How State Prices Relate to the Stochastic Discount Factor

Summary

The document explains how state prices and a stochastic discount factor express the same pricing rule in different forms. In a discrete set of possible future states, a state price is the current price of a claim paying one unit only in a particular state. Any payoff can therefore be priced by summing its state-contingent payoffs weighted by those state prices.

Dividing each state price by the physical probability of its state produces the stochastic discount factor for that state. The expected payoff multiplied by this factor, taken under the physical probability distribution, gives the same price as the state-price sum. Thus the state price equals the physical probability times the corresponding discount factor. The explanation assumes a discrete state space with nonzero probabilities and gives the algebraic relationship; it does not develop how to estimate these quantities or establish the broader conditions under which positive prices imply absence of arbitrage.

Key ideas

• State prices value payoffs by weighting each possible outcome with its state-contingent price.
• A stochastic discount factor converts physical probabilities into pricing weights.
• Each state price equals the probability of that state multiplied by its stochastic discount factor.
• The two pricing representations are equivalent in the discrete setting described.

Tags

# What is the difference between state prices and stochastic discount factor?


# What is the difference between state prices and stochastic discount factor?












I was reading a paper on arbitrage and it was mentioned that a positive SDF implies no arbitrage and later on it said that positive state prices imply no arbitrage. I am new to this topic and i am confused with the concept.

## Answer by fni (score 3, accepted)

https://quant.stackexchange.com/a/31331

The two are very similar. To understand the difference notice that given a discrete sample space $\Omega=\{\omega_1,\omega_2…\omega_S\}$ the price of any payoff can be computed if we define the state prices $q$ (or prices of Arrow-Debreu securities,i.e., securities that pay 1 in one state and 0 in all other states. For instance $q_i(\omega_i)=1$ and $q_i(\omega_j)=0$ for $i\neq j$), as $$P_t(X_{t+1})=\sum_{s=1}^S q_s X_{t+1}(\omega_s)$$ If we multiply and divide each state price by the physical probabilities $p(\omega)$ we obtain $$P_t(X_{t+1})=\sum_{s=1}^S p(\omega_s)\frac{q_s}{p(\omega_s)} X_{t+1}(\omega_s)\equiv \sum_{s=1}^S p(\omega_s)m_{t+1}(\omega_s) X_{t+1}(\omega_s)=E^p[m_{t+1}X_{t+1}]$$ where $m_{t+1}(\omega)$ is the stochastic discount factor. The relationship between state prices and SDF is, therefore, $q_s=p(\omega_s)m_{t+1}(\omega_s).$

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