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

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chapter 14 financial intermediation in the continuous time model

Chapter 14 Financial Intermediation in the Continuous-Time Model

--by Zheng Zexing

Finance Dept.of Xiamen University

  • 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
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.
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.
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.
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.
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
hakansson paradox
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.
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
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.
  • 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
  • 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 2 derivative security pricing with transactions costs
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.
tree diagram of possible stock price paths








Time 0

Time 1

Time 2

Tree diagram of possible stock-price paths


(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.









  • 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:
  • 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.
(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 .

(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.









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.









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










if instead , at , , then at , will equal either or , by the same analysis, we have that(14.3):











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


From (14.2b) and (14.3b),

The amount borrowed is independent of the level of transactions costs.









To exactly replicate the return on the option from t=0 until expiration:

It follows that:

The derivation of (14.4b)

Substitute from(14.4a),we get(14.4b):

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


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


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.

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.

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


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 .


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

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 .

Exercise price is $100, the interest rate is 5%

the array of stock prices is


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.

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
14 3 production theory for zero transaction cost financial intermediary
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.
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


pure security


  • Deriving the analog to Arrow-Debreu pure securities in the continuous-time model and demonstrate their application to the pricing of contingent-claim securities
Section13.2 ,we have gotten that


  • For
  • Subject to the boundary conditions:

denote the cash-flow rates paid to the holders

of the traded asset and its derivative security, respectively.

Assumption: the stochastic process governing the dynamics of the trade asset is such thatfor

let denote the price at time t of the particular derivative security with a payoff structure given by


For and

where for , and is infinite in such a way that , for any


Let denote the infinitesimal differential of the parameter . Consider a portfolio strategy that at time t purchased units of each of the continuum of derivative securities with parameter values , and . At time T,

if , (14.12)

= 0 otherwise

the cost of acquiring this portfolio at time t is


if and , then the value of the portfolio is $1 if and $0 otherwise. By the Mean Value Theorem, the cost of the portfolio at time t is ,

is the price at time t of an Arrow-Debreu state-contingent security that pays $1 at time T if and nothing otherwise.


About the boundary(1)

Consider the limiting portfolio strategy in which ,

and , and ,the value of the portfolio at time T will be $1 for all possible values of .

To rule out arbitrage between the riskless security and the derivative securities, their price must satisfy:


Because all possible payoff to each of the derivative securities are nonnegative, the no-arbitrage condition requires that, hence from(14.13), we have that is bounded function in the limit as

about the boundary 2
About the boundary(2)

If, at time t, we construct a portfolio that holds units of each of the continuum of derivative securities with parameter values , from(14.11), the value of the portfolio at T is given by

, for

However, in addition, that investment will also receive all payout to the asset, between t and T. hence, to avoid arbitrage opportunities between the traded asset and the derivative securities, their prices must satisfy



About the boundary(3)

From (14.14) and the nonnegativity of ,we have that

is a bounded function in the limit as , Therefore, for and fixed ,

is bounded in the limit as .

From derivation in chapter 13, to avoid arbitrage opportunities, must satisfy(14.9) ,with , , and

for all and .


About the boundary(4)

The boundary conditions required for a unique solution are that

is bounded and that (14.11) is satisfied.

Thus, under the hypothesized conditions 1-6 of chapter 13, we have determined the prices of a complete set of pure state-contingent securities, where the state space is defined by the price of the traded asset and time.



The connection between these pure state-contingent securities and the theory financial intermediation can be made apparent by examining the general class of derivative securities with pay off structure given by (14.10a)-(14.10c), with and for all t.

From the Green’s functions method of solving linear differential equations, the solution to (14.9), subject to these boundary conditions, can be written as


Once an intermediary has determined the production costs for the complete set of state-contingent securities, it can use simple quadrature to calculate the costs for any derivative security.

This equation is a powerful tool for the evaluation of derivative security cannot,however, be applied to securities for which there is a positive probability that either or for some t<T .


Let denote the early-exercise schedule of stock prices such that the put option is exercised at t if .

If is continuous function of t, then a application of (14.15) must suggest that the put price can be written as



However, this equation gives an incorrect evaluation. For time and such that ,the events that

and are not mutually exclusive. This equation implies that, if both events occur, the put holder will receive payments of at time , and at time . But , of course, if the put exercised at , it cannot also be exercised at .

Thus, when early exercise is possible. This equation overstates the value of the put.

