- Email: [email protected]
European Option Pricing under Bounded Uncertainty with Transaction Costs
EUROPEAN OPTION PRICING UNDER BOUNDED UNCERTAINTY ...
14th World Congress of IFAC
M-5e-05-4
Copyright © 1999 IFAC 14th Triennial \Vorld Congress, Beijing, P.R. China
EUROPEAN OPTION PRICING UNDER BOUNDED UNCERTAINTV WITH TRANSACTION COSTS*
Lihui Zheng
Institute a/System Engineering. Huazhong University ofScience and Technology. Wuhan, Hubei, 430074 P.R.China. email: [email protected] J
Abstract: This paper extends the worst-case robust control framework for option pricing to the case with transaction costs. The problem is studied under the assumption that disturbances in the underlying price of options are bounded while the buying and selling rates of securities are allowed unbounded~ The option price is sho\rvn to be the discounted value of a differential game. The value function of the game is characterized as the constrained viscosity (or minimax) solution of a first-order variational inequality. A numerical schelue to solve the variational inequality is also presented and computer simulations are iJnplemented. Copyright ([:} 1999lFAC Keyv"'ords: finance, bounded disturbances) differential games, dynamic programming, numerical solutions.
1. INTRODUCTION
Until recently financial theory has been primarily considered in stochastic settings; for example~ Black and Scholes (1973), Merton (1990), and Duffle (1992) etc. However, as worst-case design, or robust control theory gains a ne\v popularity in control literature (see e.g. Zames, J 98] ~ Francis~ 1987, and Basar and Bemard, 1991)~ this alternative approach to treat uncertainties has been frequently used in financial modeling and optimization. Ho~·e and Rustem (1997) studied a minimax hedging model for option pricing with a discrete-tiIne tTIodel. Continuous-time financial models based on the worst-case approach include Orszag and Yang (1995), Fleming (1995), McEneaney (1997) and Zheng (1998a) etc. The first m'o of these models investigate the optinlal consumption and investment problem under the assumption that stock price Inodels of investors have bounded L2-norms; both of them indicate that this fonnulation of financial lnarkets may lead to new capltaI asset pricing models~ The last two papers assume that disturbances in the underlying price of
options have finite) but arbitrarily high variation, and they considered the hedging and pricing of contingent claims. These researches show that this alternative fonnulation of financial problems produces parallel, but slightly different results to the stochastic methods. This paper extends the worst-case robust control approach to option pricing studied by McEneaney (1997) and Zheng (1998a) to the case with transaction costs. In this framework, disturbances in the underlying price of options are modeled as controls governed by an unfriendly nature, and the hedging of contingent claims is fonnulated as a t\Voperson zero-sum differential game; the option prices are then defined with the values of this differential game. Both McEneaney (1997) and Zheng (1998a) considered the case without transaction costs. They show that the obtained fair option price and the hedging strategy are of the stop-loss type (see. Hull, 1993)~ which are independent of the amplitude or volatility of disturbances. It \vill be shown in this paper~ however~ when transaction costs are taken into account~ a specific bound on the disturbances must be
.. This paper is partially supported by the Postdoctoral Science Foundation of China.
6308
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
EUROPEAN OPTION PRICING UNDER BOUNDED UNCERTAINTY ...
imposed to ensure the existence of the option price; although the buying and selling rates of securities can still remain unbounded. Like the optimal consumption and investment model with transaction costs (see Fleming and Soner, 1993 etc.), the differential game under these assumptions is of the singular type(for details see Zheng, 1998b). The option price is expressed as the discounted value of the differential game~ and the value of this differential game is characterized as the weak (viscosity or minimax) solution of a first-order variational inequality (VI) \vith gradient constraints. Instead of investigating the theoretical aspects of this VI, a discrete algorithm is presented to derive numerical solutions of the partial differential equation (PDE), and the effectiveness of this option pricing model is analyzed by computer simulations. The numerical scheme is constructed through discretization of the differential game and by using the discrete-time version of the dynamic programming principle. Computer simulations sho\v that the option price increases as the bound of uncertainty becomes large, and decreases, as the time to maturity becomes smalL The option price also depends on the transaction costs: lower transaction costs has lower option prices, and when transaction costs becomes zero, the option price is the same as the stop-loss option price (see HuU, J993).
The VI obtained in this paper is sirnilar to that of Davis et al. (1993), except that their VI is of second order and two variational inequalities must be used to derive the option price. The VI here describes an interesting nonstationary terminal-value problem with free boundaries. Since it is of independent significance to consider the theoretical aspects of this free-boundary problem, the related theoretical problems are studied in a companion paper (Zheng, 1998b). The results there include the value function of the differential game is the unique viscosity solution of the VI~ the numerical scheme proposed in this paper converges to the original PDE etc. Such results provide technical supports for computer simulations of this papeL Here is a more detailed description of the contents of the paper. Section 2 describes the financial market. The option pricing method is presented in Section 3, in \vhich the option price is formulated ",'ith a differential game, and then the dimension of the game is reduced. The dimension reduction results in an economic interpretation of the option price, as well as simplifies the subsequent derivations to a great extent. Section 4· is devoted to the analysis of the value function of the reduced differential game. An Isaacs equation for the value function is obtained there. Section 5 gives the discrete algorithm and implement computer simulations. Some properties of the option pricing model are discussed with the results of the simulations. The paper is concluded with Section 6,
where some advantages of the option pricing method are pointed out. Option pricing with transaction costs has been one of the central concerns of financial literature in the past few years. The researches fall into two categories: local in time and global in time (Whalley and Wilmott) 1997). The former is based on the Black.. Scholes formu la and considers transaction costs at each rehedging periods by minimizing the current level of risks (see Leland, 1985, Boyle and Vorst, 1992, and Toft, 1996 etc.). The global-in-time models are comparatively different from the BlackScholes theory of option pricing; they evaluate options by comparing the maximized utility of investors trading the option contract with that of investors not trading it (Hodges and Neuberger~ 1989, and Davis, et aI, 1993). This method thus needs solving a three-dimensional free boundary problem nvice. Other papers dealing with transaction costs in financial economics are Shreve and Saner (1994), Morton and Pliska (1995), and Davis andClark (1995) etc.
