Computing Option Reservation Prices in Incomplete Markets

Copyright IS) IFAC Computation in Economics. Finance and Engineering: Economic Systems. Cambridge. UK. 1998

COMPUTING OPTION RESERVATION PRICES IN INCOMPLETE MARKETS Claus Munk 1

Department of Management, Odense University, DK-5230 Odense M, Denmark

Abstract: With constrained portfolios contingent claims do generally not have a unique price that rules out arbitrage opportunities. Earlier studies have shown that, when there are constraints on the hedge portfolio, a no-arbitrage price interval for any contingent claim exists. Here, the more realistic case, when the constraints are imposed on the total portfolio of each investor, is defined, and reservation buying and selling prices for contingent claims are defined. The paper discusses how these reservation prices can be computed numerically and provides examples. Copyright 4l19981FAC

Keywords: Finance, stochastic control, dynamic programming, Markov decision problems, numerical solutions

any contingent claim. It seems more reasonable to consider constraints on the total portfolio of the investor. Since the investors can have diHerent preferences and endowments, and hence different optimal portfolios without considering engaging in contingent claims trading, they will typically face different constraints on their hedge portfolio. Consequently, only investor-specific (or to be more precise: preference- and endowment-specific) bounds on the price of the contingent claim can be given. In this paper, the concept of a reservation buying and a reservation selling price is introduced. The reservation buying price is the highest price, which the investor is willing to pay for a unit of the contingent claim. The reservation selling price is the lowest price, the investor is willing to accept for writing a unit of the contingent claim.

1. INTRODUCTION The pricing and hedging of contingent claims is a matter of immense interest and importance to the financial industry. Most research has focused on complete markets, in which any contingent claim can be replicated perfectly by some selffinancing dynamic portfolio strategy and, under a no-arbitrage condition, the price of the contingent claim must equal the cost of investing in this portfolio. 2 However, it is broadly recognized that financial markets are not perfectly complete. Asset prices are affected by non-tradable factors, and the investors are often restricted in the portfolios they are allowed to hold. Both facts imply that, in general, it is not possible to replicate a contingent claim perfectly.

In many cases the investor-specific bounds will presumably be much tighter than the typically wide arbitrage-based bounds. For instance, an investor writing a European call option on a stock may find it optimal to hold a share of the stock in question in any case and, therefore, she does not have to invest in a superreplicating portfolio to

Earlier studies 3 have shown that, when all investors face the same constraints on their hedge portfolio, a no-arbitrage price interval exists for 1 Financial support from the Danish Research Councils for Natural and Social Sciences is gratefully acknowledged. 2 See (Black and Scholes, 1973; Harri80n and Pliska, 1981). 3 See (Munk, 1997) for an overview.

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insure herself against bankruptcy. Of course, she will require some compensation for the promise to pay a random amount at some future point in time.

An admissible consumption process at time t is a progressively measurable process c, satisfying a standard integrability condition, and the nonnegativity constraint c(u) ~ 0, 'Vu ~ t. The set of admissible consumption processes at time t is denoted by e(t). c(u) is the rate of consumption at time u. IT an investor starts out at time t with a wealth x > 0, follows a trading strategy (a, 8) E P (t, /C) in the primary assets and a consumption process c E e(t), her wealth X:z: ,t(u) x:f'C(u) a(u) + 8(u)Tl will follow the process '

Portfolio constraints arise naturally in the case of real options, which in many cases cannot be perfectly replicated by a dynamic trading strategy in traded assets. The reservation buying price is then the adequate valuation tool for such options. The reservation buying price can also be used for valuing executive stock options. To make the incentives of an executive manager compatible with the preferences of the owners of the company, the manager's compensation scheme often involves options on the stocks of the company. To ensure that the manager cannot trade away the risk imposed this way, the manager is not allowed to trade in the underlying stocks or in stock derivatives. Hence, the manager faces portfolio constraints.