The American put option has a path-dependent payoff structure.
  • The lack of equivalence between the path-dependent and path-independent causes(14.16) to fail as a valuation formula.
The pure-securities approach can be modified to accommodate derivative securities with path-dependent payoffs.
  • To do this one can constructs a set of state-contingent securities that are also path dependent by replacing the payoff function(14.11) with the condition that
connection between the pricing theories of option and pure securities in the continuous time model
Connection between the pricing theories of option and pure securities in the continuous-time model

Consider an option investment strategy --butterfly spread

a long position ; a short positions in two options with exercise price ;a long position in an option with exercise price .

The payoff function at time T to one unit of this spread is given by


For , let denote the payoff function at time T to a portfolio containing units of the butterfly spread. It follows that can be written as


At time t, for , the cost of acquiring this portfolio is


From (14.17), we have that the limit as of is , which is exactly the payoff function of a pure state-contingent security, as given in

Therefore, to avoid arbitrage, the cost of acquiring the portfolio must, as , approach , it follows that

For European call- option, in B-S formula, taking the second derivative of their formula with respect to the exercise price. We have that:



We begin with the special case leading (14.19) in which . As described in chapter 10, the Cox-Ross procedures replaces the dynamics for the actual asset returns with is , and .

Let denote the conditional probability density function for , in the case in which and are constants, the distribution of , conditional on , is log-normal, and therefore





Hence, the price of a pure state-contingent security is equal to the product of the price of a discount bond that pays $1 at time T and the Cox-Ross risk-neutral probability density function.


With , ,and for all , we have from (14.15) and (14.21) that the price of a general derivative security can be written as


The second line is exactly the Cox-Ross option pricing formula applied to a general derivative security.

general case
General case:

Consider now the general case in which ,

with and . In chapter 3 , it was shown in (3.42) that the conditional probability density function for this diffusion process satisfies


subject to the boundary conditions bounded for all and .



by substitution of for in (14.9) with , we have that:


moreover, is bounded for all and because is so bounded.

Further, from (14.11)
  • By inspection of (14.23) and (14.24), together with their respective boundary conditions, we have that


Therefore, (14.21) and (14.22) obtain in the general case.

changing numeraire
Changing numeraire

In section 10.2 and 10.3, it was convenient to choose as numeraire, , the price at time t of one unit of a portfolio that initially invest $1 in the riskless security and retains all earnings.

Selection of that particular numeraire here leads to an interpretation of pure-security prices as probability density functions.

in this price system
In this price system,

The price of the traded asset is given by

And its dynamics can be expressed as

Because the interest rate in this system is always zero, the probability density function for the associated Cox-Ross risk-neutral equilibrium price of the asset, satisfies (14.23),with .



If denote the price of a pure state-contingent security with payoff function at time T given by for , then will satisfy (14.9),with and . but, by inspection, (14.23) and (14.9) are identical equations if and . Hence, we have that


Thus, in a price system normalized by the cumulative return on the riskless security, the prices of pure state-contingent securities are equal to the corresponding Cox-Ross risk-neutral conditional probability density functions.

14 4 risk management for financial intermediaries
14.4 risk management for financial intermediaries
  • A custom who buys a warranty on his new car from an automobile manufacturer wants the repair paid for in the even that the car is defective.
  • In fact, he has a contract that pays for repairs in the joint contingency that the car is defective and the automobile manufacturer is financially solvent.
14 4 risk management for financial intermediaries1
14.4 risk management for financial intermediaries

The success of an intermediary depends not only on charging adequate prices to cover its production costs,but also on providing adequate assurancesto its customers that promised payment will be made.

Hence, an important part of the management of financial intermediaries is the measurement and control of the risk exposures created by issuing their financial products.

CCA provides the means for intermediaries to create custom financial products in an “assembly-line”fashion.
  • CCA can be used to significantly reduce the difficulty of measurement of the risk exposure

Consider an intermediary that issues derivative securities on different traded assets. Let denote the price of asset with dynamics specified as


where and are the expected rate of return and standard deviation of return respectively. And is the payout rate on asset


For and , let denote the solution to(14.9) subject to boundary conditions(14.10) that are appropriate for the th type of derivatives security, which is contingent on the price of traded asset .

if at time .then is the unit production cost to the intermediary and it is also the value to the intermediary of owning one unit of derivative security .

From ito lemma and (14.9), we have that the dollar return to the intermediary from owning one unit of the derivative security between and can be written as


Owning one unit of derivative security between and produces the same dollar return as holding units of traded asset and of the riskless security.
If at time the intermediary owns units of this derivative security,

Adding up the exposures form all types of derivative securities , we have that the total exposure from these positions between and is equivalent to owning units of asset and dollars in the riskless security.


If denote the net value if the financial security holdings of the intermediary at time ,then we have that

The intermediary can make this conversion for all derivative securities in its portfolio.


