R N : Unlimited short-sales Portfolio payoff: x θ = D θ R S Portfolio price: q θ = q θ R 1... (synonymously, q θ = q θ) Set of marketed securities: Complete markets, S = 2,N = 2: 2 1 q = 1 3 θ = 9 12 D θ = and q θ = 34.2 Φ = {x R S : x = D θ, θ R N } = span(d) If N = S then Φ = R S, complete markets If N < S then Φ is a linear subspace of R S, incomplete markets 5 / 22 6 / 22 (continued) Incomplete markets, S = 2,N = 1: [ 2 1 ] q = [1.4] Suppose that there exist numbers, ψ s > 0 such that q j = S ψ s d js s=1 We call the ψ s numbers state prices. That is, ψ s = cost of 1 dollar in state s ψ s d js = cost of d js dollars in state s. Value additivity. s ψ sd js = cost of d j1 dollars in state 1, and d j2 dollars in state 2, and... d js dollars in state S. Assets are just things that give you state-dependent payoffs. State prices just allow us to express an asset s price as the sum of the value of each of its state-dependent payoffs. 7 / 22 8 / 22

3 State Prices How Do We Find the State Prices? : ψ R S ++ s.t. q j = q = S D js ψ s for all j s= It costs 5.25 to get 4 if state 1 occurs and 7 if state 2 occurs. The implicit state prices the ψ vector that satisfies the above equations are 0.49 ψ = 0.47 We solve systems of linear equations. Continuing with the previous example: q = The equations q j = S s=1 D jsψ s, j = 1,2, are 5.25 = 4ψ 1 +7ψ = 2ψ 1 +3ψ 2 It costs 0.49 to get 1 if state 1 occurs. It costs = 1.96 to get 4 if state 1 occurs. The solution is ψ 1 = 0.49, ψ 2 = It costs = 5.25 to get 4 if state 1 occurs and 6 if state 2 occurs. 9 / / 22 Using Matrix Algebra If markets are complete: q = Dψ ψ = D 1 q A θ such that, either (a) q θ < 0 and D θ 0 or (b) q θ 0 and D θ > 0 Theorem: given (D, q), then ψ R S ++ s.t. q = Dψ No (1) Otherwise, No unique solution Strictly positive state prices, ψ, exist if and only if there do not exist arbitrage opportunities. See Practice Problems Note: state prices are not unique if markets are not complete. Critical issue in practice. 11 / / 22

4 Risk-free asset pays the same in all states. e.g., Without loss of generality let the payoff be 1: Price, denoted b, is sum of the state prices: Risk-free interest rate: b = s ψ s = ι ψ 1+i f 1 b = 1 ι ψ Financial markets Implicit state prices: ψ = D 1 (2/3 100/303) q = ( 2/ /303) Pure contingent claims: [ (D ) 1 = Risk-free interest rate: q = /30 1/30 1/303 4/303 [ = b = ψ ι = = i f 1 1 = 0.01 b ] ] 13 / / 22 (continued) (continued) Valuation: 10 Call Price(K = 30) = ψ = Put Price(K = 30) = ψ = Expected stuff: Expected Payoffs = DP = = / Expected Returns = =, 101/ ψ = Expected Returns = What s the risk premium on the stock? Why? How about a stock that pays: / = 101/ d = [10 40] 15 / / 22

5 Valuation Algorithm 1. Observe market data: (D,q) 2. Recover state prices: ψ = D 1 q 3. Value a new claim, d new : q new = ψ d new On-line sports betting Assignment 1 Potential pitfalls of arbitrage-free pricing Practice problem Expected returns often lurking Important caveats: Complete vs. incomplete markets Risk management Corporate financial leverage Spreadsheet 17 / / 22 -Free Pricing Arb-Free Pricing Risk Arb-Free Pricing Risk Recovery and valuation. No-arbitrage implies that state prices are inherent in prices/payoffs of traded assets. Valuation theory is the theory of recovering the state prices and then adding-up. Incomplete markets means that recovery is a non-unique, tricky business. It can lead to flawed decision making if not used appropriately. Remember: the hard math is usually a sophisticated version of ψ = D 1 q (from the market) followed by q = ψ d (from your client). Risk-neutral valuation is just a alternative way to write this. It often leads to confusion between risk-neutral valuation and risk neutrality. Don t mix them up. 19 / / 22

6 State Prices and Risk Financial Innovation Arb-Free Pricing Risk Interpretation and risk. A high state price means one of two things. Either a high probability or a high valuation. We call the latter a high marginal utility. The pricing kernel scales-out the probability, leaving us with the marginal utility. What is risk? A low payoff in a high marginal utility state. Key concept: state-contingent value assessment Arb-Free Pricing Risk What is Financial Innovation Article: John Kay Creating new, unspanned payoffs? s: TIPS, CDS. Or creating linear combinations of existing payoffs? s: UST Strips, mutual funds. Which is a better business to be in? Which benefits society most? 21 / / 22