=

X:z:,t(U)

=

= + X

lU

[a(s)r(P(s))

- c(s)] ds +

lU

+ 8(s) Tb(P(s))

8(s) T u(P(s)) dw(s) .

A triple (a,8,c) is called a /C-admissible primary trading and consumption strategy at time t with wealth x> 0, if (a,8) E P(t,/C), c E e(t), and X:z:,t(u) ~ 0, U ~ t. The set of such strategies is denoted by At A(x, P, t, /C) .

=

We assume that the investor at any time t seeks to maximize the expected remaining life-time utility U(c, t) = E:z:,p,t [JtOO e- 13 (B-t)U(c(s)) ds], where U : Il4 -+ JR is a strictly concave, increasing function measuring the utility from consumption, and E:z:,p,t[-] denotes the expectation operator given X(t) = x and pet) = P. IT the investor only considers investments in the primary assets, her maximization problem is

2. THE MODEL AND BASIC DEFINITIONS Imagine a financial market where d + 1 primary assets are traded continuously. There are d risky assets with prices pet) = (P1 (t), ... ,Pd (t))T having time-homogeneous dynamics given by

dP(t)

= diag(P(t)) [b(P(t)) dt + u(P(t))] dw(t),

where w is a d-dimensional standard Wiener process on a complete probability space (n, F, JP). We let F = {Ft}t>o be the filtration generated by w . The remainiiig asset is an instantaneously riskless asset called the savings account with price poet) satisfying dPo(t) = r(P(t))Po(t) dt, where r is the short-term interest rate process.

Vex, P, t)

=

sup

U(c, t).

(1)

(a,8,c)EA.

Due to the time homogeneity, we have Vex, P, t) vex, P, 0) == vex, P) for all t E 114 .

=

Now suppose that the investor also has the opportunity to buy or sell (write) units of some contingent claim at time t. We assume that the claim expires at time T, is ofthe European type, and its terminal value is a the realization of some non-negative random variable t has a wealth of x, follows the trading strategy (a,8) in the primary assets, and consumes according to the consumption rate process c, her wealth X; rC·) will follow the process '

We assume that the d x d volatility matrix u(P) has full rank d for all values of P. This implies the dynamic completeness of the market, at least potentially. As we shall see below, the market can be incomplete due to restrictions on the set of admissible trading strategies. A trading strategy in the primary assets is a pair (a,8) of progressively measurable processes, where a( u) E JR denotes the dollar amount invested in the savings account at time u, and 8(u) = (8 1 (u), ... ,8d (u))T with 8i (u) denoting the dollar amount invested at time u in the i'th risky asset. Let /C be a non-empty, closed, convex subset of JRd+1. A trading strategy (a, 8) is called /C-admissible at time t, if (a(u),8(u)) E /C, and standard integrability conditions are satisfied. /C is called the portfolio constraint set. The set of /Cadmissible trading strategies at time t is denoted by P(t,/C).

X;,rCU) = x

+

l

- c(s)] ds +

u

[a(s)r(P(s))

l

+ 8(s)Tb(P(s))

u

8(s) T u(P(s)) dw(s)

+ c
T

~ u.

IT the unit buying and selling price of the contingent claim is z, her wealth will decrease by cz

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at time t due to the investment in the contingent claim: X;.t(H) = x - gz.

While (1) is of the well-known infinite horizon controlled diffusion type, (2) is seemingly very complex, due to the jumps in wealth at the time of investment in the contingent claim and at its expiration date. Suppose the investment in the contingent claim takes place at time t < T. Between time t and the expiration date T, the wealth of the investor follows a controlled diffusion process. At time T, when the contingent claim expires, the wealth of the investor will jump by gcpP(T). The dynamic programming property implies that

The question is at which prices the investor is willing to buy, resp. sell, the contingent claim. Suppose, the investor is restricted to a buy and hold position of g units of the contingent claim at a unit price of z. Given g and z, a triple (a, 0, c) is K-admissible with time t < T wealth x and prices P, if (a, 0) E P(t,K), c E C(t), and X;.t(u) ~ 0, Vu ~ t. The set of such triples is denoted by ~.z = Ae.z(x, P, t, K), and the value function of this problem is

Ve.z(x,P,t) =

sup

U(c,t).