2. DESCRIPTION OF THE FINANCIAL MARKET Consider a European call option with maturity T and strike price K. The option gives its holder the right to buy one share ofa specified stock at price K. Assume that the underlying price x(t) is governed by an ordinary differential equation xC!) == a x(t) + P x(t)w(t) . (1) Here a denotes the detenninistic part of the stock~s spot rate of return. In the uncertainty part of the spot rate; p denotes the bound of disturbances, and li-{') ~ an arbitrary function such that w(t) E [-1,1] for any t € [0, T] , denotes the way disturbances move.
Note that the underlying price model has the same structure as that of the commonly used geometric Brownian motion, except that the uncertainty here is modeJed by a unifonnJy bounded function rather than by a standard Wiener process. Uncertainties of this kind are usually referred to as the Knightian uncertainty (see Knight, 1921). There are at least two reasons to study option pricing problems under the Knightian uncertainty: one is that such uncertainty can include a larger class of model errors that are unsuitable to be considered as Wiener processes~ the other is the fact that in many circumstances, agents in an economy do not have subjective probabjJity distribution over the economic variables. Assume that a riskless asset (called bank account) is also available in the market; the price of the bank account z(t) follov~'s z(t)
=r
z(t) ,
\vhere r is the risk less interest rate.
6309
Copyright 1999 IFAC
ISBN: 0 08 043248 4
EUROPEAN OPTION PRICING UNDER BOUNDED UNCERTAINTY ...
Consider a dynamic portfolio consisting of the stock and the ri'skless asset. Denote the number of stock shares by y(t) and the amount of riskless asset by z(t). Let let) and met) be the instantaneous rates of buying and selling the stock respectively. Then the stock shares in the portfolio is given by ~v(t) = l(t) - m(t) . (2) Assume that (A) the buying and selling rates let) and m(t) are integrable on [0, TJ~ such that yet) is unifonnly bounded on [0,71. Suppose that the transaction costs involved in the trading of securities are proportional to the value of the traded assets, that is, a trade of one share of the stock at price x incurs a transaction cost of b; the transaction fee is paid from the bank account. Thus the amount of riskless asset z(t) follo\\'s i(t) == r z(t) - (1 + A )l(t)x(t) + (1- A )m(t)x(t) (3) In the stochastic theory of option pricing, the central ideal is to construct a self-financing portfolio such that its tenninal value replicates the contractual payoff of the contingent claim with probability one. Then~ by non-arbitrage arguments, the initial investment of the portfolio can be considered as a fair option price. However~ when transaction costs are taken into account, perfect-replicating portfolios on Jonger exist. Therefore~ it is necessary to investigate the option pricing problem in other frameworks.
14th World Congress of IFAC
uncertainty. Denote by C(x, y) the cash value of y shares of the stock at price x, (l- .l)xy, Y 2 O. C(x,y) { (1 + A)XY, Y < O. =:;
Consider the surplus of the hedging portfolio's tenn inal value against the option payoff r(s, X; 1(·), nl(·)~ ~{.)) == z(T) + C[x(T), yeT)] ' (4) -max {(l-A)x(T)-K,O } where S E [0, T] denotes initial time, X = (x, y, z) denote initial values of the underlying price and the hedging investments. Assume that the dynamics of a differential game are given by (1)-(3), in which the control variables [l(t) , m(t)] and the uncertainty w(t) are governed by two antagonistic players, a hedger and an unfriendly nature. The hedger tries to maximize the surplus functional (4); the unfriendly nature, on the contrary, wishes to minimize the functional. Before going to the defmition of option prices, some terms are needed from the theory of differential games. '
The theory of pas itional (feedback) differential games is adapted here to give hedging strategies in closed-loop forms; for full explanations on the theory) see Krassovskii and Subbotin (1988), and Subbotin (1995). For the moment, assume that the trading rates ICt) and n~(t) are bounded by k, and km respectively, such that the differential game is well-defmed. This bound restriction is left, in Section 4, by sending k, and km to infinity. Denote D I", =[O,k,Jx{O,kmJ and R. = [0, GO). According to the theory of positional games, a feedback strategy of the hedger is an
3. STATEMENT OF THE METHOD
arbitrary function LM(t, X); [s" T]x R+ X R 2 ~ Drill ' such that [let), /n(t)] = LM[t, X(t)] for any t E [s, T] .
In this section, the robust control framework for option pricing is applied to the problem \vith transaction costs. First the option price is fonnulated with a differential game, and then dimensionality reduction ':is perfonned for the differential game to simply the fOffilulation.
Similarly, a feedback strategy of the nature is also an arbitrary function W(t, X): [s, T] x R+ X R 2 ~ [-1,1], such that li-{t) = W(t, X(t)] for any t E [s, T]. Both LM(t, .x) and W(t, X) are not necessarily continuous. The lower and upper values of the differential game are defined as r. (s~ X) ~ x ~in yCs, X; LM, W) ~
3. J. Forn2ulation of the Option Price
Ille
When investors of an option are faced with the Kn ightian· uncertainty in stock prices, their decisions on option prices are usually based on the worst case of the uncertainty. The writers prefer an option price, with which as an initial investment a dynamic portfolio could be constructed to hedge the option even in the worst case. The buyers, on the other hand, may accept the option price only ",,"hen the portfolio is not an arbitrage, because otherwise they v,.'ould construct such portfolios by themselves. This argument can be used to give a new definition of option prices.
r,(s~X) = -
minmaxy(s,X;LM,W). H" Lt]
It is kno\vn from the theory of differential games that inequality r 1 (s, X) ~ r 2 (s~ X) is always valid for the upper and lower values. In the case that equality takes place, this equality defines the value Va)(s~ X) of the galne, i.e. Val(s, X) = r. (8, X) = r 2 (s, X) .
Definition 1 For any time t and un.derlying price x,
the writing (buying) price of a European option is defined as elelnents in the set ~I' (t, x) = {z: r l (t,x,O, z) ~ o} . (~(t,x) = {z: r 2 (t,x,O,z) 5: o}).
First introduce a differential game model for the hedging of European options under the Knightian
6310
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
EUROPEAN OPTION PRICING UNDER BOUNDED UNCERTAINTY ...
where . .¥ (1) and y(1) are terminal values of x(-) and Note that the option is evaluated, in this definition~ with only cash and the initial share of stocks is set to O. This definition indicates that the writing price of options is always linked to a trading strategy that optim izes the worst-case hedging effects expressed by (4)~ If the resultjng lower value of the differentia) game is non-negative, the option writers may accept the initial portfolio value as the option price because they can use the trading strategy to hedge their investment in the option. The buying price~ on the other hand, is always linked to a strategy of the nature that selects the worst case of the hedging effect. If the resulting upper value of the game is non-positive, it will be impossible for the hedger to make a riskless arbitrage. In this case, the buyers may accept the initial value of the portfolio as an option price,. and buy the option to remove their risk exposure in the underlying asset. Definition 2 For any tin1e t and underlying price x} the fair price of a European option is defined as elements in the set
Pr (t, x)
=P
w
(t, x)
n p;. (t, x) .
