Trading in Convergent Markets Mattias Jonsson, University of Michigan, Department of Mathematics, Ann Arbor, MI 48108 Jan Veˇ ceˇ r, Columbia University, (
[email protected])
Department of Statistics,
New York,
NY 10027
March 20, 2002 Preliminary version, any comments welcomed
1
Introduction
The purpose of this paper is to find an optimal trading strategy for trading in stocks of two companies (or currencies of two countries) which will merge in the future (or join monetary union in the case of currencies). If the merger actually happens at time T , we assume that the ratio of the two stock prices ST1 /ST2 will be equal to some pre-specified constant C: ST1 = C. ST2
(1)
The constant C is the ratio in which the two companies exchange their old stocks for the newly created merged company. Generally, there are many examples of convergent markets when there are two or more processes (stock prices, exchange rates or interest rates) for which we have some partial information about their relative future evolution in the above form. Given the merger, the arbitrage free market would restrict the dynamics of the two stocks in such a way that no risk free profit is possible. However, in many cases the merger is often legally challenged and it is not obvious that it would indeed happen for sure. Therefore we can still observe uncertainty governing the prices of stocks of the two companies. In this sense, we are studying the situation of the conditional arbitrage: the risk-free profit is possible only if the merger actually takes place. Some authors refer to similar situations as statistical arbitrage. In our paper, we assume that the two stocks are driven by two possibly correlated Brownian motions. We show that the condition (1) is equivalent to the restriction on the final position of the linear combination of the two Brownian motions. It is very similar to the Brownian bridge process, with the exception that it is two dimensional. We refer to it as a planar Brownian bridge process. First, we study the correlation structure and stochastic differential equation governing such a process. Next, we apply these results to the dynamics of the two convergent stocks. An investor trading in these stocks can obtain optimal trading strategy which maximizes his expected profit. Interestingly enough, this optimal strategy is often different from simply locking the arbitrage opportunity.
1
1.1
Brownian Bridge
Suppose that the underlying process is a Brownian Bridge with the starting point X0 = a and the end point XT = b. The dynamics of such process can be written in the following form: dXt =
2 2.1
b − Xt dt + dWt , T −t
X0 = a.
(2)
Planar Brownian Bridge Distribution of Planar Brownian Bridge
In this subsection, we will use the general result valid for jointly normal vectors (X1 , X2 ). Suppose that X1 µ1 Σ11 Σ12 ∼N , . (3) X2 µ2 Σ21 Σ22 Then the conditional distribution of X1 given X2 is −1 (X1 |X2 ) ∼ N (µ1 − Σ12 Σ−1 22 (X2 − µ2 ), Σ11 − Σ12 Σ22 Σ21 ).
(4)
We can use this result to get the conditional distributions of Wt1 and Wt2 given that a1 WT1 + a2 WT2 = b. We easily conclude that for s < t, ! a21 s2 a21 st a1 b·s s − s − 2 2 )T (a21 +a22 )T (a21 +a22 )T 2 (Ws1 , Wt1 |a1 WT1 + a2 WT2 = b) ∼ N (a1a+a , , (5) 1 b·t a21 st a21 t2 2 2 s − (a2 +a2 )T t − (a2 +a 2 )T (a1 +a2 )T 1
! a2 b·s s− 2 2 2 )T (Ws2 , Wt2 |a1 WT1 + a2 WT2 = b) ∼ N (a1a+a , 2 b·t s− (a21 +a22 )T
2
a22 s2 (a21 +a22 )T a22 st (a21 +a22 )T
1
s− t−
2
a22 st (a21 +a22 )T a22 t2 (a21 +a22 )T
,
(6)
and (Ws1 , Wt2 |a1 WT1 + a2 WT2 = b) ∼ N
a1 b·s (a21 +a22 )T a2 b·t (a21 +a22 )T
! a21 s2 s − (a2 +a 2 )T 1 2 , a1 a2 st − (a2 +a2 )T 1
2
a2 st − (aa21+a 2 )T 1
t−
2 a22 t2 2 (a1 +a22 )T
.
(7)
In terms of the correlation structure, we have ρ11 (s, t) ρ22 (s, t) ρ12 (s, t)
a21 st , + a22 )T a2 st = E(Ws2 Wt2 ) = s ∧ t − 2 2 2 , (a1 + a2 )T −a1 a2 st = E(Ws1 Wt2 ) = 2 . (a1 + a22 )T = E(Ws1 Wt1 ) = s ∧ t −
(a21
(8) (9) (10)
Now the question is what kind of dynamics would match the above correlation structure.
