VALUING TRINOMIAL OPTION PRICING WITH PSEUDOINVERSE MATRIX
J. Indones. Math. Soc. (MIHMI) Vol. 12, No. 2 (2006), pp. 131–140.
VALUING TRINOMIAL OPTION PRICING
WITH PSEUDOINVERSE MATRIX
Abdurakhman, Subanar, S. Guritno, and Z. Soejoeti
Abstract.In this paper we investigate the trinomial model for European call option pricing theory using pseudoinverse matrix. Here we use pseudoinverse matrix to find the risk neutral pseudoprobability (rnpp). We prove that this rnpp yield least square error portfolio. Our result generalizes the risk neutral probability of CRR binomial model.
1. INTRODUCTION
The obvious difference between the pricing and hedging problem in completemarkets as compared to incomplete one is that in incomplete case the price of
option is not unique, see Kamrad and Ritchken [3] , Melnikov [4]. Unfortunately
actual markets are incomplete. There are many reasons for that. A number of
prominent examples are transaction costs, short sale, or even credit risk. A player
in the market appear both as a seller and as a buyer of the option. Both of them
want to make some profits. In this paper we propose trinomial model as an example
of incomplete model and we obtain a unique prices that minimizes the difference
profit both a seller and a buyer.For a rectangular matrix A m×n we can find pseudoinverse matrix notified by
- A that fulfill four conditions below >a. AA A
b. A AA = A , Received 7 March 2005, Revised 20 October 2005, Accepted 28 February 2006.
2000 Mathematics Subject Classification : 62P05. Key words and Phrases : trinomial, pseudoprobability, minimizing risk.
′
Abdurakhman, et al.
c. (AA ) = AA , ′
d. (A
A) = A A. Pseudoinverse of matrix A can be obtained by using algebra theorem below: B
Theorem 1.1. Suppose we have a matrix A m×n and fulfill P AQ = ,
B is a r × r non singular matrix, the pseudoinverse of matrix A m×n is
−1 B U
- A = Q P, with
n×m
V W
− − −
1
1
+ + +
1 U P P ,
V Q B , W Q B P P ,
1 1 = Q
1
1
2 = −B 2 = −Q
2
2 ✔ ✕ − − ✏ ✑
1 ✏ ✑
1 ′ ′ ′ ′
P 1 ✂ ✄ m ×m n ×n + + P = , Q = Q
1 Q 2 , P = P P
2 P , Q = Q Q
2 Q ,
2
2
2
2
2
2 P
2
where P and Q is an elementary row (column) matrix operation of matrix A m×n ,
r is rank of matrix A m×n .
2. ONE-PERIOD OPTION
Here stock prices are assumed to follow trinomial model: the initial stockprice which is denoted by S . At each time step the stock price move by factors
of ρ has three possibilities value (1 + a ) , (1 + a ) , (1 + a ) with probabilities
1
1
2
3
p , p , p respectively with p + p + p = 1. The bank account process is non-
1
2
3
1
2
3
random and is given by B = 1 and B = (1 + r), r is an interest rate positive
1
constant. The movement of the stock change can be represented with the following
diagram:(1 + a
1 ) S , prob p
1 S ≺ (1 + a 2 ) S , prob p
2
(1 + a ) S , prob p
3
3
t = 0 t = 1 First we start with one period away. Let C be the current of the call,
and C , C , C be its value at the end of the period if the stock price goes
1
2
3
to (1 + a ) S , (1 + a ) S , and (1 + a ) S respectively. Then we have payoffs
1
2
3
function at the end of period for the option
Option pricing with pseudoinverse and K is contract value. Therefore, C
1 = max {0, (1 + a 1 ) S − K} , prob p
1 C ≺ C = max {0, (1 + a ) S − K} , prob p
2
2
2 C = max {0, (1 + a ) S − K} , prob p
3
3
3 t = 0 t = 1.
