Download Competitive Analysis of On-line Securities Investment

Competitive Analysis of On-line Securities
Investment
Shuhua Hu1 , Qin Guo1 , and Hongyi Li2
1
School of Management, Xi’an Jiaotong University, Xi’an, China
[email protected], [email protected]
2
Dept of Decision Sciences and Managerial Economics,
The Chinese University of Hongkong, Hongkong, China
[email protected]
Abstract. Based on the unidirectional conversion model, we investigate a practical buy-and-hold trading problem. This problem is useful
for long-term investors, we use competitive analysis and game theory to
design some trading rules in the securities markets. We present an online
algorithm, Mixed Strategy, for the problem and prove its competitive
, where n is the trading horizon and t is the daily fluctuaratio 1 + (n−1)t
2
tions of securities prices. The Dynamic-Mixed Strategy is also presented
to further reduce the competitive ratio. An investing example is simulated with the Mixed Strategy and Dollar Average Strategy based on the
actual market data.
1
Introduction
In many situations, we are forced to choose between different alternatives without
knowledge of each alternative’s future worth. The finance management problem
has attracted substantial attention lately due to the coexistence of high returns
and high risks. Like many other ongoing financial activities, securities investing
must be carried out in an on-line fashion, with no secure knowledge of future
events. Faced with this lack of knowledge, the on-line investors of this financial
activity often use models based on assumptions about the future distribution of
relevant quantities such as securities prices, and aim for acceptable results on the
average. The on-line algorithm and competitive analysis[1] are important tools
for making decision. The approach we follow here is to use competitive analysis,
which was first applied to on-line algorithms by Sleator and Tarjan in[2].
Competitive analysis has been extensively discussed in the domain of financial
management since Cover[3] first investigated the portfolio problem with it in
1991. Cover presented a simple on-line strategy that dynamically changed the
distribution of its current wealth among the stocks based on the market history.
El-Yaniv[4][5][6] explored the on-line algorithm for the unidirectional conversion
problem under the assumption that the whole exchange rates, instead of the daily
Supported by NSF Grant No.70471035 and No.10371094.
N. Megiddo, Y. Xu, and B. Zhu (Eds.): AAIM 2005, LNCS 3521, pp. 224–232, 2005.
c Springer-Verlag Berlin Heidelberg 2005
Competitive Analysis of On-line Securities Investment
225
ones, are between a pair of upper and lower bounds, no matter how erratically or
unfortunately the rates vary from day to day. Xu et al.[7] further studied the same
setting in a framework with commission and interest rate. Chou[8] investigated
the bidirectional currency trading problem against a weak statistical adversary
to make a money-making strategy.
In most studies, it is inevitable for the researchers to make some assumptions
about future events to design a competitive strategy to maximize the future
gain. Although much research has been devoted to the investing problem, little
research has been done based on the real trading rules. Instead of knowing about
the distribution of future market prices, an online strategy might be based only
on the knowledge of the daily bounds on possible market prices. In this paper we
use the framework of the daily price constraints model, in which the next day’s
price e depends on the current day’s price e with e · (1 − t) ≤ e ≤ e · (1 + t) for
some fixed t < 1, to investigate the buy-and-hold trading problem using game
theory and competitive analysis based on the unidirectional conversion model in
[4]. We obtain an online algorithm Mixed Strategy for the problem and prove its
. We then present the Dynamic-Mixed Strategy
exact competitive ratio 1+ (n−1)t
2
to further reduce the competitive ratio. The Mixed Strategy and Dollar Average
Strategy are also compared and an investing example is simulated based on
the actual market data of CBM in 2003. Our results could be beneficial to the
long-term trading players in the securities market to make a decision.
This paper is organized as follows. In Section 2, we present the on-line securities investment problem, and derive the Mixed Strategy and the Dynamic-Mixed
Strategy using the Game Theory and Competitive Analysis. Section 3 refers to
compare the Mixed Strategy and the popular Dollar Average Strategy, and an
investing example is stimulated using the two strategies based on the actual
market data of CBM in 2003. Conclusions and work-in-progress are reported in
section 4.
2
2.1
On-line Securities Investment, OSI
Problem Statement
Suppose that an investor has initial capital of w0 = 1, and invests it for a certain
security in n days, which is referred to as the trading horizon (assume n ≥ 2 to
avoid triviality). We also assume that, on each trading day, the security has only
one price, which refers to the number of shares of the securities one unit capital
could buy. After each price is realized, the investor executes one transaction for
that day and invests all or part of his capital. All the capital must be invested
on the n-th trading day, and the holding securities are not permitted converting
back to the capital. The performance of the investing is the accumulation of
shares of the securities that the investor holds at the end of the investment.