The fair price of options is defined as the initial investment that adlnits both super-replicating portfolios for the hedger and arbitrage-eliminating strategies for the nature In this case, investors of the option can neither get a profit nor incur a loss when they use the optimal hedging strategy and the worst case appears. This situation is referred to as nonarbitrage in the ttlorst case. SpecificaUy the financial market admits no arbitrage in the worst case if and only if o ~ r l (t, x~O, z) '$ 1 2 (t, x~O~ z) ~ 0 is valid. This leads to the follo\ving result.
of0 at initial value
rl Ct, x, y; /(-), m(-); w(.) = g(x(T), yeT»~ + Jh(r, xCr); 1(1'), m(r»)dr
(6)
In the sequel, a reduced differential game will be concentrated on with payoff functional (6) and state equations rewritten from (1)-(2) as .i:(/) = a x(t) + P x(t)w(t) , (7a) (7b)
},(t) = let) - met) . It is easy to see from (5) and (6) that y(t,X;/(), me); lV(» =
zer(T-1)
+ rr (t,x,y;l(-),m(-); w(·).
This relation also holds for the value function V(t,.A) of the original differential game and that of the reduced game VI (t, x, y), i.e. Y~(t~.JY)
== ze,,(T-n + VI (t:,x,y) .
When the reduced differential game has a value, Theorem 1 can be extended to the following result by setting the above equality to 0 at (t,x,O,z), Le.V(I~x,OJZ) = 0, and by solving it for z.
This result sho\v that just like option prices in the stochastic theory, option prices determined in the y,.·orst-case frame\'vork are also discounted values of certain payoffs, \vhich, in the stochastic settings, are expected option payoffs under the so..caUed riskneutralized probability. In this worst-case framework, the payoff is the negative value of the reduced differential game. Moreover, this theorem transfonns the option pricing problem into deriving the value function of a differential game. Therefore, our next task is to construct an lsaacs equation for the differential game, and to solve it for the value function.
3.2. Difnension Reduction of the DWerential Gante
In this subsection, the dimension of differential game (I )-(4) is reduced from four to three. First introduce some notations. Let het ,x~ n1.J) == -(1 + ...l)! x e d ] -I) + (1- /l)n2 x e 1
TtT
-
) ,
4. THE ISAACS EQUA TION
g(x, y) = C(x, y) - max {(I ~ A)X - K,O}.
The following can be obtained by solving (3) for z(.) and substituting z( 1) into (4)
This section investigates the value function of differential galne (6)-(7) and derives a variational inequality for the value function. This goal is achieved in two steps: First the problem with bounded trading rates is considered, and an Isaacs equation is obtained. Then the VI is reached by sending the bounds to infinity. A result from Zheng (1998b) is cited~ which claims that the value function
ret, X;/(·), m(-); It{·)) = zer(T-n + g(x(T), y(T))
r
= y(t, x, Y,O; . ) ,
(t,x,y~l(·},m(·); wC'») -= g(x(T),y(T)) + IT h(r,x(r);l(r)~m(r))d,
J
(t,x~O,z).
+ h[ r ~ x ( r ), I ( T)~ 1n ( r )] d r
YI
has a value at initial value (t,x,O). Moreover, in the case it has, thefair price can be expressed as z" (t,x) = _e-r(T-1) V] (t, x, 0) . (8)
Proposition 1 For any· tinle I and stock price x, an initial investrnent z is a fair price .for the option, i. e. Z E Pt (t, x), if and only if differential galne (1)-(4) val~,e
new payoff functional Y l (t, X, Y; . )
Proposition 2 For an}' time t and stock price x, an initial investnlent z is a fair price for the option, i. e. Z E Pj(t,x) if and only if differential game· (6)-(7)
4
has a
y(.) ~'ith initial values x and .y respectively. Define a
(5)
6311
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
EUROPEAN OPTION PRICING UNDER BOUNDED UNCERTAINTY ...
is the unique viscosity solution of the VI; proofs are
UP(l~X,y)=
given in that paper. When the trading rates are bounded, i.e. [let), nl{t)] E D rm for any t E [0, T], the theory of positional differential games (see Krassovskii and Subbotin,1988, and Subbotin,1995) can be used to give the following Isaacs equation. ~ + (XX V~ + ~:~? ttrw V~ + max If.m)eJ}I~,
{1[V,,'- (I + A)xe1"fT-r;.]+ m[- Vi' + (1-.4 )xe,,(T-r) ~:= 0 .
.
where the subscripts denote partial derivatives. With the same arguments as those in Davis et al (1993, p.477), when the bounds k, and km on the trading rates tend to infinity, it is reasonable to conjecture that the above Isaacs equation converges to the variational
inequality belo~' max{V/ +axVx -P-x]Vx ]' (9) J~v - (I + 2) xe l"u-,) ,-V~. + (1- 2)xe rIT - n } = 0 Therefore the follo~~ing result can be obtained; proofs are given in Zheng (1998b). Proposition 3 Under the assumption (A)~ the value function ofdifferential game (6)-(7) is the unique viscosity solution ofPDE (9) with terminal condition V(T,x,y)
maxsk min{U [l+ p, X + px(a+ J1wty+ pCI-m)] $) p
r
Ski m
m
».
11'
+
p[- (1 + 2)!xe
F
(T-r)
+ (1- A) mxe
t-(7-f)
Let 17 be the discrete variable of y, the interval of which is Bp with B a constant real number. As the bounds of control variables I and m tend to infinity, the following discrete equation can be obtained UP (l, x, 17) = nlax{U P (I, x, TJ + (Jp) - (1 + A)()%er(T-J) ,
U p (1, X, ,., - Bp) + (1- A )f)pxeT(T-,) ,
n,;iH UP [1+ p, x + J2t(a + j1w), 1]]} which is a finite-difference approximation to the original PDE. Now a result of Zheng (1998b) is stated without proof: Proposition 4 As the discretization parameter p
~
0~
the solution [II' of (10) converges uniformly to the viscosity solution of PDE (9).