In the reason for seeking rational option price, from the value of call C we
form a portfolio that containing γ shares of stock and the dollar amount β in riskless
bonds. So we have the value of this portfolio is γS + β. At the end period the
value of this portfolio will have three possibilitiesγ (1 + a ) S + (1 + r) β, probability p
1
1
(γS + β) ≺ γ (1 + a ) S + (1 + r) β, probability p
2
2
γ (1 + a ) S + (1 + r) β, probability p
3
3 t = 0 t = 1.
We choose the value of (γ, β) to equate the end of period values of the portfolio and the payoff function. So we have hedging equation for trinomial model:
S (1 + a ) 1 + r C
1
1
γ S (1 + a ) 1 + r = C (1)
2 2 .
β
S (1 + a ) 1 + r C
3
3 Equation (1) is overdetermined, or the market is incomplete. It means that the
value of (γ, β) is not always satisfy the hedging. Special case it can be satisfied if
the hedging equation is in a linear plot. Here we propose a special technique using
pseudoinverse-matrix in Theorem (1.1) to find a solution with special property, by
taking a = b, a = c, a = a we get1
2
3 1 − # c−
" 2 2a+b ( )
1 2 2 2 γ 1 0 − − −
ca ab bc (c +a +b )S
= S (1+c) × Q
1 1 (a−b)(−b+c)
β − 2 2 2
1+r 1+r 0 1 − − − c ca ab bc 2 +a +b
with
1
1
− C
S (b−c) S (b−c)
1
Q =
1 C
2 c−a a−b
C
1
3 b−c b−c
Taking C = k , C = l, and C = m, we have
1
2
3
1 kc + ka − 2kb − 2lc + la + lb + mc − 2ma + mb γ = −
2
2
2
Abdurakhman, et al.
and 2− − −
2 2 2 2 2k (c bc +c−2b+a+a ab )+l(−bc−ca−2c+a +b+b +a)+m(c−2a+b+c ca−ab +b ) 2 2 2
β = −
2(1+r)(c +a +b ca−ab−bc )
The value of (γ, β) is not free from risk like in binomial model, see Cox, Ross,
Rubeinstein [2] , Boyle [1] . Automatically the payoff function and the value of the
portfolio is not always same. By the way, pseudoinverse has interesting property
that is it can minimize the risk. Finally we can obtain the option price after we
have (γ, β) as follows: γS
1 C =
(2) − − β 2 2 2 2
1 kab−lar−kar +2kbr+2lcr−lca−lbc+kc +ka +la +lb kbc−kcr−lbr
= 2
− − −
2
2 +ca ab bc 2 (1+r)(c +a +b ) − − 2 2 mcr mbr 1 +2mar−mab−mca+mc +mb 2 − − − 2 2 ca ab bc
2 (1+r)(c +a +b ) − 2 2 2 2 1 − − 1 2rb−ra−rc−ab−bc+a +c 2 2 2 1 rb +2rc−ra+b bc−ac +a 2 2 2
= (1 + r) C C
− − − + +1 − − −
2 c ca ab bc c ca ab bc 2 +a +b 2 2 2 +a +b 1 2ra−rb−rc−ac−ab+c +b 2 2 2 − − − C
3 c ca ab bc 2 +a +b The option price in equation (2) can be simplified by taking E(ρ − r) = 0.
1 This assumption is realistic, due to both ρ and r come from one term, return. For
1 above assumption we unfortunately have two equation for three unknown.
p
1
a
1 a 2 a 3 r
p
2 =
1
1
1
1
p
3 A p = r 2×3 e 3×1 e 2×1
- p = A r e 3×1 e 2×1
3×2
Gauss elimination method is not working here to find a unique solution for p
1 , p 2 , p 3 .
