Arrow-Debreu Security

Study guide: Exam INV-201

An Arrow-Debreu security (a pure security or state-contingent claim) pays 1 at time $T$ if one particular state of the world $\omega$ occurs and nothing otherwise. Its price $\psi_\omega$ is the state price of that state. Any payoff is a bundle of these securities, so state prices price every claim, and scaled up by the risk-free growth factor they are the risk-neutral probabilities. A market is complete when its traded assets can replicate every Arrow-Debreu security, and incomplete when they cannot.

$V_0 = \sum_{\omega} \psi_\omega\,X_\omega$

$\sum_{\omega} \psi_\omega = e^{-rT}$

$q_\omega = e^{rT}\,\psi_\omega$

$\psi(K) = e^{-rT}\,\frac{\partial^2 C}{\partial K^2}$

Symbols: $X_\omega$ is the payoff of a claim in state $\omega$ , and $\psi_\omega > 0$ is the price today of 1 paid in that state. The riskless bond pays 1 in every state, so it is the sum of all the Arrow-Debreu securities, which gives the second block. Dividing each state price by that sum gives probabilities $q_\omega$ that are positive and sum to 1, and under them $V_0 = e^{-rT}E^{\mathbb{Q}}[X]$ .
Finding state prices: With $n$ states and traded assets of known prices and payoffs, the state prices solve the linear system that prices each traded asset, $\sum_\omega \psi_\omega X^{(j)}_\omega = V^{(j)}_0$ . A replicating portfolio for a single Arrow-Debreu security is the same calculation viewed from the other side.
Fundamental theorem of asset pricing: There is no arbitrage if and only if strictly positive state prices exist, which is the same as the existence of an equivalent martingale measure. The market is complete if and only if those state prices, and hence $\mathbb{Q}$ , are unique.
Complete markets: In a one-period market with $n$ states, completeness requires $n$ traded assets with linearly independent payoffs. In the binomial model a stock and a bond span the two states at each node, and in the Black-Scholes-Merton model continuous trading in the stock and bond spans every payoff, so every option has one price.
Incomplete markets: When there are more sources of risk than traded assets, many sets of state prices fit the traded prices, so a claim has a range of no-arbitrage prices rather than one. Jumps in the asset price, stochastic volatility, untraded variables such as weather, and insurance risks such as mortality, lapse and policyholder behaviour all cause incompleteness. Pricing there requires a choice of measure, such as a market price for the unhedgeable risk, a utility function or a cost-of-capital margin.
Continuous states: When the state is the terminal stock price, the state-price density is the second derivative of the call price with respect to strike (fourth block), the Breeden-Litzenberger result. This is how a volatility smile is read as a risk-neutral distribution of $S_T$ .

State Prices in a Two-State Market and the Price of a Call A non-dividend stock at 100 will be worth 120 or 90 in one year, and $r = 0.05$ continuously compounded. Find the two state prices and the risk-neutral probabilities, then price a one-year European call with strike 105.
Answer The bond and the stock give two equations in $\psi_u$ and $\psi_d$ .
 $\begin{align*} \psi_u + \psi_d &= e^{-0.05} = 0.951229 \\ 120\,\psi_u + 90\,\psi_d &= 100 \\ 30\,\psi_u &= 100 - 90(0.951229) \\ \psi_u &= 0.479645 \\ \psi_d &= 0.951229 - 0.479645 \\ &= 0.471584 \end{align*}$ 
The risk-neutral probabilities are $q_u = e^{0.05}(0.479645) = 0.50424$ and $q_d = 0.49576$ . The call pays 15 in the up state only.
 $\begin{align*} C &= 15\,\psi_u + 0\,\psi_d \\ &= 15(0.479645) \\ &= 7.19 \end{align*}$ 
The call is worth 7.19, the same as the binomial risk-neutral price, because state prices are the risk-neutral probabilities discounted at $r$ .

An Incomplete Market Gives a Range of Prices A stock at 100 will be worth 125, 105 or 85 in one year, and a zero-coupon bond paying 1 in every state costs 0.95. Only the stock and the bond trade. Find all positive state prices consistent with these prices, and the range of no-arbitrage prices for a call with strike 105.
Answer Two equations constrain three state prices, so one is free. Write the others in terms of $\psi_1$ , the price of the top state.
 $\begin{align*} \psi_1 + \psi_2 + \psi_3 &= 0.95 \\ 125\,\psi_1 + 105\,\psi_2 + 85\,\psi_3 &= 100 \\ \psi_2 &= 0.9625 - 2\psi_1 \\ \psi_3 &= \psi_1 - 0.0125 \end{align*}$ 
Positivity needs $\psi_3 > 0$ and $\psi_2 > 0$ , so $0.0125 < \psi_1 < 0.48125$ . The call pays 20 in the top state only, so its price is $20\psi_1$ .
 $\begin{align*} 20(0.0125) &< C < 20(0.48125) \\ 0.25 &< C < 9.625 \end{align*}$ 
Any price in that interval is free of arbitrage, and the market alone cannot choose one. If the call itself traded at, say, 4.00, it would fix $\psi_1 = 0.20$ , then $\psi_2 = 0.5625$ and $\psi_3 = 0.1875$ , and the market would become complete.

How it appears on the exam

INV-201: The syllabus asks candidates to understand Arrow-Debreu securities and the distinction between complete and incomplete markets, next to the objective on no-arbitrage and replication. The readings are Campolieti's chapter 5 and Hull's chapters on forward pricing and binomial trees.

State-price calculations: A question gives a one-period market with two or three states and asks for the state prices, the risk-neutral probabilities, or the price of a call, put or digital payoff built from them. It may also ask candidates to show an arbitrage when a state price would be negative.

Completeness: A question asks whether a given set of assets spans all the states, or for the range of no-arbitrage prices of a claim that they do not span.

Written answers: Candidates may be asked why insurance liabilities, jump risk or stochastic volatility make a market incomplete, and what that means for pricing and hedging a guarantee.

Exam traps

State prices are not probabilities: State prices sum to the discount factor $e^{-rT}$ , not to 1. Treating them as probabilities and discounting again counts the discounting twice.

A negative state price means arbitrage: If the system gives a negative $\psi_\omega$ , the market is not arbitrage-free. Report the arbitrage rather than pricing with a negative weight.

Counting assets, not states: Three assets do not make a three-state market complete if their payoffs are linearly dependent. A call that is a combination of the stock and the bond adds nothing.

Incomplete does not mean any price: In an incomplete market the price of an unspanned claim is still bounded by the cheapest super-replicating and the dearest sub-replicating portfolios. A price outside those bounds is an arbitrage.