In many securities markets, it is regulated that the next day’s closing price
ei+1 depends on the current day’s closing price ei with ei+1 ∈ [ei (1−t), ei (1+t)],
for some fixed t < 1, and it means the daily price fluctuation. Let E denote
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S. Hu, Q. Guo, and H. Li
the set of all the feasible price sequences, O denote the optimal off-line trading
algorithm, and A be the investor’s on-line trading algorithm. On the i-th trading
day, the investor must decide the investing amount with only the information
of ei . In this model, the optimal off-line return is O(ē) = max 1/ei , and the
1≤i≤n
n
on-line investor’s return is A(ē) = i=1 ai /ei , where ai is the amount of capital
that the investor invests on the i-th trading day.
Let Si be the trade-once algorithm for i ∈ Zn , i.e., the investor trades all his
initial capital on the i-th trading day, and his return is Si (ē) = 1/ei . Note that
Si is static because converting the holding securities back to cash is prohibited
once the trade realized. Let S be a randomized static algorithm, and si be the
expected
invested by S on the i-th day. For all i, si ≥ 0
n amount of capital
and i=1 si = 1. Let S denote the deterministic static algorithm that invests
si on the i-th day. Since the value of si defines a probability density function
in Φ(Zn ), let S denote the randomized static algorithm that applies Si with
probability si .
Lemma 1. [9] S, S , and S are equivalent in the sense that for all ē ∈ E,
S(ē) = S (ē) = S (ē).
Let rs∗ be the smallest possible competitive ratio of the on-line static algorithm, then according to Lemma 1,
rs∗ =
2.2
inf
O(ē)
.
i=1 f (i)Si (ē)
sup n
f ∈Φ(Zn ) ē∈E
(1)
Mixed Strategy and Competitive Analysis, MIX
The OSI problem could be considered as a two-person game G(m, n) [4]. For any
integer k > 0, we set Zk = {1, 2, · · · , k}. The maximizing player is the on-line
investor, whose pure strategies are the deterministic on-line algorithm Ai of G
indexed with i ∈ Zm . The minimizing player is the adversary in G, whose pure
strategies are the input price sequences σ̄j of G with i ∈ Zn . If the vectors of
the payoff matrix are denoted by the reciprocal of the competitive ratio, then
the payoff matrix is
H(i, j) =
Ai (σ̄j )
> 0, i ∈ Zm , j ∈ Zn .
Oi (σ̄j )
(2)
Let Φ(Zk ) be the set of all probability density functions defined on Zk . For
k = m, or k = n, each h ∈ Φ(Zk ) could be considered as a point in the kdimensional Euclidean space and represents a Mixed Strategy that applies the
l-th pure strategy indexed by Zk with probability h(l). Let r∗ denote the smallest
possible competitive ratio of any randomized on-line algorithm for G(m, n), by
equation (1)
O(σ̄j )
.
(3)
r∗ = min max
f ∈Φ(Zm ) j∈Zn f (i)Ai (σ̄j )
A randomized on-line algorithm is optimal if its competitive ratio is r∗ .
Competitive Analysis of On-line Securities Investment
227
e1
e2
1
t
e3
1
t
e4
e5
e6
day
Fig. 1. The adversary’s dominating sequences
The buy-and-hold trading of OSI is an infinite game because the adversary’s
pure strategies are infinite. However, the on-line investor has n pure strategies
Si . Using the strictly inferior method of the game theory and the worst-case
analysis, we can delete the non-worst-case price sequences of the adversary and
make the game be finite. In other words, we neglect all the sequences that dose
not rise to or lower to the limit. For j = 1, · · · , n, let ēj be the adversary’s
dominating sequences, then
ēj = [(1 − t), (1 − t)2 , · · · , (1 − t)j ,
j
(1 − t)j (1 + t), (1 − t)j (1 + t)2 · · · , (1 − t)j (1 + t)n−j ].
(4)
n−j
See figure 1 for its illustration.
Lemma 2. [9] Given a static algorithm S, each ē ∈ E is dominated by
O(ēj )
ēj , i.e., O(ē)
S(ē) ≤ S(ēj ) , where ej = min ei .
1≤i≤n
Theorem 1. If the investor know the trading horizon n and the daily price
fluctuation t at the beginning, there is Nash Balance of the Mixed Strategy
⎧
1+t
i=1
⎪
⎨ (n−1)t+2
t
∗
i ∈ [2, n − 1]
xi = (n−1)t+2
⎪
1
⎩
i=n
(n−1)t+2
n
and i=1 x∗i = 1, which is the optimal on-line investing strategy. Its competitive
ratio is 1 + 2t (n − 1).