So we use pseudoinverse matrix. Once again by taking a
1 = b, a 2 = c, a 3 = a, we
obtain − c c−a
1
b−c (b−c) b b−a
Q 3×3 = −
b−c b−c
1 −
1
1 B
1 = 2 2 V −
(b−c)(−c+a) c
1 2 2 2 1 +ac−b +ab 2 2 2 − − − − − c ac ab−bc c ac ab−bc 2 +a +b
2 +a +b
" #
1 b
P =
1
2×2
−
1 b Option pricing with pseudoinverse −
1 Here U and W are null matrix because from B and V are enough to get
pseudoinverse of A. Further we obtain pseudoinverse matrix A:
2
2
−2b + a + c −ab − bc + a + c
1
1
−
2
2
2
2
2
2
2
2
c − ac + a + b − ab − bc c − ac + a + b − ab − bc
2 2
- 1
b − 2c + a b − bc − ac + a
1 A =
−
3×2
2
2
2
2
2
2
2
2
c − ac + a + b − ab − bc c − ac + a + b − ab − bc
2
2
−c + 2a − b −c + ac − b + ab
1
1
−
2
2
2
2
2
2
2
2
c − ac + a + b − ab − bc c − ac + a + b − ab − bc Finally the new probabilities are obtained:
2
2
2rb − ra − rc − ab − bc + a + c
1
2
2
2
2
c − ac + a + b − ab − bc p
1
2
2
2rc − rb − ra + b − bc − ac + a
1
p p =
2 =
(3)
e
2
2
2
2
c − ac + a + b − ab − bc p
3
2
2
2ra − rc − rb + c − ac + b − ab
1
2
2
2
2
c − ac + a + b − ab − bc
called pseudoprobabilities. From Gauss elimination method we have relationship
between a i and rr − c (c − a) p
1
p = ≥ 0 +
3
b − c b − c
r − c + (c − a) p
1
=
b − c
b − c > 0 ⇐⇒ r − c + (c − a) p ≥ 0
1
c − r p
1 ≥ ≥ 0 ⇐⇒ c ≥ r
c − a So we must take take c ≥ r. Finally the condition below is right
a < 0 ≤ r ≤ c < b ⇔ a < 0 ≤ r ≤ a < a
3
2
1 Using these probabilities the option price in equation (2) can be simplified as an expectation of the payoffs function in the pseudoprobability world.
3
−
X
1 C = (1 + r) p i × C i i − =1
1 = (1 + r) E P (C T ) .
=1
From above calculation we can derive a theorem about trinomial option pricing as
follows:Theorem 2.1. For a one-period trinomial european option pricing model with
payoff functions C i = max{0, S (1 + a i ) − K} the fair price is determined as follows
−1
Abdurakhman, et al.
Resetting the notation of a i , the pseudoprobability in equation (3) can be represented as general formula
3
3
3 P P P
2
a − r a i + 3ra j − a j a i
i i i i =1 =1 =1
p j = , j = 1, 2, 3 . (4)
2
3
3 P P
2
3 a − a i
i i i =1 =1
Until now we have find a general formula for trinomial model. Take a
look the risk neutral pseudoprobability in trinomial case. We have an interesting
relationship between risk neutral probability in binomial model by CRR [2] and
our pseudoprobability. The result is represented in the next propositionProposition 2.1. Risk neutral probability in CRR binomial model [2] can be
generalized as a pseudoprobability formulaFor the binomial model, take ρ = {a , a } , a < 0 ≤ r < a and r as positive
1
1
2
2
1 2 r−a
interest rate. We know that in binomial risk neutral probability p = p = ,
1 − − a 1 a 2 a 1 r 1 − p = p = .
