State prices

In financial economics, a state-price security, also called an Arrow-Debreu security (from its origins in the Arrow-Debreu model), a pure security, or a primitive security is a contract that agrees to pay one unit of a numeraire (a currency or a commodity) if a particular state occurs at a particular time in the future and pays zero numeraire in all the other states. The price of this security is the state price of this particular state of the world, which may be represented by a vector. The state price vector is the vector of state prices for all states.[1][2][3]

As such, any derivatives contract whose settlement value is a function of an underlying whose value is uncertain at contract date can be decomposed as a linear combination of its Arrow-Debreu securities, and thus as a weighted sum of its state prices.

The Arrow-Debreu model (also referred to as the Arrow-Debreu-McKenzie model or ADM model) is the central model in the General Equilibrium Theory and uses state prices in the process of proving the existence of a unique general equilibrium.

Example

Imagine a world where two states are possible tomorrow: peace (P) and war (W). Denote the random variable which represents the state as ω; denote tomorrow's random variable as ω1. Thus, ω1 can take two values: ω1=P and ω1=W.

Let's imagine that:

The prices qP and qW are the state prices.

The factors that affect these state prices are:

Application to financial assets

If the agent buys both qP and qW, he has secured £1 for tomorrow. He has purchased a riskless bond. The price of the bond is b0 = qP + qW.

Now consider a security with state-dependent payouts (e.g. an equity security, an option, a risky bond etc.). It pays ck if ω1=k -- i.e. it pays cP in peacetime and cW in wartime). The price of this security is c0 = qPcP + qWcW.

Generally, the usefulness of state prices arises from their linearity: Any security can be valued as the sum over all possible states of state price times payoff in that state: c_0 = \sum_k q_k\times c_k.

Analogously, for a continuous random variable indicating a continuum of possible states, the value is found by integrating over the state price density.

See also

References

  1. economics.about.com Accessed June 18, 2008
  2. Rebonato, R. (2005). Volatility and correlation: the perfect hedger and the fox. John Wiley & Sons. http://books.google.co.uk/books?id=HD9P6L2zX48C&pg=PA323&dq=arrow+debreu+state+price&hl=en&sa=X&ei=U2xwVJjAA9P3aqWJgMAH&redir_esc=y#v=onepage&q=arrow%20debreu%20state%20price&f=false
  3. Dupire, B. (1997). Pricing and hedging with smiles (pp. 103-112). in: Mathematics of derivative securities. Dempster and Pliska eds., Cambridge Uni. Press.