Proof. Let Si be the investor’s i-th pure strategy and ēj be the j-the pure
strategy of the adversary. According to equations (2) and (4), the payoff matrix
Si (ēj )
, where O(ēj ) = (1 − t)−j and
of the on-line game is defined by H(i, j) = O(ē
j)
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S. Hu, Q. Guo, and H. Li
Si (ēj ) = (1 − t)−i if i ≤ j or Si (ēj ) = (1 − t)−j (1 + t)j−i otherwise, i.e.,
⎛
⎞
1
(1 − t)
(1 − t)2 · · · (1 − t)n−1
⎜ (1 + t)−1
1
(1 − t) · · · (1 − t)n−2 ⎟
⎜
⎟
⎜ (1 + t)−2 (1 + t)−1
1
· · · (1 − t)n−3 ⎟
H=⎜
⎟
⎜
⎟
..
..
..
..
..
⎝
⎠
.
.
.
.
.
(1 + t)1−n (1 + t)2−n (1 + t)3−n · · ·
for n ≥ 2,
det(H) = (
(5)
1
2t n−1
)
> 0.
1+t
(6)
Let X = (x1 , x2 , . . . , xn )T denote the Mixed Strategy for the on-line investor,
and π be the payoff of the algorithm. The investor could get the same payoff,
i.e., the same competitive ratio, with the Mixed Strategy whatever strategies the
adversary choose
(7)
π1 = π2 = · · · = πn .
The payoff of the algorithm is
Π = HX = (π1 , π2 , · · · , πn )T
⎞
⎛
1 · x1
+ (1 − t) · x2 + · · · + (1 − t)n−1 · xn
⎜ (1 + t)−1 · x1 +
1 · x2
+ · · · + (1 − t)n−2 · xn ⎟
⎟
⎜
−2
−1
⎜
= ⎜ (1 + t) · x1 + (1 + t) · x2 + · · · + (1 − t)n−3 · xn ⎟
⎟
⎝ ..................................................... ⎠
1 · xn
(1 + t)1−n · x1 + (1 + t)2−n · x2 + · · · +
The undermentioned solution follows from
⎧
1+t
⎪
⎨ (n−1)t+2
t
xi = (n−1)t+2
⎪
1
⎩
and
(n−1)t+2
n
(8)
equation (7).
i=1
i ∈ [2, n − 1]
i=n
(9)
i=1 xi = 1. Equation (9) means that the on-line investor should invest
1+t
1
(n−1)t+2 unit on the first trading day, (n−1)t+2 unit on the last trading day and
t
(n−1)t+2 on each of the other trading days starting with one unit capital initially.
The payoff of the algorithm is
2
.
(n − 1)t + 2
(10)
t
(n − 1)t + 2
= 1 + (n − 1).
2
2
(11)
π1 = π2 = · · · = πn =
Combining r =
1
π,
it suffices to show that
rmix =
The unique optimality of the Mixed Strategy among the static algorithms for
the price constraints model follows from Lemma 2.
Competitive Analysis of On-line Securities Investment
229
According to Theory 1, the on-line investor’s investing decision depends on
the trading horizon n. Though the competitive ratio and the daily investing
amount take O(1) time to evaluate, lim rmix = lim [1 + 2t (n − 1)] = ∞.
n→∞
n→∞
The algorithm is not competitive when the trading horizon goes to infinite.
However, the algorithm is a practical buy-and-hold strategy for a fund manager
who decides to change the position of some portfolio, and those who invest in
the stock-index fund.
2.3
Dynamic-Mixed Strategy, D-MIX
The Dynamic-Mixed Strategy is an improved strategy against a clumsy adversary. The simple investment strategy above is overly pessimistic since it fixes
the competitive ratio based on the assumption of worst-case input sequences of
prices, and does not change it thereafter. However, the on-line investor could
delete the not-occurred worst-case sequences mentioned in the equation 6 according to the history data whenever some sequences are realized. Thus, we
could strictly improve the off-line to on-line ratio by recalculating the achievable
competitive ratio, and this is a dynamic process. At the start of each trading
day, the on-line player has some number w of capital. The investor knows the
number n of days remaining, and is given a price e . The investor acts as if the
current day were the first trading day of an n -day trading horizon in which the
adversary starts with one unit capital and the player with w unit capital. By
arguments similar to those used in section 2.2, an expression for the competitive
ratio is determined and maximized over the remaining market prices. For the
k-th realizing sequence the amount xj to invest is given by
⎧
1+t
⎪
⎨ (n −1)t+2 · wk
t
xj = (n −1)t+2 · wk
⎪
⎩ 1
(n −1)t+2 · wk
j=1
j ∈ [2, n − 1]
j = n
(12)
The corresponding competitive ratio is
rdm = 1 +
2.4
(n − 1)t
.
2
(13)
Dollar Average Strategy, DA
The Dollar Average Strategy is a static algorithm which invests an equal amount
of capital, i.e., n1 unit capital on each trading day. The following theory gives
the competitive ratio of the static algorithm.
