This paper investigates an investment/risk-hedging problem for a stochastic diffusion
model of a security market consisting of a risk-free bond and a finite number of risky
stocks. Associated with a given contingent claim, an investment strategy is said to have
bounded risk if the claim is replicated with an error not exceeding a given level. In a
broad sense, a bounded risk investment strategy is also called a hedging strategy. The most
well-known hedging strategies for this model were obtained by Black and Scholes (1973)
and Merton (1969, 1973). For the Black and Scholes model, the strategies were used to
hedge given claims exactly (without any error). For the Merton model, the strategies were obtained for an optimization problem of maximizingEU(X(T)), where X(T) is the wealth at
the expiration time T and U() is a utility function. For various variants and extensions, see,
e.g. Samuelson (1969), Hakansson (1971), Perold (1984), Karatzas et al. (1987), Dumas
and Liucinao (1991), Zhou (1998), and Khanna and Kulldorff (1999). However, in the
literature explicit formulae for optimal strategies have been established only for the cases
when appreciation rates of the stocks are non-random and known, and U() has quadratic
form, log form or power form. In the general case of random appreciation rates, solution of
the optimal investment problem calls for using the so-called backward stochastic differential
equations (for a most updated account of this theory see Chapter 7 ofYong and Zhou (1999)),
which unfortunately is difficult to solve explicitly and computationally.
Another problem of wide interest is a mean–variance hedging, or a problem of minimizing
EjX.T / − j2, where is a given random claim. For this problem, explicit solutions were
obtained for the case of observable appreciation rates, see, e.g. Föllmer and Sondermann
(1986), Duffie and Richardson (1991), Pham et al. (1998), Kohlmann and Zhou (1998),
Pham et al. (1998), and Laurent and Pham (1999). The resulting optimal hedging strategies
are combinations of the Merton strategy and the Black and Scholes strategy, which depend
on the direct observation of the appreciation rates. Unfortunately, as well known the appreciation
rates are usually hard to observe in real-time market, especially when the volatility
coefficients are larger than the average deviations of the appreciation rates per unit time.
Moreover, in these studies the error between the terminal wealth and the claim is bounded
in the mean–variance sense, rather than in the almost-surely one.
In this paper,we consider an investment/hedging model with several newfeatures. First of
all, in our model we do not assume that the appreciation rates of the stocks are non-random
and observable; we only assume that the distributions of the appreciation rates are known
based on the observation of the stock prices. The finally derived strategy depends only on
the current stock prices, the distributions of the appreciation rates, and one scalar parameter
which can be calculated numerically. Second, our model involves the replication of a given
claim with a guaranteed error bound (gap). More precisely, our admissible strategies ensure
that the replication errors do not exceed a given level almost surely. Note that in the classical
problem (for a complete market) of an exact replication, the strategy is uniquely determined
by the claim. In an incomplete market, where an exact replication is no longer generally
possible, it is sensible to consider replications with some gap, which in turn makes it possible
to choose among many possible strategies. Finally, the utility function under consideration
in our model is a fairly general one, covering the mean–variance criterion, non-continuous
functions, and nonlinear concave functions as special cases. In particular, our general utility
function incorporates the so-called goal achieving problem. A goal achieving problem is to
maximize the probability that the event of reaching a prescribed goal happens before the
event of a failure. Specifically, it is to maximize P( 1 2), where i is the first time that
the discounted process O X.t/ , er.T−t/X.t/ reaches the level ki , i D 1; 2. Here r is the risk
free interest rate, k2 is the goal level while k1 is a level considered to be a failure (certainly,
k1 < k2).
The goal achieving problem is interesting in its own right. The problem for a single-stock
market model with only additive stochastic disturbanceswas first solved by Karatzas (1997),
where the constructed optimal strategy depends only on the distribution of the stock appreciation
rate. In the present paper, using an approach completely different than that of Karatzas (1997), we are able to derive optimal strategies that are independent of the appreciation rate
estimations, for a general model with multiple, correlated stocks and additional constraints
of bounded risks. In particular, it is concluded that for an optimal strategy the levels k1 and
k2 will never be achieved before the expiration time.
It should also be noted that for a general problem with non-random appreciation rates,
Khanna and Kulldorff (1999) showed that the so-called Mutual Fund Theorem holds,
namely, the optimum can be achieved on a set of Merton type strategies. But, this theorem
does not hold for the case of random and non-observable rates that is being studied
here, nor does the method of Khanna and Kulldorff (1999) apply.
The rest of this paper is organized as follows. In Section 2, the general model under
consideration is formulated and necessary preliminaries are given. In Section 3, an optimal
solution to the general problem is presented. Section 4 discusses several important special
cases of the general problem, including the goal achieving problem. Numerical results are
reported in Section 5. In Section 6, some concluding remarks are given. Finally, in Appendix,
proofs of the results are supplied.