Where is the dollar amount of the riskless security held at and is the number of units of traded asset held at

from 14 26 14 27 14 28 between and the dollar return to the intermediary s portfolio can be written
From (14.26), (14.27), (14.28), between and ,the dollar return to the intermediary’s portfolio can be written


We have that the intermediary’s risk exposures to the different derivative securities and the traded assets can be expressed in terms of risk exposures to the traded assets alone.

The derivation of bid-ask price spreads in section 14.2 overstates the effect of transactions costs on an active financial intermediary. Because it assumes no inventories of either traded assets or other derivative securities.
  • To perfectly hedge its entire portfolio, the intermediary need only hold inventories of traded assts that are sufficient to hedge the net exposures created by its financial product
  • In (14.29), a perfect hedge require only that
Consider an intermediary with current exposures that are completely hedged. Now, suppose it issues call option on asset with an equivalent exposure of and issues put options on asset with an equivalent exposure of ,

Because and , The dollar transaction in asset required to jointly hedge these new exposures, is less than

, the dollar amount required if each new exposure is treated as an isolated transaction.

In the absence of new business, to maintain a hedged position against its current inventory of financial products, transactions in traded assets are required. If the policy is to maintain , then by Ito’s lemma we have that:


the volume of transactions induced by unexpected changes in trade-asset price is proportional to the absolute magnitude of .

hence, the size of transactions caused by changes in traded-asset prices can be minimized if the intermediary can maintain a hedged portfolio with close to zero.


If we neglect the effect of new business, then from the definition of ,we have that


If the production cost for a derivative security of type is a strictly convex function of ,then . if it is a strictly concave function, then .

Thus, by offering a mix of “convex” and “concave” products ,the intermediary can reduce the volume of transactions in traded assets that is required to maintain a hedged portfolio.


is called the “delta” of derivative security

is called the “gamma” of derivative security .

The gamma of the intermediary’s aggregate position, characterizes the degree of local convexity or concavity of the position with respect to .

If, however, this locally hedged position is such that , then the intermediary will gain form any large (“nonlocal”) changes in , whether up or down.


If denotes the value of the intermediary’s position, then by Taylor’s theorem,

where is some number satisfying . If , then . Hence, the change in the value is positive(negative), if

Similarly, if , the intermediary will lose from any large change in . Thus,for ,the gamma of the aggregate position measures the intermediary’s exposure to large moves in .

Our assumptions about the dynamics of traded-asset prices are such that nonlocal movements cannot occur. Such assurances cannot, of course,be given for the behavior of real-world asset prices

So, the assessment of one’s exposure to convexity is an important element of risk management for practitioners.

alternative diversity
Alternative: diversity
  • Perfect hedging of an intermediary’s entire portfolio can be suboptimal policy, because of transactions costs.
  • Diversification can be used as a risk-management alternative to transacting in the traded assets.
  • Two ways:
    • Diversity among the products
    • Diversity among the underlying traded assets

From(14.29), the instantaneous variance of the dollar return on an intermediary’s portfolio can be expressed as

Among traded assets with , a product mix that leads to will also reduce the volatility.

For products mixed that lead to , the intermediary can reduce portfolio variance by offering these products on traded assets with relatively small correlations among their returns.

portfolio insurance
Portfolio insurance
  • One common form of regulation is to require the intermediary to maintain a minimum level of net worth of capital if it is to continue in business
  • The strategy is to maintain a put-option position on the entire portfolio of financial assets.
  • By selecting an exercise price larger than the minimum capital requirement and a time until expiration of sufficient duration, the intermediary can protect itself against violation of the regulation.
  • This strategy is call insured-equity or protective-put, but most commonly called portfolio insurance
To implement, the intermediary could purchase insurance form another firm.
  • As a minimum-cost transactor, it is more cost-effective to create its own portfolio insurance by using the CCA method.
    • the intermediary retains the mark-up or spread
    • Self-insuring allows greater flexibility in the intermediary’s investment policy.
The main disadvantage :
    • the self-insurer retains the risk
      • the dynamic hedging strategies fail to replicate the payoffs to a put option(nonlocal movement) on the portfolio.
      • nonlocal movements cannot occur. But if it happen, then use to create the desired gamma exposure.
14 5 on the role of efficient financial intermediation in the continuous time model
14.5 On the role of efficient financial intermediation in the continuous-time model
  • The continuous-time model provides a rich analytical framework for developing a theory of financial intermediation
  • The existence of a well-functioning financial intermediation sector can do much to justify the continuous-time model as a relevant mode of analysis for the study of general financial economic behavior.
Real-world financial markets in well-developed economies are open virtually all the time, and therefore the assumption of continuous trading is a reasonable approximation
  • In the real word, individual investors face significant transactions costs for trading.
  • a question about the the robustness of the model prescriptions as an approximation to feasible behavior in the real world.
Under the existing of efficient financial intermediation , we show that:
    • all investor can achieve the identical consumption-bequest allocations that they would have chosen if they could have traded continuously without cost.
    • the aggregate demands for traded assets are the same as the ones that obtain in the frictionless-market models of Chapters 4-6.
  • Thus, the consumption and asset demands derived in these earlier chapters are shown to apply in a more realistic model with transactions costs.