2 − a 1 a 2 Next both of them will be proved follow pseudoprobability formula
r − a a − a
2
1
2
p = ×
1
a − a a − a
1
2
1
2
2
2
a + a + ra
1 − ra 2 − a 1 (a 1 + a 2 )
1
2
=
2
2
a + a − 2a a
1
2
1
2 P P P
2
a − r a i + 2ra − a a i
i
1
1
= , i = 1, 2
P P2
2
2 a − ( a i )
i
a
1 − r a 1 − a
2
p = ×
2
a − a a − a
1
2
1
2
2
2
a + a − ra + ra − a (a + a ) − ra + ra
1
2
2
1
2
2
2
1
2
=
2
2
a + a − 2a a
1
2
1
2 P P P
2
a − r a i + 2ra − a a i
i
2
2
= , i = 1, 2 .
P P
2
2
2 a − ( a i )
i Option pricing with pseudoinverse
3. n-PERIOD OPTION
After finishing one period option pricing trinomial model, let see the twoperiods expiration time. The stock can take on six possible values after two periods:
2
(1 + a ) S ,
1
(1 + a ) (1 + a ) S (1 + a ) S ≺
1
2
1
(1 + a ) (1 + a ) S
1
3
(1 + a ) (1 + a ) S
2
1
2 S ≺ (1 + a 2 ) S ≺ (1 + a ) S ,
2
(1 + a ) (1 + a ) S
2
3
(1 + a ) (1 + a ) S
1
3
(1 + a
3 ) S ≺ (1 + a ) (1 + a ) S
2
3
2
(1 + a ) S
3
t = 0 t = 1 t = 2 . Similarly, the payoff function for the call after two periods:
2 C = max{0, (1 + a ) S − K}
11
1 C ≺ C 12 = max{0, (1 + a 1 ) (1 + a 2 ) S − K}
1 C 13 = max{0, (1 + a 1 ) (1 + a 3 ) S − K}
C
21 = max{0, (1 + a 2 ) (1 + a 1 ) S − K}
2 C ≺ C ≺
2 C 22 = max{0, (1 + a 2 ) S − K}
C
23 = max{0, (1 + a 2 ) (1 + a 3 ) S − K}
C
31 = max{0, (1 + a 3 ) (1 + a 1 ) S − K}
C ≺
3 C 32 = max{0, (1 + a 3 ) (1 + a 2 ) S − K}
2 C 33 = max{0, (1 + a 3 ) S − K}
t = 0 t = 1 t = 2
C ij stands for the value of a call two periods from the current time if the stock
move by factor (1 + a i ) × (1 + a j ) ; i, j = 1, 2, 3. From the previous analysis we have
the relations between one period payoff function and two period payoff function
−1 C 1 = (1 + r) (C − 11 × (p 1 ) + C 12 × (p 2 ) + C 13 × (p 3 )) (5)
1 C = (1 + r) (C × (p ) + C × (p ) + C × (p ))
2 −
21
1
22
2
23
3
1 C = (1 + r) (C × (p ) + C × (p ) + C × (p ))
3
31
1
32
2
33
3
Abdurakhman, et al.
equation (5) into equation (2) we obtain3 −
X
1 C = (1 + r) p i × C i
(6)
i − =1
2
2
2
2
= (1 + r) p C + 2p p C + p C + 2p p C + p C + 2p p C
11
1
2
12
22
1
3
13
33
2
3
13
1
2
3
2
2
n o P P j i j
i 2−i−j 2−i−j 2!
p p p max 0, (1 + a ) (1 + a ) (1 + a ) S − K
1
2
3 i
1
2
3 !j!(2−i−j)! i =0 j =0
=
2
(1 + r) ≫ for i + j ≤ 2
By backward method we have a recursive procedure for obtaining the value of a
call with any number to go. We can generalize equation (6) with n period and give:
n n
n o P P j n−i−j i j n−i−j
n ! i
p p p max 0, (1 + a ) (1 + a ) (1 + a ) S − K
1
2
3 i
1
2
3 !j!(n−i−j)! i j
=0 =0
C =
n
(1 + r) ≫ for i + j ≤ n
The result can be represented as a theorem for general trinomial option pricing
model.Theorem 3.1. For a trinomial european option pricing model with payoff functions
max (S n − K, 0), and n periods time, the fair price is determined by the formula
below :n n
n o P P j n−i−j i j n−i−j
n i !