Based on this result numerical solution of PDE (9) can be obtained; then option prices can be calculated from (8). Computer simulations of the algorithm are implemented in the next subsection.
= g(x,y). 5.2. Computer Sitnulations
5. NUMERICAL SCHEMES AND COMPUTER SIMULATTONS
In this subsection, computer simulations are performed with the numerical scheme presented above~ The purpose is to anaJyze the behavior of option prices, as well as to show the correctness of the algorithm. Here three parameters that affect the option price in the option pricing model are examined. It is shown that option prices decrease as time approaches maturity~ It is also obtained that the higher transaction costs are, the higher option prices become, and larger uncertainty amplitudes in stock prices result in higher option prices. Because of the limitation of space, the figures are omitted. Further simulations are also needed to test the effects of other factors on the option price.
In this section~ a numerical scheme is presented to solve the Isaacs equation obtained in Section 4, and computer simulations are performed based on the algorithm.
5.1. l\/umerical Schemes
The numerical scheme for PDE (9) starts from discretization of differential game (6)-(7). By solving this discrete-time version of the differential game, the discrete time dynamic programming equation can be obtained, which is a discrete approximation to the original PDE; proofs of the convergence of this numerical,;scheme is provided in Zheng (1998b).
6. CONCLUSIONS The worst-case robust control framework for option
Let the discrete time variable l takes values in {O, 81 ~ 2&, ... , 1'1t5t } ~ where ,5 t is the time interval
pricing initialized by McEneaney (1997) and Zheng (1998a) is extended to the case with transaction costs. The option price is expressed as the discounted value
and satisfies T = N6 t. Discretization of differential game (6)-(7) yields difference equations X(1 + p) = XCi) + .ox(l)[a + ffiv(t)], Y(l + p) = yet) + p[l(l) - m(l)]. where p = 0 t. In the case that control variables I and m take finite values, the discrete-time dynamic programming principle can be employed to get the following finite difference equation for the value function UP of the discrete-time problem
of a differential game, and the vaJue function is shown to follow a first-order variational inequality in the viscosity sense. A numerical scheme is presented to solve the VI, and simulation results are also provided. This option pricing method has several advantages. First the option pricing model is based on the optimization behavior of option investors, and the option price reflects both the need of option writers to
6312
Copyright 1999 IFAC
ISBN: 0 08 043248 4
EUROPEAN OPTION PRICING UNDER BOUNDED UNCERTAINTY ...
construct super-replicating portfolios and the requirement of option buyers to eliminate arbitrage. Second the method is independent of investors' utility and consequently both the writers and the buyers can accept the option price. Third the resulting PDE is not difficult to compute since it is of first order and the option price can be obtained by solving the equation only once. Finally the option pricing model allows disturbances to move in an arbitrary way rather than folIo"'" some probability distributions~ this is important in the case when stationarity of the financial market is not assumed. These advantages demonstrate at least the theoretical value of this option pricing method.
REFERENCES Basar, T. and P. Bernhard (1991). H~-Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach. Birkhauser, Boston~ MA. Black, F. and M. Scholes (1973). The pricing of options and corporate liabilities. J. Political Economy, 81~ 637-54. Boyle, P.P. and T. Vorst (1992). Option replication in discrete time with transaction costs. J. Finance, 47, 271 - 293~ Davis, M.H.A., V.G. Panas, and T. Zariphopouloll (1993). European option pricing with transaction costs. SIAM J. Control Optim., 31, 470 - 493. Davis, M.H.A. and J.M.C. Clark (1995). A note on super-replicating strategies. in Mathematical Models in Finance, eds. S.D. Howison et at. Chapman & Hall, London: 35-44~ Duffie, D. (1992). Dynafnic Assets Pricing Theory. Princeton University Press, Princeton,New Jersey. Fleming, W.H. (] 995). Optimal investment models and risk sensitive stochastic controL in Jvlathe;matical Finance, eds~ M.H.A. Davis et al. Springer-Verlag, New York. Francis, B.A. (1987). A Course in H:f: Control Theory, Lecture Notes in Control and lnforntation Sciences, 88. Springer-Verlag~ New York. Hodges, S.D.and A. Neuberger (1989). Optimal replication of contingent claims under transaction costs. Review a/Futures Markets~ 8, 222-239. Toft, K.B. (1996). On the mean-variance tradeoff in option replication with transaction costs. J of Financial and Quantitative Analysis, 31, 233 263. Howe'l M.A. and B. Rustem (1997). A robust hedging algorithm. J. ofEcononlic Dynamics and Control, 21~ 1065- 1092. Hull.. l.C. (1993). Options, Futures, and Other Derivative Securities. Prentice-Hall Inc., Englewood Cliffs, NJ. Knight, F. (1921). Risk, Uncertainty and Profit. Houghton-Miffiin, Boston.
14th World Congress of IFAC
Krassovskii, N.N. and A.I.Subbotin (1988). GameTheoretical Control Problems. Springer-Verlag, New York. Leland, H.E. (1985). Option pricing and replication with transaction costs. JFinance, 40, 1283-301. McEneaney, W.M. (1997). A robust control frame\\i'ork for option pricing. Mathematics of Operations Research, 22, 202 - 221. Merton, R.e. (1990). Continuous-Time Finance. Basil Blackwell, Cambridge. Mortan, A. and S. Pliska (1995). Optimal portfolio lnanagement with fixed transaction costs. Math. Finance, 5, 337 - 356. Orszag, J.M. and H. Yang (1995). Portfolio Choice with Knightian Uncertainty. J. of Economic DynaJ11ics And Control, 19, 873-900. Shreve, S.E. and H.M.Soner (1994). Optimal investment and consumption with transaction costs. Ann..4pplied Prob., 4, 680 - 692. Subbotin, A.1. (1995). Generalized Solutions ofFirstOrder PDEs: The Dynamical Optimization Perspective. Birkhauser, Boston, MA. Whalley, A.E. and P. Wilmott (1997)~ An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Mathematical Finance, 7, 307 - 324. Zames, G( ]981). Feedback and optimal sensitivity: model reference transformation, multiplicative seminonns, and approximate inverses. IEEE Trans. AutoNlatic Control, 26, 301-320. Zheng, L. (1998a). A worst-case approach to the hedging and pricing of European contingent claims. The 3'" International Confer~nce on Jvlanagelnent, Shanghai, China: R393. Zheng, L. (1998b). Numerical analysis of a freeboundary problem in option pricing under bounded uncertainty. Submitted~
6313
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
M-5e-05-4
Copyright © 1999 IFAC 14th Triennial \Vorld Congress, Beijing, P.R. China
EUROPEAN OPTION PRICING UNDER BOUNDED UNCERTAINTV WITH TRANSACTION COSTS*
Lihui Zheng
Institute a/System Engineering. Huazhong University ofScience and Technology. Wuhan, Hubei, 430074 P.R.China. email: [email protected] J
Abstract: This paper extends the worst-case robust control framework for option pricing to the case with transaction costs. The problem is studied under the assumption that disturbances in the underlying price of options are bounded while the buying and selling rates of securities are allowed unbounded~ The option price is sho\rvn to be the discounted value of a differential game. The value function of the game is characterized as the constrained viscosity (or minimax) solution of a first-order variational inequality. A numerical schelue to solve the variational inequality is also presented and computer simulations are iJnplemented. Copyright ([:} 1999lFAC Keyv"'ords: finance, bounded disturbances) differential games, dynamic programming, numerical solutions.
