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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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--by Zheng Zexing
Finance Dept.of Xiamen University
(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.
(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 .
,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.
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
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:
Substitute from(14.4a),we get(14.4b):
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
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.
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
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 .
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
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.
denote the cash-flow rates paid to the holders
of the traded asset and its derivative security, respectively.
let denote the price at time t of the particular derivative security with a payoff structure given by
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.
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
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
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
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 .
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.
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.
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.
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
For European call- option, in B-S formula, taking the second derivative of their formula with respect to the exercise price. We have that:
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.
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.
Therefore, (14.21) and (14.22) obtain in the general case.
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.
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.
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.
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 .
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
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.
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.
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 .
So, the assessment of one’s exposure to convexity is an important element of risk management for practitioners.
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.
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.
We then showed that, to rule out arbitrage, for all .where is the solution to the partial differential equation (13.1)
a particular derivative security
feasible derivative security
particular investor’s optimal portfolio
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 .
Continuous dividend rate:
at the time T
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)
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