2.2
Dynamics of PBB
Theorem 2.1 Suppose that Wt1 , Wt2 are two Brownian motions with W01 = W02 = 0, dWt1 ·dWt2 = ρdt and suppose that a1 WT1 + a2 WT2 = b. (11) 2
Than the dynamics of Wt1 , Wt2 can be written as 1
√
2
dWt1
b−a Wt −a2 Wt dt + √ = (a1 + ρa2 ) (T −t)(a12 +2ρa 2 1 a2 +a )
dWt2
=
1
2
1
2
b−a Wt −a2 Wt (a2 + ρa1 ) (T −t)(a12 +2ρa dt + √ 2 1 a2 +a ) 1
2
a1 +ρa2 a21 +2ρa1 a2 +a22
dBt1 − √
a2 +ρa1 a21 +2ρa1 a2 +a22
dBt1 + √
a2
1−ρ2
a21 +2ρa1 a2 +a22
√
a1
1−ρ2
a21 +2ρa1 a2 +a22
dBt2 , (12) dBt2 , (13)
where Bt1 and Bt2 are two independent standard Brownian motions. Proof. Let us introduce two new processes: Ut1 Ut2
= a1 Wt1 + a2 Wt2 , = −(a2 + ρa1 )Wt1 + (a1 + ρa2 )Wt2 .
(14) (15)
It is easy to show that Ut1 and Ut2 are independent, Ut1 is a Brownian bridge starting at 0 and ending at b, and Ut2 is a Brownian motion. Thus we can write dynamics of Ut1 and Ut2 , which turns out to be q b − Ut1 dUt1 = dt + a21 + 2ρa1 a2 + a22 dBt1 , (16) T −t q p 1 − ρ2 a21 + 2ρa1 a2 + a22 dBt2 , (17) dUt2 = where Bt1 , Bt2 are independent standard Brownian motions. If we express these equations in terms of Wt1 and Wt2 , we get q b−a1 Wt1 −a2 Wt2 dt + a21 + 2ρa1 a2 + a22 dBt1 , (18) d(a1 Wt1 + a2 Wt2 ) = T −t q p d(−(a2 + ρa1 )Wt1 + (a1 + ρa2 )Wt2 ) = 1 − ρ2 a21 + 2ρa1 a2 + a22 dBt2 . (19) Now we can solve for dWt1 and dWt2 to obtain dWt1 dWt2
3
1
=
1
=
√
2
b−a Wt −a2 Wt (a1 + ρa2 ) (T −t)(a12 +2ρa dt + √ 2 1 a2 +a )
(a2 +
2
b−a Wt1 −a2 Wt2 ρa1 ) (T −t)(a12 +2ρa 2 1 a2 +a2 ) 1
dt + √
a1 +ρa2 a21 +2ρa1 a2 +a22
dBt1 − √
a2 +ρa1 a21 +2ρa1 a2 +a22
dBt1
a2
√
+√
1−ρ2
a21 +2ρa1 a2 +a22 a1
1−ρ2
a21 +2ρa1 a2 +a22
dBt2 , (20) dBt2 . (21)
Stock Merger
Theorem 3.1 Assume that the dynamics of two stock prices is given by dSt1 dSt2
= St1 (µ1 dt + σ1 dWt1 ), = St2 (µ2 dt + σ2 dWt1 ),
(22) (23)
where Wt1 and Wt2 are two correlated Brownian motions with dWt1 · dWt2 = ρdt. Moreover, if the condition ST1 = C · ST2 (24)
3
is given, then the dynamics of stock prices can be expressed as dSt1 St1
=
1 1 σ1 (σ1 −ρσ2 ) log(CSt2 /St1 )+(T −t) µ1 σ2 (σ2 −ρσ1 )+(µ2 + 2 σ12 − 2 σ22 )σ1 (σ1 −ρσ2 ) (T −t)(σ12 −2ρσ1 σ2 +σ22 )
+ √ σ21 (σ1 −ρσ2 )
√ σ σ dBt1 + √ 21 2 2
dt
1−ρ2
σ1 −2ρσ1 σ2 +σ22 1 2 1 2 1 2 σ2 (σ2 −ρσ1 ) log(St /CSt )+(T −t) µ2 σ1 (σ1 −ρσ2 )+(µ1 + 2 σ2 − 2 σ1 )σ2 (σ2 −ρσ1 ) σ1 −2ρσ1 σ2 +σ2
dSt2 St2
=
(25)
dt
(T −t)(σ12 −2ρσ1 σ2 +σ22 )
− √ σ22 (σ2 −ρσ1 )
dBt2 ,
√ σ σ dBt1 + √ 21 2 2
1−ρ2
σ1 −2ρσ1 σ2 +σ22
σ1 −2ρσ1 σ2 +σ2
dBt2 ,
(26)
where Bt1 and Bt2 are two independent standard Brownian motions. Proof. For geometric Brownian motion we have Sti = S0i exp((µi − 21 σi2 )t + σi Wti ).
