1. Security Markets V Miloslav S Vošvrda Theory of Capital Markets

2. Competitive Market Equilibrium A feasible allocation is a collection of choices, with allocated to agent i, satisfying An equilibrium is a feasible allocation and a non-zero linear price function p on L satisfying, for each agent i, solves subject to

3. An optimal feasible allocation A feasible allocation is optimal if there is no other feasible allocation such that for all i.

4. A concept of competitive markets The vector choice space L was taken to be the commodity space, for some number of commodities, with a typical element representing a claim to units of the c-th commodity, for

5. Uncertainty Uncertainty can be added to this model as a set of states of the world, one of which will be chosen at random. The vector choice space L can be treated as the space of matrices. The (s, c)-element of a typical choice x represents consumption of units of the c-th commodity in state s. A given linear price functional p on L can be represented by an matrix, taking as the unit price of consumption of commodity c in s. That is, writing for the s-th row of any matrix x, there is a unique price matrix such that for all x in L.

6. A contingent commodity market equilibrium Trading in these contracts occurs before the true state is revealed; then contracted deliveries occur and consumption ensues. An equilibrium, in this case, is called a contingent commodity market equilibrium.

7. Financial security markets are an effective alternative to contingent commodity markets. We take the (S states, C commodities). Security markets can be characterized by an S x N dividend matrix d, where N is the number of securities. The n-th security is defined by the n-th column of d, with representing the number of units of account paid by the n-th security in a given state s. Securities are sold before the true state is resolved, at prices given by a vector

8. Spot markets are opened after the true state is resolved. Spot prices are given by an matrix with representing the unit price of the c-th commodity in state s. Let denote the endowments of the I agents, taking as the endowment of commodity c to agent i in state s. An agent‘s plan is a pair, where the matrix is a consumption choice and is a security portfolio.

9. The crown payoff of portfolio in state s is. Given security and spot prices, a plan is budget feasible for agent i if (1) and (2) A budget feasible plan is optimal for agent i if there is no budget feasible plan such that

10. A security-spot market equilibrium is a collection with the property: for each agent i, the plan is optimal given the security-spot price pair ; markets clear:

11. The effectiveness of financial securities Suppose that is a contingent commodity market equilibrium, where p is a price functional represented by the price matrix. Take N = S securities and let the security dividend matrix d be the identity matrix, meaning that the n-th security pays one dollar in state n and zero otherwise. Let the security price vector be and take the spot price matrix

12. A security-spot market equilibrium For each agent i and state s, let equating the number of units of the s-th security held with the spot market cost of the net consumption choice for state s. Then is a security-spot market equilibrium.

13. Budget feasibility Budget feasibility obtains since, for any agent i, and

14. Proof: Suppose is a budget feasible plan for agent i and. By optimality in the given contingent commodity market equilibrium we have or is an optimal couple

15. If is budget feasible for agent i, then implies that and thus that since

16. But then implies that, which contradicts. Thus is indeed optimal for agent i. Spot market clearing follows from the fact that is a feasible allocation. Security market clearing obtains since also because is feasible.

17. More general model Suppose for some number N of goods. If is differentiable,

18. The first order necessary condition for optimality of in problem is where is a scalar Lagrange multiplier. The gradient of at a choice x is the linear functional defined by

19. for any y in Thus optimality of for agent i implies that a fundamental condition

20. Since for some in, for any two goods n and m with, a well known identity equating the ratio of prices of two goods to the marginal rate of substitution of the two goods for any agent.

21. A two period model For a two period model with one commodity and S different states of nature in the second period. We can take L to be, where represents units of consumption in the first period and units in state s of the second period,

22. Suppose preferences are given by expected utility, or where denotes the probability of state s occurring, and is a strictly increasing differentiable function. For any two states m and n, we have

23. We also have (3) Suppose there are securities defined by a full rank dividend matrix d, whose n-th column is the vector of dividends of security n in the S states.

24. Convert a commodity market equilibrium into a security spot market equilibrium We can convert a commodity market equilibrium into a security spot market equilibrium in which the spot price for consumption is one for each for each and which The market value of the n-th security is

25. From, for any agent i, the market value of the n-th security is The market value of a security in this setting is the expected value of the product of its future dividend and the future marginal utility for consumption, all divided by the current marginal utility for consumption. Relation (**) is a mainstay of asset pricing models. (**)

26. Example Consider an economy in which the state of the world on any given day is good or bad. A given security, say a stock, appreciates in market value by 20 % on a good day and does not change in value on a bad day. A bond, has a rate of return of r per day with certainty. Third security, say a crown, will have a market value of if the following day is good and a value of if the following day is bad. We can construct a portfolio of shares of the stock and of the bond whose market value after one day the value of the crown.

27. The equations determining and are: where S 0 and B 0 are the initial market values of the stock and bond resp. The solutions are shares of stock and of bond.

28. The initial market value C of the crown must therefore be. The supporting argument, one of the most commonly made in finance theory, is that implies the following arbitrage opportunity.To make an arbitrage profit of M, one sells M/k units of crown and purchases M /k shares of stock and M /k of bond. The selection of M as a profit is arbitrary.

29. The value of the portfolio held will be if good day and if bad day.The selection of M as a profit is arbitrary. Given a riskless return r of 10%, we calculate from the solution for and that

30. Let denote the market value of the crown after two good days, denote its value after a good day followed by a bad day, and so on. Then the market value of the crown is After T days, we have (5) where denotes the random market value of the crown after T days and prob. of G and B days is equal.

31. We are able to price an arbitrary security with random terminal value by relation because there is a strategy for trading the stock and bond through time that requires an initial investment of and that has a random terminal value of. The argument is easily extended to securities that pay intermediate dividends. There are different states of the world at time T.

32. Precluding the re-trade of securities, we would thus require different securities for spanning. With re-trade, as shown, only two securities are sufficient. Any other security, given the stock and bond, is redundant.

33. The classical example of pricing a redundant security is the Black-Scholes Option Pricing Formula. We take the crown to be a call option on a share of the stock at time T with exercise price K. Since the option is exercised only if, and in that case nets an option holder the value the terminal value of the option is where denotes the random market value of the stock at time T.

34. Given n good days out of T for example, and the call option is worth the larger of and zero, since the call gives its owner the option to purchase the stock at a cost of K. From, we obtain

35. This formula evaluates by calculating given n good days out of T, then multiplies this payoff by the binomial formula for the probability of n good days of out of T, and finally sums over n.

36. The Central Limit Theorem tells us that the normalized sum of independent binomial trials converges to a random variable with a normal distribution as the number of trials goes to infinity. The limit of as the number of trading intervals in approaches infinity is not surprisingly, then, an expression involving the cumulative normal distribution function, making appropriate adjustments of the returns per trading interval.

37. The limit is the Black-Scholes Option Pricing Formula where The scalar represents the standard deviation of the rate of return of the stock per day.