1. INTRODUCTION
Until recently financial theory has been primarily considered in stochastic settings; for example~ Black and Scholes (1973), Merton (1990), and Duffle (1992) etc. However, as worst-case design, or robust control theory gains a ne\v popularity in control literature (see e.g. Zames, J 98] ~ Francis~ 1987, and Basar and Bemard, 1991)~ this alternative approach to treat uncertainties has been frequently used in financial modeling and optimization. Ho~·e and Rustem (1997) studied a minimax hedging model for option pricing with a discrete-tiIne tTIodel. Continuous-time financial models based on the worst-case approach include Orszag and Yang (1995), Fleming (1995), McEneaney (1997) and Zheng (1998a) etc. The first m'o of these models investigate the optinlal consumption and investment problem under the assumption that stock price Inodels of investors have bounded L2-norms; both of them indicate that this fonnulation of financial lnarkets may lead to new capltaI asset pricing models~ The last two papers assume that disturbances in the underlying price of
options have finite) but arbitrarily high variation, and they considered the hedging and pricing of contingent claims. These researches show that this alternative fonnulation of financial problems produces parallel, but slightly different results to the stochastic methods. This paper extends the worst-case robust control approach to option pricing studied by McEneaney (1997) and Zheng (1998a) to the case with transaction costs. In this framework, disturbances in the underlying price of options are modeled as controls governed by an unfriendly nature, and the hedging of contingent claims is fonnulated as a t\Voperson zero-sum differential game; the option prices are then defined with the values of this differential game. Both McEneaney (1997) and Zheng (1998a) considered the case without transaction costs. They show that the obtained fair option price and the hedging strategy are of the stop-loss type (see. Hull, 1993)~ which are independent of the amplitude or volatility of disturbances. It \vill be shown in this paper~ however~ when transaction costs are taken into account~ a specific bound on the disturbances must be
.. This paper is partially supported by the Postdoctoral Science Foundation of China.
6308
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
EUROPEAN OPTION PRICING UNDER BOUNDED UNCERTAINTY ...
imposed to ensure the existence of the option price; although the buying and selling rates of securities can still remain unbounded. Like the optimal consumption and investment model with transaction costs (see Fleming and Soner, 1993 etc.), the differential game under these assumptions is of the singular type(for details see Zheng, 1998b). The option price is expressed as the discounted value of the differential game~ and the value of this differential game is characterized as the weak (viscosity or minimax) solution of a first-order variational inequality (VI) \vith gradient constraints. Instead of investigating the theoretical aspects of this VI, a discrete algorithm is presented to derive numerical solutions of the partial differential equation (PDE), and the effectiveness of this option pricing model is analyzed by computer simulations. The numerical scheme is constructed through discretization of the differential game and by using the discrete-time version of the dynamic programming principle. Computer simulations sho\v that the option price increases as the bound of uncertainty becomes large, and decreases, as the time to maturity becomes smalL The option price also depends on the transaction costs: lower transaction costs has lower option prices, and when transaction costs becomes zero, the option price is the same as the stop-loss option price (see HuU, J993).
The VI obtained in this paper is sirnilar to that of Davis et al. (1993), except that their VI is of second order and two variational inequalities must be used to derive the option price. The VI here describes an interesting nonstationary terminal-value problem with free boundaries. Since it is of independent significance to consider the theoretical aspects of this free-boundary problem, the related theoretical problems are studied in a companion paper (Zheng, 1998b). The results there include the value function of the differential game is the unique viscosity solution of the VI~ the numerical scheme proposed in this paper converges to the original PDE etc. Such results provide technical supports for computer simulations of this papeL Here is a more detailed description of the contents of the paper. Section 2 describes the financial market. The option pricing method is presented in Section 3, in \vhich the option price is formulated ",'ith a differential game, and then the dimension of the game is reduced. The dimension reduction results in an economic interpretation of the option price, as well as simplifies the subsequent derivations to a great extent. Section 4· is devoted to the analysis of the value function of the reduced differential game. An Isaacs equation for the value function is obtained there. Section 5 gives the discrete algorithm and implement computer simulations. Some properties of the option pricing model are discussed with the results of the simulations. The paper is concluded with Section 6,
where some advantages of the option pricing method are pointed out. Option pricing with transaction costs has been one of the central concerns of financial literature in the past few years. The researches fall into two categories: local in time and global in time (Whalley and Wilmott) 1997). The former is based on the Black.. Scholes formu la and considers transaction costs at each rehedging periods by minimizing the current level of risks (see Leland, 1985, Boyle and Vorst, 1992, and Toft, 1996 etc.). The global-in-time models are comparatively different from the BlackScholes theory of option pricing; they evaluate options by comparing the maximized utility of investors trading the option contract with that of investors not trading it (Hodges and Neuberger~ 1989, and Davis, et aI, 1993). This method thus needs solving a three-dimensional free boundary problem nvice. Other papers dealing with transaction costs in financial economics are Shreve and Saner (1994), Morton and Pliska (1995), and Davis andClark (1995) etc.
2. DESCRIPTION OF THE FINANCIAL MARKET Consider a European call option with maturity T and strike price K. The option gives its holder the right to buy one share ofa specified stock at price K. Assume that the underlying price x(t) is governed by an ordinary differential equation xC!) == a x(t) + P x(t)w(t) . (1) Here a denotes the detenninistic part of the stock~s spot rate of return. In the uncertainty part of the spot rate; p denotes the bound of disturbances, and li-{') ~ an arbitrary function such that w(t) E [-1,1] for any t € [0, T] , denotes the way disturbances move.