Ve.z(x, P, t+) =

(2)

(a.9.c)E.A~··

Note that Vo.z(x, P, t)

sup

= V(x, P, t).

(a.9.c)E.A~··

Starting with wealth x and prices P at time t < T, the investor will be willing to sell one unit of the contingent claim (cp, T) at a price of z, if

V-1.z(x, P, t)

~

V(x, P, t),

In the remaining part of the paper, it is assumed that the utility function is of the CRRA type U(c) = c'Y with 0 < 'Y < 1, that there is a single risky asset (d = 1), and that the market coefficients b, IT, and r are constant. Under these assumptions, the dynamics of the wealth process between the date of the purchase/sale of the contingent claim and its expiration date is dX(t) = [r X(t) + O(t)(b - r) - c(t)] dt + O(t)lT dw(t), and the dynamics of the price of the risky asset is dP(t) = P(t) [bdt + lTdw(t)] . Since the price does not enter the wealth dynamics, wealth is the only state variable for the no contingent claim problem. For the fixed g problem, however, the risky asset price must be included as a state variable due to its influence on the terminal reward through the payment gcp(P(T» upon expiration of the claim.

Conversely, the investor will be willing to buy one unit of the contingent claim (cp, T) at a price z, if ~

V(x, P, t),

i.e., the investor is better off sacrificing the amount z of initial wealth for a random payment cp(P(T» at time T. The reseroation buying price of the investor at time t is defined as the maximum price, the investor is willing to pay for the claim, i.e.

g(x,P,t)

= sup{z I V1.z(x,P,t)

~

(3)

There is also a jump in wealth at time t: Ve.z(x, P, t) = Ve.z(x -gZ, P, t+). We can consider Ve.zL·, t+) as the value function for a finite horizon controlled diffusion problem with a running reward given by U (c, s) and a terminal reward given by the function (X, P) I-t V(X +gCP(P), P) . Note that the terminal reward function is itself the value function for a no contingent claim problem.

= inf{z I V-1.z(x, P, t) ~ V(x, P, t)}.

V1.z(x, P, t)

t

+ e-P(T-t)V(X;.t(T-) + gcpP(T), P(T»].

i.e., if the wealth transfer z now is sufficient compensation - in terms of utility - for committing to pay cp(P(T» at the expiration of the contingent claim. The reseroation selling price of the investor at time t is defined as the minimum price, the investor will accept, i.e.

G(x, P, t)

Ez.p.t [iT e- P(8-t)U(c(s» ds

V(x,P,t)}.

Of course, the reservation prices depend on the contingent claim (cp, T), the portfolio constraint set K, and also on the preferences of the investor. From the concavity of the utility function U, it follows that G(x, P, t) ~ g(x, P, t) for all (x, P) .

For this simple model, the value function for the no contingent claim problem and the terminal reward function for the fixed g problem are known in closed form, when the only portfolio constraint is the non-zero wealth constraint, cf. (Merton, 1969).4 For more general constraints they have to be computed numerically. The appearance of the random variable cp(P(T» in (3) spoils any hope of solving the fixed g problem

3. COMPUTATIONAL ASPECTS To compute the reservation buying and selling prices and the corresponding hedging policies, two types of stochastic control problems are to be solved: The no contingent claim problem (1) associated with the situation, where the investor only invests in primary assets, and the fixed g problem (2) for g = ±1 associated with the purchase/sale of one unit of the contingent claim.

4 Assuming A > 0, where A = (fJ - r-y)/(1 - -y) -y(b - r)2/[2(12(1 - -y)2). the value function is given by V(x) A"Y- 1 x"Y, and the optimal controls are given by c(x, t) Ax, 8(x. t)/x (b - r)/(12(1 - -y».

