Chapter 14 Financial Intermediation in the Continuous-Time Model

1. Chapter 14 Financial Intermediation in the Continuous-Time Model --by Zheng Zexing Finance Dept.of Xiamen University

2. Introduction • The core of financial economic theory is the study of the microbehavior of agents in the intertemporal deployment of their resources in an environment of uncertainty. • Economic organization are regarded as existing primarily to facilitate these allocations and therefore endogenous to the theory

3. In this chapter, the continuous model is used to analyze the risk-pooling and risk-sharing roles of financial intermediaries. • The focus is on the economic function of financial intermediaries rather than on their specific institutional structure.

4. Under specified condition, all possible optimal portfolios can be generated by combinations of a relatively small and select set of portfolios. • These generating portfolios have an institutional interpretation as mutual funds or investment companies • So spanning theorems provide a basis for a beginning theory of financial intermediation.

5. Section 5.5 shows, with joint log-normally distributed security prices, with or without a riskless security, only two mutual funds are required to span the set of optimal portfolios. • In the model of Section 11.7, three funds are sufficient for spanning. • Theorems 15.5 and 15.3 expands to an m-fund spanning theorem in the general case of the continuous-time model.

6. As with the theorems derived in the static environment of Chapter 2, These spanning theorems create a theoretical foundation for the role of financial intermediaries in a dynamic economic system with continuous-trading opportunities.

7. Contingent-claims analysis(CCA) has a broad range of application to the pricing financial instruments. • The contribution of CCA to the enrichment of that theory is deeper than just the pricing of financial instruments issued or purchased by intermediaries. • Contingent-claim securities with payoffs that can be expressed as functions of other traded-securities prices are called derivative securities

8. Hakansson Paradox: • In the standard model, because investor can themselves use dynamic strategies of CCA to replicate the payoff patterns to these securities, investors are indifferent as to whether or not derivative securities are created. • Hakansson paradox: CCA only provides the production technology and production cost for creating securities that are of no consequence. • Applied to the mutual-fund theorems:Investors are indifferent between selecting their portfolios form a group of funds that span the optimal portfolio set and selecting from all available securities.

9. Why do the financial intermediaries exit • Why do investors still need derivative securities? • Some types of transaction-cost structure in which financial intermediaries and market makers have a comparative advantage with respect to other investors and corporate issuers

10. Even the simple binomial model is greatly complicated by the explicit recognition of transactions costs. • So introduce a continuous-time model in which many investors cannot trade costless,but the lowest-cost transactors can. • Under this model, standard CCA can be used to determine the production costs for financial products issued by intermediaries • Unlike in the standard zero-cost model, these products can significantly improve economic efficiency.

11. 其实就是说，金融中介能够在类似无成本，可连续的交易环境中对金融产品定价，而其他投资者由于无法做到无成本连续地交易资产来复制payoff，所以又需要这些“冗余的”金融产品来规避风险。这样，也就说明的金融中介的存在必要性和重要性。其实就是说，金融中介能够在类似无成本，可连续的交易环境中对金融产品定价，而其他投资者由于无法做到无成本连续地交易资产来复制payoff，所以又需要这些“冗余的”金融产品来规避风险。这样，也就说明的金融中介的存在必要性和重要性。

12. Content • Section 14.2, using the binomial model derives the production technology and cost for creating a derivative security in the presence of transactions costs. • Section 14.3, introducing the production theory of zero-transaction-cost financial intermediaries

13. Content • Section 14.4, examine how the CCA, with general dynamic portfolio theory, can be used to measure and control the total risk of an intermediary’s entire portfolio. • Section 14.5, introduce the role of efficient intermediary in the continuous-time model. • Section 14.6, afterword about application of continuous-time model to policy and strategy issues in intermediation.

14. 14.2 Derivative-security pricing with transactions costs • In this section, we examine the effects of transactions costs on derivative security pricing by using the two-period version of the Cox-Ross Rubinstein Binomial option model as analyzed in Section 10.2.

15. $S24 $S12 $S23 $S0 $S22 $S11 $S21 Time 0 Time 1 Time 2 Tree diagram of possible stock-price paths

16. Assumption: (1)The commission rate is a fixed proportion of the dollar amount of the transaction. (2) Investors pay the ask price for the stock, ,when they buy; receive the bid price, ,when they sell. (3)There are no costs for transacting in the riskless security. denotes the return per dollar invested in the riskless security and is constant over both periods.

17. S24 S12 S23 S0 S22 S11 S21 • To rule out the possibility of arbitrage or dominance opportunities between the stock and the riskless security, the corresponding set of restrictions in the presence of transactions costs can be written as:

18. Assumption: • In determining the cost, we assume that the intermediary has no position in the underlying stock and that all stock held at the expiration date of the option is sold in the market.