Note that the underlying price model has the same structure as that of the commonly used geometric Brownian motion, except that the uncertainty here is modeJed by a unifonnJy bounded function rather than by a standard Wiener process. Uncertainties of this kind are usually referred to as the Knightian uncertainty (see Knight, 1921). There are at least two reasons to study option pricing problems under the Knightian uncertainty: one is that such uncertainty can include a larger class of model errors that are unsuitable to be considered as Wiener processes~ the other is the fact that in many circumstances, agents in an economy do not have subjective probabjJity distribution over the economic variables. Assume that a riskless asset (called bank account) is also available in the market; the price of the bank account z(t) follov~'s z(t)
=r
z(t) ,
\vhere r is the risk less interest rate.
6309
Copyright 1999 IFAC
ISBN: 0 08 043248 4
EUROPEAN OPTION PRICING UNDER BOUNDED UNCERTAINTY ...
Consider a dynamic portfolio consisting of the stock and the ri'skless asset. Denote the number of stock shares by y(t) and the amount of riskless asset by z(t). Let let) and met) be the instantaneous rates of buying and selling the stock respectively. Then the stock shares in the portfolio is given by ~v(t) = l(t) - m(t) . (2) Assume that (A) the buying and selling rates let) and m(t) are integrable on [0, TJ~ such that yet) is unifonnly bounded on [0,71. Suppose that the transaction costs involved in the trading of securities are proportional to the value of the traded assets, that is, a trade of one share of the stock at price x incurs a transaction cost of b; the transaction fee is paid from the bank account. Thus the amount of riskless asset z(t) follo\\'s i(t) == r z(t) - (1 + A )l(t)x(t) + (1- A )m(t)x(t) (3) In the stochastic theory of option pricing, the central ideal is to construct a self-financing portfolio such that its tenninal value replicates the contractual payoff of the contingent claim with probability one. Then~ by non-arbitrage arguments, the initial investment of the portfolio can be considered as a fair option price. However~ when transaction costs are taken into account, perfect-replicating portfolios on Jonger exist. Therefore~ it is necessary to investigate the option pricing problem in other frameworks.
14th World Congress of IFAC
uncertainty. Denote by C(x, y) the cash value of y shares of the stock at price x, (l- .l)xy, Y 2 O. C(x,y) { (1 + A)XY, Y < O. =:;
Consider the surplus of the hedging portfolio's tenn inal value against the option payoff r(s, X; 1(·), nl(·)~ ~{.)) == z(T) + C[x(T), yeT)] ' (4) -max {(l-A)x(T)-K,O } where S E [0, T] denotes initial time, X = (x, y, z) denote initial values of the underlying price and the hedging investments. Assume that the dynamics of a differential game are given by (1)-(3), in which the control variables [l(t) , m(t)] and the uncertainty w(t) are governed by two antagonistic players, a hedger and an unfriendly nature. The hedger tries to maximize the surplus functional (4); the unfriendly nature, on the contrary, wishes to minimize the functional. Before going to the defmition of option prices, some terms are needed from the theory of differential games. '
The theory of pas itional (feedback) differential games is adapted here to give hedging strategies in closed-loop forms; for full explanations on the theory) see Krassovskii and Subbotin (1988), and Subbotin (1995). For the moment, assume that the trading rates ICt) and n~(t) are bounded by k, and km respectively, such that the differential game is well-defmed. This bound restriction is left, in Section 4, by sending k, and km to infinity. Denote D I", =[O,k,Jx{O,kmJ and R. = [0, GO). According to the theory of positional games, a feedback strategy of the hedger is an
3. STATEMENT OF THE METHOD
arbitrary function LM(t, X); [s" T]x R+ X R 2 ~ Drill ' such that [let), /n(t)] = LM[t, X(t)] for any t E [s, T] .
In this section, the robust control framework for option pricing is applied to the problem \vith transaction costs. First the option price is fonnulated with a differential game, and then dimensionality reduction ':is perfonned for the differential game to simply the fOffilulation.
Similarly, a feedback strategy of the nature is also an arbitrary function W(t, X): [s, T] x R+ X R 2 ~ [-1,1], such that li-{t) = W(t, X(t)] for any t E [s, T]. Both LM(t, .x) and W(t, X) are not necessarily continuous. The lower and upper values of the differential game are defined as r. (s~ X) ~ x ~in yCs, X; LM, W) ~
3. J. Forn2ulation of the Option Price
Ille
When investors of an option are faced with the Kn ightian· uncertainty in stock prices, their decisions on option prices are usually based on the worst case of the uncertainty. The writers prefer an option price, with which as an initial investment a dynamic portfolio could be constructed to hedge the option even in the worst case. The buyers, on the other hand, may accept the option price only ",,"hen the portfolio is not an arbitrage, because otherwise they v,.'ould construct such portfolios by themselves. This argument can be used to give a new definition of option prices.
r,(s~X) = -
minmaxy(s,X;LM,W). H" Lt]
It is kno\vn from the theory of differential games that inequality r 1 (s, X) ~ r 2 (s~ X) is always valid for the upper and lower values. In the case that equality takes place, this equality defines the value Va)(s~ X) of the galne, i.e. Val(s, X) = r. (8, X) = r 2 (s, X) .
Definition 1 For any time t and un.derlying price x,
the writing (buying) price of a European option is defined as elelnents in the set ~I' (t, x) = {z: r l (t,x,O, z) ~ o} . (~(t,x) = {z: r 2 (t,x,O,z) 5: o}).
First introduce a differential game model for the hedging of European options under the Knightian
6310
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
EUROPEAN OPTION PRICING UNDER BOUNDED UNCERTAINTY ...
where . .¥ (1) and y(1) are terminal values of x(-) and Note that the option is evaluated, in this definition~ with only cash and the initial share of stocks is set to O. This definition indicates that the writing price of options is always linked to a trading strategy that optim izes the worst-case hedging effects expressed by (4)~ If the resultjng lower value of the differentia) game is non-negative, the option writers may accept the initial portfolio value as the option price because they can use the trading strategy to hedge their investment in the option. The buying price~ on the other hand, is always linked to a strategy of the nature that selects the worst case of the hedging effect. If the resulting upper value of the game is non-positive, it will be impossible for the hedger to make a riskless arbitrage. In this case, the buyers may accept the initial value of the portfolio as an option price,. and buy the option to remove their risk exposure in the underlying asset. Definition 2 For any tin1e t and underlying price x} the fair price of a European option is defined as elements in the set
Pr (t, x)
=P
w
(t, x)
n p;. (t, x) .