Theorem 2. rda =
nt
1−(1−t)n .
Proof. By Theorem 1, DA is the uniform Mixed Strategy for the on-line investor
in the game G(m, n). That is to say, the on-line investor’s investing strategy
is X̃ = (x̃1 , x̃2 , · · · , x̃n ) = ( n1 , n1 , · · · , n1 )T with one unit capital initially. The
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S. Hu, Q. Guo, and H. Li
associated payoff is Π̃ = (π̃1 , π̃2 , · · · , π̃n )T . The payoff
⎛
1
(1 − t)
(1 − t)2
⎜ (1 + t)−1
1
(1 − t)
−2
1⎜
⎜
(1 + t)−1
1
Π̃ = H X̃ = ⎜ (1 + t)
n⎜
..
..
..
⎝
.
.
.
matrix of DA is
⎞
· · · (1 − t)n−1
· · · (1 − t)n−2 ⎟
⎟
· · · (1 − t)n−3 ⎟
⎟
⎟
..
..
⎠
.
.
(1 + t)1−n (1 + t)2−n (1 + t)3−n · · ·
(14)
1
The competitive ratio of the Dollar Average Strategy is
rda = max(
1 1
1
nt
, ,··· ,
)=
.
π̃1 π̃2
π̃n
1 − (1 − t)n
(15)
3
Simulation
The competitive ratios of DA and MIX were plotted against trading horizon in
Figure 2 for t = 0.1. It is obvious that the Mixed Strategy is better than the
Dollar Average Strategy with increase of the trading horizon.
competitive rario
10
DA
MIX
8
6
4
2
20
40
60
80
100
n
Fig. 2. Competitive ratios of MIX and DA (t=0.1)
This seems inconsistent with El-Yani’s[4] result, who demonstrated that the
Dollar Average Strategy is the best plan of the threat-based algorithm. One
reason for this could be that the two models were based on different constraints of
the input price sequences. El-Yani’s model fixes a static upper bound and a static
lower bound on the daily exchange rates for the entire investing horizon, while
our model sets new dynamic price constraints for every trading day. According
to the constraints of daily price fluctuation in many securities markets, the latter
one should be more practical.
Based on the market data of China Merchants Bank Co., Limited (CMB) in
2003, we simulated an investing example using MIX and DA. Figure 3 shows the
daily closing prices of CMB in 2003. All stock prices are quoted in RMB. One
Competitive Analysis of On-line Securities Investment
231
investment plan is executed to buy the shares of CBM in one year with an initial
capital of one RMB. For easy comparison, the daily accumulation is expressed
by the summation of the remaining cash and the currency values of the holding
share. The currency values of the holding share are evaluated by the price of the
current trading day. Figure 4 shows the daily accumulations of MIX, DA and the
optimal off-line strategy based on the data of CBM in 2003. At the end of the
plan, the return of the off-line strategy, MIX, and DA are 1.34, 1.054 and 1.044,
respectively. The corresponding realized competitive ratio of MIX and DA are
1.27 and 1.28, respectively. The Mixed Strategy outperforms the Dollar Average
Strategy as a whole but the difference between the two strategies is not distinct.
The main reason is that the real price sequence of CMB in 2003 is not given in
the worst-case, but one of the infinite non-worst-cases.
1.6
13
12
accumulation
1.4
price
11
10
9
OPT
1.2
MIX
DA
1.0
8
1-1
3-2
5-1
6-30
8-29
10-28
12-27
date
Fig. 3. CMB’s daily closing stock
prices in 2003
4
1-1
3-2
5-1
6-30
8-29
10-28
12-27
date
Fig. 4. The accumulations of OPT, MIX
and DA (t=0.1)
Conclusions
Considering the trading rules of daily price constraints in securities markets,
we obtained an on-line algorithm, the Mixed Strategy, and proved its optimality
among the static algorithms in the daily price constraints model for the buy-andhold trading problem in a discrete case. This method can provide a valuable and
quantitative analysis tool for securities investment. However, the Mixed Strategy
is pessimistic because of the assumption of the worst-case input sequences and
the invariability of the competitive ratio. Taking the not-occurred worst-cases
into account, the Mixed Strategy is improved further to the Dynamic-Mixed
Strategy. Based on the market data of China Merchants Bank Co., Limited
(CMB) in 2003, we compared the Mixed Strategy and the Dollar Average Strategy. The results show that the former strategy outperforms the latter one as a
whole. But the difference is not so obvious as in the worst-case.
However, it must be pointed out that the Mixed Strategy is not competitive
when the trading horizon goes to infinity. Therefore, how to design a competitive
algorithm not related to n is still an open problem for this model. In addition,
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S. Hu, Q. Guo, and H. Li
it would be interesting to investigate the bidirectional trading problem and the
trading problem with transaction cost using this model.
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