= =

13

=

analytically. Therefore, to compute the reservation prices, generally both a finite-horizon and an infinite horizon controlled diffusion problem have to be solved with numerical methods.

eo 50

:30 "l5

~

"'20

Res. buy

50

eo

70

eo 90 100 110 120 130 Price oI ... ..-tyIng rioky _ _

140

150

Fig. 1. The reservation prices compared with the Black-Scholes price starting with a wealth of x = 1000, when there are no portfolio constraints.

In the following sections, time 0 reservation prices for a European call option on the risky asset are studied. Hence, cp(P(T» = (P(T) - X)+, where X is the exercise price. The complete markets price is the Black-Scholes price G{P) = PN(T7+) Xe-rT N(T7-), whereT7± = log(p/[Xe-rT])/[erVT]± ~erJT, and N is the cumulative normal distribution function. The parameter values are taken to be: f3 = 1, b = 0.1, r = 0.05, er = 0.3, 'Y = 0.5, X = 100,T = 0.5.

50 BS price

<10

~

Sal.' . 50

:30 0

.,

Buy•• = 50

~2O

Sal.'. 100 Buy •• _ 100 50

60

70

eo 90 100 110 120 130 Price 01110 ~ rioky __

1<10

150

Fig. 2. The reservation prices compared with the Black-Scholes price starting with a wealth of x, when there are no portfolio constraints.

4. LIQUIDITY CONSTRAINTS In this section, the only constraint on the portfolios is the liquidity (or no-bankruptcy) constraint: Wealth has to stay non-negative, i.e. K = ]R2 . In Figure 1 the numerically computed reservation selling price G(x, P) and reservation buying price g(x, P) are compared with the Black-Scholes price G(P) for an investor with an initial wealth of x = 1000. Over the entire range of P's shown in the figure, the reservation selling price is slightly higher than the Black-Scholes price, which again is slightly higher than the reservation buying price.

will be more willing to consume out of her wealth before that date. She will even accept a relatively low price for the option, since she is desperate for present consumption. If she buys a (deep-in-the-money) call with low initial wealth, there will be a considerable probability that she defaults, before she can cash in the prospective exercise value of the option. Hence, she will not hedge the option completely, and she is willing to consume less than before to increase the probability of getting the exercise payoff. The more she pays for the option, the higher the probability of going bankrupt before the expiration date. Therefore, her reservation buying price is relatively low compared to the case, where her initial wealth is comparatively high. Her reservation buying price is anyhow bounded from above by her initial wealth. Both her reservation selling and her reservation buying price are significantly smaller than the Black-Scholes price.

If the initial wealth is of the same magnitude as the price of the underlying, the picture is different. Figure 2 shows the reservation selling and reservation buying price of the investor starting with a wealth of x = 50 and x = 100, respectively, compared to the Black-Scholes price. For relatively high prices of the underlying risky asset, the reservation prices differ significantly from the Black-Scholes price. The lower the initial wealth x, the higher the difference. When the investor sells a (deep-in-the-money) call with low initial wealth, there will be a considerable probability of default at the expiration date of the option. Therefore, she will not hedge the option completely, and she 5

El

~
The Markov chain approximation approach is a well-documented and intuitive technique for numerical solution of such problems. 5 The main idea of the method is to approximate the state variables of the problem with a discrete time, discrete state Markov chain, and define a control problem for this Markov chain, such that the solution (the value function and optimal controls) is a good approximation to the solution of the original continuous time, continuous state controlled diffusion problem. For details on the implementation, see (Munk, 1998).

5. BORROWING CONSTRAINTS Suppose now that the investor is not allowed to borrow funds, so the amount invested in the risky asset is bounded from above by her wealth:

See, e.g., (Kushner and Dupuis, 1992; Munk, 1997).