The fair price of options is defined as the initial investment that adlnits both super-replicating portfolios for the hedger and arbitrage-eliminating strategies for the nature In this case, investors of the option can neither get a profit nor incur a loss when they use the optimal hedging strategy and the worst case appears. This situation is referred to as nonarbitrage in the ttlorst case. SpecificaUy the financial market admits no arbitrage in the worst case if and only if o ~ r l (t, x~O, z) '$ 1 2 (t, x~O~ z) ~ 0 is valid. This leads to the follo\ving result.
of0 at initial value
rl Ct, x, y; /(-), m(-); w(.) = g(x(T), yeT»~ + Jh(r, xCr); 1(1'), m(r»)dr
(6)
In the sequel, a reduced differential game will be concentrated on with payoff functional (6) and state equations rewritten from (1)-(2) as .i:(/) = a x(t) + P x(t)w(t) , (7a) (7b)
},(t) = let) - met) . It is easy to see from (5) and (6) that y(t,X;/(), me); lV(» =
zer(T-1)
+ rr (t,x,y;l(-),m(-); w(·).
This relation also holds for the value function V(t,.A) of the original differential game and that of the reduced game VI (t, x, y), i.e. Y~(t~.JY)
== ze,,(T-n + VI (t:,x,y) .
When the reduced differential game has a value, Theorem 1 can be extended to the following result by setting the above equality to 0 at (t,x,O,z), Le.V(I~x,OJZ) = 0, and by solving it for z.
This result sho\v that just like option prices in the stochastic theory, option prices determined in the y,.·orst-case frame\'vork are also discounted values of certain payoffs, \vhich, in the stochastic settings, are expected option payoffs under the so..caUed riskneutralized probability. In this worst-case framework, the payoff is the negative value of the reduced differential game. Moreover, this theorem transfonns the option pricing problem into deriving the value function of a differential game. Therefore, our next task is to construct an lsaacs equation for the differential game, and to solve it for the value function.
3.2. Difnension Reduction of the DWerential Gante
In this subsection, the dimension of differential game (I )-(4) is reduced from four to three. First introduce some notations. Let het ,x~ n1.J) == -(1 + ...l)! x e d ] -I) + (1- /l)n2 x e 1
TtT
-
) ,
4. THE ISAACS EQUA TION
g(x, y) = C(x, y) - max {(I ~ A)X - K,O}.
The following can be obtained by solving (3) for z(.) and substituting z( 1) into (4)
This section investigates the value function of differential galne (6)-(7) and derives a variational inequality for the value function. This goal is achieved in two steps: First the problem with bounded trading rates is considered, and an Isaacs equation is obtained. Then the VI is reached by sending the bounds to infinity. A result from Zheng (1998b) is cited~ which claims that the value function
ret, X;/(·), m(-); It{·)) = zer(T-n + g(x(T), y(T))
r
= y(t, x, Y,O; . ) ,
(t,x,y~l(·},m(·); wC'») -= g(x(T),y(T)) + IT h(r,x(r);l(r)~m(r))d,
J
(t,x~O,z).
+ h[ r ~ x ( r ), I ( T)~ 1n ( r )] d r
YI
has a value at initial value (t,x,O). Moreover, in the case it has, thefair price can be expressed as z" (t,x) = _e-r(T-1) V] (t, x, 0) . (8)
Proposition 1 For any· tinle I and stock price x, an initial investrnent z is a fair price .for the option, i. e. Z E Pt (t, x), if and only if differential galne (1)-(4) val~,e
new payoff functional Y l (t, X, Y; . )
Proposition 2 For an}' time t and stock price x, an initial investnlent z is a fair price for the option, i. e. Z E Pj(t,x) if and only if differential game· (6)-(7)
4
has a
y(.) ~'ith initial values x and .y respectively. Define a
(5)
6311
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
EUROPEAN OPTION PRICING UNDER BOUNDED UNCERTAINTY ...
is the unique viscosity solution of the VI; proofs are
UP(l~X,y)=
given in that paper. When the trading rates are bounded, i.e. [let), nl{t)] E D rm for any t E [0, T], the theory of positional differential games (see Krassovskii and Subbotin,1988, and Subbotin,1995) can be used to give the following Isaacs equation. ~ + (XX V~ + ~:~? ttrw V~ + max If.m)eJ}I~,
{1[V,,'- (I + A)xe1"fT-r;.]+ m[- Vi' + (1-.4 )xe,,(T-r) ~:= 0 .
.
where the subscripts denote partial derivatives. With the same arguments as those in Davis et al (1993, p.477), when the bounds k, and km on the trading rates tend to infinity, it is reasonable to conjecture that the above Isaacs equation converges to the variational
inequality belo~' max{V/ +axVx -P-x]Vx ]' (9) J~v - (I + 2) xe l"u-,) ,-V~. + (1- 2)xe rIT - n } = 0 Therefore the follo~~ing result can be obtained; proofs are given in Zheng (1998b). Proposition 3 Under the assumption (A)~ the value function ofdifferential game (6)-(7) is the unique viscosity solution ofPDE (9) with terminal condition V(T,x,y)
maxsk min{U [l+ p, X + px(a+ J1wty+ pCI-m)] $) p
r
Ski m
m
».
11'
+
p[- (1 + 2)!xe
F
(T-r)
+ (1- A) mxe
t-(7-f)
Let 17 be the discrete variable of y, the interval of which is Bp with B a constant real number. As the bounds of control variables I and m tend to infinity, the following discrete equation can be obtained UP (l, x, 17) = nlax{U P (I, x, TJ + (Jp) - (1 + A)()%er(T-J) ,
U p (1, X, ,., - Bp) + (1- A )f)pxeT(T-,) ,
n,;iH UP [1+ p, x + J2t(a + j1w), 1]]} which is a finite-difference approximation to the original PDE. Now a result of Zheng (1998b) is stated without proof: Proposition 4 As the discretization parameter p
~
0~
the solution [II' of (10) converges uniformly to the viscosity solution of PDE (9).
Based on this result numerical solution of PDE (9) can be obtained; then option prices can be calculated from (8). Computer simulations of the algorithm are implemented in the next subsection.