19. (2)the commission rate is, (3) denote the number of shares of stock held in the portfolio at time t after adjusting the portfolio to the desired position. If , the portfolio is short shares .

20. (4) denote the amount of the riskless security held in the portfolio after the payment of the transaction costs associated with adjustments to the portfolio at time t, if ,then the portfolio borrowed . (5) denote the value of the portfolio before payment of transactions costs incurred at time t.

21. S24 S12 S23 S0 S22 S11 S21 If ， then to exactly match the payoff to the option at t=2, the portfolio composition must satisfy , in the event . , in the event is the schedule of payments to the customer at expiration and we have taken account of commissions paid on the sale of the stock in the portfolio.

22. From the match conditions ,we have that: (14.2a) and (14.2b)

23. S24 S12 S23 S0 S22 S11 S21 Because , in the event , the portfolio holdings of the stock should be reduced from the initial position , so the intermediary will incur a transaction cost of to adjust the portfolio. • The total resources required in the portfolio at time 1 to support this strategy can be written as (14.2c)

24. Form(14.2a)and (14.2b),and ,

25. S24 S12 S23 S0 S22 S11 S21 if instead , at , , then at , will equal either or , by the same analysis, we have that(14.3): (14.3a) (14.3b)

26. S24 S12 S23 S0 S22 S11 S21 Because ，in the event ，the intermediary will incur a transaction cost of to adjust the portfolio. From (14.3a) and (14.3b),the total value required at t=1 is

27. by inspection of (14.2a) and (14.3a)

28. From (14.2b) and (14.3b), The amount borrowed is independent of the level of transactions costs.

29. From (14.2c) and (14.3c),

30. S24 S12 S23 S0 S22 S11 S21 To exactly replicate the return on the option from t=0 until expiration: It follows that:

31. By substitution from (14.2c) and (14.3c),we have that: (14.4a) and (14.4b)

32. The derivation of (14.4a)

33. (14.4a)

34. The derivation of (14.4b) Substitute from(14.4a),we get(14.4b):

36. Because and , from (14.4a) and (14.4b), we have that : Hence, the presence of transactions costs causes a larger long position in the stock and additional borrowing in the replicating portfolio

37. The initial investment in the portfolio required to undertake these position (including the transaction cost ) can be written as (14.4c)

38. Because and , we have that : We thus verify that an increase in the cost of producing a call option caused by commissions charged in the stock market increases the option price charged by the intermediary.

39. Now consider a customer who would like to sell a call option to the intermediary. In this case, without cost intermediary holds reverse position. • If considering the transaction cost, the magnitudes of the positions held will not be the same because the intermediary must pay the commissions no matter which side of the transaction it undertakes.

40. We have that: Hence, the number of shares held short to hedge a long call position is fewer than the number held long to hedge a short call position

41. We have that: ] That is, the minimum price at which the intermediary would sell a call option exceeds the maximum price at which the intermediary would buy a call option.the zero-transactions-cost price the option is between the two .

42. In the competitive financial-service industry, the bid price for the call option is: the ask price for the call option is: The average of the bid and ask prices of the call option is equal to

44. That is, the average of the bid and ask prices of the option is a biased-high estimate of its zero-transactions-cost price. • Symmetry of the bid and ask prices of the stock around its zero-transactions-cost price does not imply a corresponding symmetry for the bid and ask prices of the call option .

45. Example: Exercise price is $100, the interest rate is 5% the array of stock prices is

46. Table 14.1 bid and ask call-option prices with transactions costs

47. The percentage premium of the ask price above the zero-cost price is approximate linear in and equal to ,similar results hold for the percentage discount of the bid price below the zero-cost price. Hence, the percentage spread between the bid and ask price is approximately . Although the price of the stock is much larger than the option price , the dollar spread between the bid and ask prices of the option is larger than the corresponding spread for the stock.

48. In summary , the two-period binomial model illustrates how bid and ask prices for derivative securities can be endogenously determined form the transaction-cost structure of their underlying securities • This table overstate the actual costs to intermediaries • The analysis show that the percentage spreads in the production costs of derivative securities can be many times larger than the spreads in their underlying securities

49. 14.3 Production Theory For Zero-transaction-cost Financial Intermediary • Assumption: • some agents face significant transactions costs, but that the financial intermediaries do not • An Arrow-Debreu pure state-contingent security that pays its holder $1 if a particular point in time, and otherwise pays nothing.

50. Ross,etc shows that the combination of options could be used to create pure securities and that these pure securities could be used to price derivative securities. B-S approach Option pricing derivative security Ross,etc pure security Arrow