14

1.35 , - - - - - - - - - - - - - - - , 1.3

r~

... . _. _.... _.. _. . . _...... ... . ~-

.:--:

0.8

~.- I~ I

:_:""::lII ••::0.........- .••• - • • • • •• ••• -• •- . -••-.-..•• -. -••

1.05

0.9

~~--+

i

O7 •

EJ

~0.6

BS DeIIII

8.0.5 04

• •-1

'

.. - _ . . - .. . . . . . . . . . _ . . . . - - - - - - " . -

Sal h.. Xzl000

0.3 . 0.2 0.1

1+--+-r-~-~~~-~_+-r-~

50

60

70

80 80 100 110 120 130 Price 01 the ~rtoIcy asset

140

150

50

60

70

80

90

100

110

120

130 140 150

Priceol"~r1IIcy_

Fig. 3. The fraction of wealth optimally invested in the risky asset, when one call option is written at the reservation selling price, and there are no portfolio constraints.

Fig. 4. The number of units of the underlying risky asset in the hedge of a written call option, when the investor faces a borrowing constraint. 4.0 % - . c - - - - - - - - - - - - - - - ,

O(t) ~ X;(t), Vt ~ O. The optimal risky investment without portfolio constraints is O(x, t) = (br)/(a 2 (1 - 'Y»x ::::: 1.11x, cf. fn. 4. Therefore, the

3.5%

t

i

borrowing constraint is binding. As expected, the numerical computations show that the optimal investment policy under the constraint is to invest the entire wealth in the risky asset, i.e. to be at the boundary of the set of admissible policies. The constrained optimal consumption rate is not substantially different from the unconstrained rate, and the value function is only slightly reduced due to the borrowing constraint.

. . .. .

••..• - . . . . . . . . . . . . . . . . . . . . .. . . . . .

3 .0% 2 .5%

.s;2.0%

1

1.5%

1.0%

0.5%

- - - _ ..• _ . . . . -' - _ •. . ..

O.O%+---O---+--~--+---+--~-~

80

90

100 110 120 130 Price 01 the ~ rtoIcy_

140

150

Fig. 5. The percentage increase in the reservation selling price due to the borrowing constraint.

When the investor writes an option, she would like to buy a fraction of the risky asset to hedge his position. In the unconstrained case (Section 4), she buys a number of units of the risky asset equal to the Black-Scholes Delta N(TJ+)' except for combinations of a low wealth and a high price of the underlying, where the results are influenced by bankruptcy considerations. In the following discussion such combinations are ignored to separate the bankruptcy effects from the portfolio constraint effects. Figure 3 shows the fraction of wealth optimally invested in the risky asset, O-l,G(z,P)(x , P,O)/x, when there is no borrowing constraint and the investor has written a call option at her reservation selling price G(x, P). Obviously, the borrowing constraint is even more restrictive, when the investor has written an option.

unconstrained case, the investor does not borrow funds to finance the purchase of the underlying asset, which hedges the writing of the call, because it is not allowed. Obviously, it is impossible to hedge the written option perfectly, and, therefore, the investor will require a higher price for the option before she is willing to sell it, i.e., the reservation selling price will be higher in the constrained case than in the unconstrained case. In Figure 5, we have graphed the percentage increase in the reservation selling price relative to the unconstrained case. As it can be seen from Figure 3, the borrowing constraint is more restrictive, the lower the initial wealth x is. Therefore, it is to be expected that the increase in the reservation selling price relative to the unconstrained case is decreasing in initial wealth, and this is exactly what the results show.

The hedge portfolio of the option writer consists of G(x,P)/P units of the underlying risky asset, i.e. the investor is only able to increase her risky investment by exactly the proceeds from selling the option. Figure 4 depicts the number of risky assets in the hedge portfolio of an investor with initial wealth x = 1000. The Black-Scholes Delta is also shown in the figure. As in the unconstrained case, the optimal consumption rate, when the investor has written an option, is not significantly different from the no-options case. Contrary to the

When the investor buys the option, she wants to hedge her position by selling a fraction of the underlying risky asset. The unconstrained optimal risky investment, when buying the call, will therefore be lower than in the no-options case. Figure 6 shows the fraction of wealth optimally invested in the risky asset, Ol,9(Z ,P) (x, P)/x, when one call is bought at the reservation buying price g(x, P), and there is no borrowing constraint.