= g(x,y). 5.2. Computer Sitnulations
5. NUMERICAL SCHEMES AND COMPUTER SIMULATTONS
In this subsection, computer simulations are performed with the numerical scheme presented above~ The purpose is to anaJyze the behavior of option prices, as well as to show the correctness of the algorithm. Here three parameters that affect the option price in the option pricing model are examined. It is shown that option prices decrease as time approaches maturity~ It is also obtained that the higher transaction costs are, the higher option prices become, and larger uncertainty amplitudes in stock prices result in higher option prices. Because of the limitation of space, the figures are omitted. Further simulations are also needed to test the effects of other factors on the option price.
In this section~ a numerical scheme is presented to solve the Isaacs equation obtained in Section 4, and computer simulations are performed based on the algorithm.
5.1. l\/umerical Schemes
The numerical scheme for PDE (9) starts from discretization of differential game (6)-(7). By solving this discrete-time version of the differential game, the discrete time dynamic programming equation can be obtained, which is a discrete approximation to the original PDE; proofs of the convergence of this numerical,;scheme is provided in Zheng (1998b).
6. CONCLUSIONS The worst-case robust control framework for option
Let the discrete time variable l takes values in {O, 81 ~ 2&, ... , 1'1t5t } ~ where ,5 t is the time interval
pricing initialized by McEneaney (1997) and Zheng (1998a) is extended to the case with transaction costs. The option price is expressed as the discounted value
and satisfies T = N6 t. Discretization of differential game (6)-(7) yields difference equations X(1 + p) = XCi) + .ox(l)[a + ffiv(t)], Y(l + p) = yet) + p[l(l) - m(l)]. where p = 0 t. In the case that control variables I and m take finite values, the discrete-time dynamic programming principle can be employed to get the following finite difference equation for the value function UP of the discrete-time problem
of a differential game, and the vaJue function is shown to follow a first-order variational inequality in the viscosity sense. A numerical scheme is presented to solve the VI, and simulation results are also provided. This option pricing method has several advantages. First the option pricing model is based on the optimization behavior of option investors, and the option price reflects both the need of option writers to
6312
Copyright 1999 IFAC
ISBN: 0 08 043248 4
EUROPEAN OPTION PRICING UNDER BOUNDED UNCERTAINTY ...
construct super-replicating portfolios and the requirement of option buyers to eliminate arbitrage. Second the method is independent of investors' utility and consequently both the writers and the buyers can accept the option price. Third the resulting PDE is not difficult to compute since it is of first order and the option price can be obtained by solving the equation only once. Finally the option pricing model allows disturbances to move in an arbitrary way rather than folIo"'" some probability distributions~ this is important in the case when stationarity of the financial market is not assumed. These advantages demonstrate at least the theoretical value of this option pricing method.
REFERENCES Basar, T. and P. Bernhard (1991). H~-Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach. Birkhauser, Boston~ MA. Black, F. and M. Scholes (1973). The pricing of options and corporate liabilities. J. Political Economy, 81~ 637-54. Boyle, P.P. and T. Vorst (1992). Option replication in discrete time with transaction costs. J. Finance, 47, 271 - 293~ Davis, M.H.A., V.G. Panas, and T. Zariphopouloll (1993). European option pricing with transaction costs. SIAM J. Control Optim., 31, 470 - 493. Davis, M.H.A. and J.M.C. Clark (1995). A note on super-replicating strategies. in Mathematical Models in Finance, eds. S.D. Howison et at. Chapman & Hall, London: 35-44~ Duffie, D. (1992). Dynafnic Assets Pricing Theory. Princeton University Press, Princeton,New Jersey. Fleming, W.H. (] 995). Optimal investment models and risk sensitive stochastic controL in Jvlathe;matical Finance, eds~ M.H.A. Davis et al. Springer-Verlag, New York. Francis, B.A. (1987). A Course in H:f: Control Theory, Lecture Notes in Control and lnforntation Sciences, 88. Springer-Verlag~ New York. Hodges, S.D.and A. Neuberger (1989). Optimal replication of contingent claims under transaction costs. Review a/Futures Markets~ 8, 222-239. Toft, K.B. (1996). On the mean-variance tradeoff in option replication with transaction costs. J of Financial and Quantitative Analysis, 31, 233 263. Howe'l M.A. and B. Rustem (1997). A robust hedging algorithm. J. ofEcononlic Dynamics and Control, 21~ 1065- 1092. Hull.. l.C. (1993). Options, Futures, and Other Derivative Securities. Prentice-Hall Inc., Englewood Cliffs, NJ. Knight, F. (1921). Risk, Uncertainty and Profit. Houghton-Miffiin, Boston.
14th World Congress of IFAC
Krassovskii, N.N. and A.I.Subbotin (1988). GameTheoretical Control Problems. Springer-Verlag, New York. Leland, H.E. (1985). Option pricing and replication with transaction costs. JFinance, 40, 1283-301. McEneaney, W.M. (1997). A robust control frame\\i'ork for option pricing. Mathematics of Operations Research, 22, 202 - 221. Merton, R.e. (1990). Continuous-Time Finance. Basil Blackwell, Cambridge. Mortan, A. and S. Pliska (1995). Optimal portfolio lnanagement with fixed transaction costs. Math. Finance, 5, 337 - 356. Orszag, J.M. and H. Yang (1995). Portfolio Choice with Knightian Uncertainty. J. of Economic DynaJ11ics And Control, 19, 873-900. Shreve, S.E. and H.M.Soner (1994). Optimal investment and consumption with transaction costs. Ann..4pplied Prob., 4, 680 - 692. Subbotin, A.1. (1995). Generalized Solutions ofFirstOrder PDEs: The Dynamical Optimization Perspective. Birkhauser, Boston, MA. Whalley, A.E. and P. Wilmott (1997)~ An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Mathematical Finance, 7, 307 - 324. Zames, G( ]981). Feedback and optimal sensitivity: model reference transformation, multiplicative seminonns, and approximate inverses. IEEE Trans. AutoNlatic Control, 26, 301-320. Zheng, L. (1998a). A worst-case approach to the hedging and pricing of European contingent claims. The 3'" International Confer~nce on Jvlanagelnent, Shanghai, China: R393. Zheng, L. (1998b). Numerical analysis of a freeboundary problem in option pricing under bounded uncertainty. Submitted~
6313
Copyright 1999 IFAC
ISBN: 0 08 043248 4