Let denote the consumption rate of the investor

Let be the fraction of the investors’ portfolio allocated to the risky traded asset.

If denotes the value of the investor’s portfolio at time t, then from (13.4),we have that


Where and are assumed to be functions of and only.

To derive the replicating portfolio for a particular derivative security, we required that

We then showed that, to rule out arbitrage, for all .where is the solution to the partial differential equation (13.1)



Derivation process:


feasible portfolio

a particular derivative security

Instead :

feasible derivative security


particular investor’s optimal portfolio

Suppose that a blueprint for constructing such a security can be found and that the price charged by a financial intermediary for the security dose not exceed the investor’s budget constraint.
  • Then, as alternative to continuously trading assets to achieve his optimal consumption-bequest allocation, the investor can buy the appropriate derivative security from the intermediary and achieve his optimal allocation without ever trading again.
If the information-acquisition and transactions costs are smaller for the intermediary than for the investor, then the availability of such derivative securities makes it possible for the investor to achieve welfare-improving allocations that would not otherwise be feasible if the investor had to trade directly in the asset markets.
we use the Cox-Ross approach to the optimal consumption-investment problem as analyzed in chapter 6.

Let denote the price per share of a mutual fund that pays no dividends to its shareholders.



let denotes the investor’s optimal consumption rate at time .

let denotes his optimal bequest of wealth.


Then , as shown in Chapter 6, there exist functions and such that:



where and depend parametrically on , , and the investor’s initial wealth . Moreover, for any feasible consumption plan, and are nonnegative, and for all .

To match the optimal consumption-bequest allocation of the investor,the terms of the derivative security must be that

Continuous dividend rate:

for t<T

Final payment:

at the time T


subjection to the boundary conditions




As Chapter 13, define to be solution to the linear partial differential equation



Let denote the value of a portfolio that makes continuous payouts at the rate and follows an investment strategy of allocating dollars to the growth-optimum portfolio, dollars to the risky asset, and the balance of the portfolio to the riskless asset, where



it follows that the dynamics of can be written as



because is twice-continuously differentiable , we can use Ito’s lemma to express the stochastic process for



but satisfies (14.35), Hence we can rewrite (14.39) as



Let . From (14.38) and (14.40), we have that


the solution to (14.41) for any is ,

Therefore, if we choose the initial investment in the portfolio so that then and

for all . It follows that for all





(1) Thus, we have derived a dynamic portfolio strategy that exactly replicates the payoffs of a derivative security that makes continuous payments at the rate for and has a final payout of at time .

(2) The production technology for creating this security is given by (14.37a) and (14.37b)

The payoffs of the derivative security exactly match the consumption-bequest allocation of the investor’s optimal continuous-trading strategy.
  • Hence, the investor can achieve this allocation by simply buying the security, provided that the price of the security does not exceed his budget constrain.
As discussion in Chapter 6, C-H have shown that the investor’s optimally invested wealth can be expressed a function of and the function satisfies our partial differential equation(14.35) with boundary conditions (14.36). Therefore, in particular, .
  • Then, purchase of the derivative security by the investor is thus always feasible.
information acquisition
  • Information needed must be reasonable.
  • For intermediary, it need only know the schedules and . It doesn\'t’t require monitoring of the investor’s endowment or creditworthiness.
  • For investors, the information needed is how to determine the function , but this information is no larger than the one he would use to solve for his optimal consumption and investment program in the absence of such intermediation opportunities.
default risk
Default risk:
  • Two important issues:
    • Derivative securities with identical promised that are issued by different intermediation are no longer perfect substitutes----for investor, they need additional information about the creditworthiness of the issuing intermediary.
    • It reduces the functional efficiency of the derivative securities. Which are contingent on and can match their optimal consumption-bequest allocations, but now it is also contingent on the value of the issuing intermediary and its entire structure of outstanding liabilities
the role of financial intermediation in making the continuous model more robust
The role of financial intermediation in making the continuous-model more robust
  • Assume an institutional environment in which financial intermediaries pay no transactions costs and consider an investor who must pay significant transactions costs.

Cannot trade directly without cost

Optimal consumption-investment program with G and H



Buy derivative securities to achieve the optimal allocation

Aggregate its individual exposure and hedge