15

12.---------------------------~

1.15

o~~~~~----------------. ~. 1

••• . • . •. • • . ••• • • • ••••• •• • •• • • • • • • •• • •••

~2

1.1

~.3

BShadga

i~·4

'1. 05 ! ill 1°·95

1

~. 5

f~·6

Buy 1\., X=8OO

~.7

0.9

~.8

0.85

BuyI\., x-I200

~.9

· 1 +--+--+--+---+--+--+---+--+---'i~ 50 60 70 110 110 100 110 120 130 140 150 PrIce oI1he ..-.yIng risky _

0.8 +--+---+---+---+---+----.,f---+---+---+----l 50 60 70 110 110 100 110 120 130 140 150 PrIce oI1he ~ rIsky_

Fig. 6. The fraction of wealth optimally invested in the risky asset, when one call option is bought, and there are no portfolio constraints.

Fig. 7. The number of units of the underlying risky asset in the hedge of a bought call option, when the investor faces a borrowing constraint. 12%.---------------------------~

The lower the initial wealth and the higher the price of the underlying, the less restrictive is the constraint. In fact, for combinations of relatively low initial wealth and relatively high prices of the underlying, the amount optimally invested in the risky asset, will be lower than the initial wealth. 6

1.0%

ifo.8% ,1;;0.6%

1°·4%

For those combinations of initial wealth and risky asset price, where the optimal risky investment, when buying a call, is at the upper bound, the hedge portfolio of the investor consists of selling g(x, P)/ P units of the underlying asset. The proceeds from this sale finance the purchase of the call option exactly. Consider Figure 7. For an initial wealth of x = 1200 and the range of prices shown, it is optimal to invest the entire wealth in the risky asset, even when the investor is long one call. The graphs corresponding to an initial wealth of x = 400 and x = 800 follow the x = 1200 line until a certain risky asset price. Above this price, the portfolio constraint is not binding, when buying an option, and therefore the number of risky assets sold in the hedge exceeds g( x, P) / P. Still, it will be smaller than the Black-Scholes Delta. The optimal consumption rate is again practically unchanged relative to the unconstrained problem.

'.

... - .......... . ...... . . . ... . ..... _.- .. . .......... '~ ""'"

_

................. -. - ..

o.O%L--I----+-....:==~===+==+=~ 110

90

100

110

120

PrIce oI1he InIar1ytng risky _

130

140

150

Fig. 8. The increase in the reservation buying price due to the borrowing constraint. 6. REFERENCES Black, Fischer and Myron Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81(3),637-654. Harrison, J. Michael and Stanley R . Pliska (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11, 215260. Kushner, Harold J . and Paul G. Dupuis (1992). Numerical Methods for Stochastic Control Problems in Continuous Time. Vol. 24 of Applications of Mathematics. Springer-Verlag. New York. Merton, Robert C. (1969) . Lifetime portfolio selection under uncertainty: The continuoustime case. Review of Economics and Statistics 51, 247-257. Munk, Claus (1997). Optimal ConsumptionPortfolio Policies and Contingent Claims Pricing and Hedging in Incomplete Markets. PhD thesis. School of Business and Economics, Odense University. DK-5230 Odense M, Denmark. Munk, Claus (1998) . The valuation of options in incomplete markets: Computing reservation buying and selling prices. Working paper. Odense University.

Based on the above considerations, it is to be expected that the reservation buying price will increase relative to the unconstrained case, and that the increase is largest for high values of wealth and low risky asset prices, since the borrowing constraint is most restrictive for such wealth/price combinations. This is confirmed by Figure 8.

6 In these situations, the fraction of wealth optimally invested in the risky asset under the borrowing constraint, when one call option is bought at the reservation buying price, is practically indistinguishable from the minimum of the unconstrained optimal risky asset weight Ol,g(z,P)(x, P,O)/x and the upper bound of 1.

16