(27)
Therefore the condition ST1 = C · ST2 translates as 2 CS σ1 WT1 − σ2 WT2 = µ2 − µ1 − 12 (σ22 − σ12 ) T + log S 10 .
(28)
0
This identity defines planar Brownian bridge with a1 = σ1 , a2 = −σ2 , and
2 CS b = µ2 − µ1 − 21 (σ22 − σ12 ) T + log S 10 . 0
The dynamics of the planar Brownian bridge was given in Theorem 2.1 in terms of two independent standard Brownian motions Bt1 and Bt2 . Plugging expressions (12) and (13) in (22) and (23), we conclude the proof.
3.1
Special Cases
One important special case is when we have just a single convergent stock, i.e., condition ST1 = C is given. This is just a special case of the above analysis with S02 = 1, σ2 = µ2 = 0. Substituting back to (25), we conclude: Corollary 3.2 Suppose that dSt = St (µ1 dt + σ1 dWt1 ),
(29)
log ST − log St 1 2 1 + σ1 dt + σ1 dBt . T −t 2
(30)
and ST is known. Then dSt = St
Remark 3.3 Notice that the actual drift µ1 has absolutely no relevance to the dynamics of the stock price if we know its terminal price.
4
4
Optimal Strategy
Suppose that there is a trader who trades in the two stocks of companies which will merge and their stock prices evolve according to (25) and (26). The wealth of the investor Ytq follows dYtq = qt1 dSt1 + qt2 dSt2 ,
(31)
where qt1 and qt2 are investor’s positions in the market. The objective of the investor is to maximize his expected wealth (or utility of his expected wealth) subject to some trading constraints. We might assume that his absolute number of shared purchased is limited (say |qti | ≤ 1, or |qt1 St1 | + |qt2 St2 | ≤ K), or we may assume that his wealth gets never below to some specific level (Yt ≥ L for all t). Let’s assume for the following analysis that the investor wants to maximize the expected profit resulting from his trading subject to the trading constraint |qti | ≤ 1. We have the following result in this case: Theorem 4.1 The optimal strategy maximizing EYTq subject to |qti | ≤ 1 is given by 2 CS qt1 = sgn(σ1 − ρσ2 ) · sgn S 1t − Rt1 , t 1 St 2 qt = sgn(σ2 − ρσ1 ) · sgn CS 2 − Rt2 ,
(32) (33)
t
where Rt1
=
Rt2
=
h i 2 (σ2 −ρσ1 ) − µ2 , exp (T − t) − 12 (σ12 − σ22 )) − µσ1 σ1 (σ 1 −ρσ2 ) i h 1 (σ1 −ρσ2 ) − µ . exp (T − t) − 12 (σ22 − σ12 ) − µσ2 σ2 (σ 1 2 −ρσ1 )
(34) (35)
Proof. We can write Z T EYTq = Y0 + E (qt1 dSt1 + qt2 dSt2 )dt
(36)
0
Z
T
= Y0 + E 0
Z +E
T
qt2
qt1
1 1 σ1 (σ1 −ρσ2 ) log(CSt2 /St1 )+(T −t) µ1 σ2 (σ2 −ρσ1 )+(µ2 + 2 σ12 − 2 σ22 )σ1 (σ1 −ρσ2 )
(T −t)(σ12 −2ρσ1 σ2 +σ22 ) 1 1 σ2 (σ2 −ρσ1 ) log(St1 /CSt2 )+(T −t) µ2 σ1 (σ1 −ρσ2 )+(µ1 + 2 σ22 − 2 σ12 )σ2 (σ2 −ρσ1 )
0
(T −t)(σ12 −2ρσ1 σ2 +σ22 )
dt
dt. (37)
The investor then chooses long position if the dt term is positive, or short position if it is negative. The signum of the dt term is given by formulas (32) and (33), which concludes the proof.
4.1
Special Cases
Corollary 4.2 When we know that ST = C, the optimal strategy reduces to qt∗ = sgn(log( SSTt ) + 12 σ 2 · (T − t)). Proof. Immediately follows from the expression (30).
5
(38)
Remark 4.3 Notice that this strategy qt∗ is different from the strategy which locks the arbitrage coming from the knowledge of the terminal stock price ST . For example if r = 0, then in order to lock the arbitrage, we would sell the stock when St > ST , and we would buy the stock if ST > St , which would guarantee us the profit |ST − St |. But the strategy which maximizes the expected profit q ∗ suggests that we should buy the stock when ST · exp( 12 σ 2 · (T − t)) > St and sell the stock otherwise. This is rather surprising, because when ST · exp( 12 σ 2 · (T − t)) > St > ST , one should buy the stock in order to maximize the expected profit, even knowing that the stock price would drop.
6