p p p max 0, (1 + a ) (1 + a ) (1 + a ) S − K
1
2
3 i
1
2
3 !j!(n−i−j)! i =0 j =0
C =
n
(1 + r) ≫ for i + j ≤ n It is clear that the one period valuation are valid for any number number of
periods. The value of a call is an expectation of the payoff function in a pseudo-
probability world. Example 3.1.We have a stock price S = 1$ = 8000 rp. Stock change price −
1
1
movement follows trinomial movement a = , a = , and a = 0, with
1
3
2
10
10 K = 8000 rp . Take that B = 1 rp and r = 0. We have pseudoprobability ∗ p = [1/3, 1/3, 1/3]. Option price can be calculated:
e
1
1 C = (8800 − 8000) × + (8000 − 8000) × + 0
3
3 800 = rp. Option pricing with pseudoinverse With least square error strategy we have γ ∗
1
f a
1
= −3.7333. That means
the option seller gets 800/3 rp and borrows 3733.3 rp to gets 0.5 dollars. We will
show that strategy (γ = 0.5, β = −3733.33) has minimum sum square of error.γ β X a
1
f a
2
f a
= 0.5 and β ∗
3
1 X a
3 SSE
2 X a
26.7 157866.7 0.35 −2533.3 546.7 266.7 26.7 135466.7 0.4 −2933.3 586.7 266.7 −13.3 119466.7
0.45 −3333.3 626.7 266.7 −53.3 109866.7 0.5 -3733.3 666.7 266.7 -93.3 106666.7
0.55 −4133.3 706.7 266.7 −133.3 109866.7 0.6 −4533.3 746.7 266.7 −173.3 119466.7 0.65 −4933.3 786.7 266.7 −213.3 135466.7 0.7 −5333.3 826.7 266.7 −253.3 157866.7
0.8 −6133.3 906.7 266.7 −333.3 221866.7 0.9 −6933.3 986.7 266.7 −413.3 311466.7 1 −7733.3 1066.7 266.7 −493.3 426666.7
4. CONCLUSION
Using pseudoinverse matrix to solve incomplete market problem we obtainunique solution. In trinomial option pricing formula there are no risk-neutral
probability, but pseudoprobability that minimize the risk.Acknowledgement.
The financial support from the government of Indonesia via
BPPS Project is gratefully acknowledged. I also would like to thank to a referee
for his or her comment.
REFERENCES
1. P.P. Boyle, Options and The Management of Financial Risk, Society of Actuaries,
1992.
2. J.C. Cox, S.A. Ross, M. Rubeinstein, “Option pricing : a simplified approach”,
Journal Financial of Economics, 1979, 7, 229-263.
3. B. Kamrad, and P. Ritchken, “Multinomial approximating models for options with
k state variables”, Management Science, 37(1991), 21-31.
4. A.V. Melnikov, “Financial markets stochastic analysis and the pricing of derivative
securities”, Translations of Mathematical Monographs, 1999, 184, pp. 63-72.266.6 266.7 800 266.7 266.7 426666.7 0.1 −533.3 346.7 266.7 186.7 311466.7 0.2 −1333.3 426.7 266.7 106.7 221866.7 0.3 −2133.3 506.7 266.7
Abdurakhman, et al.
Abdurakhman : Department of Mathematics, Universitas Gadjah Mada, Yogyakarta 52281, Indonesia. E-mail: rachmanstat@ugm.ac.id.Subanar : Department of Mathematics, Universitas Gadjah Mada, Yogyakarta 55281,
Indonesia. E-mail: subanar@yahoo.com S. Guritno: Department of Mathematics, Universitas Gadjah Mada , Yogyakarta 52281, Indonesia. Z. Soejoeti : Department of Mathematics, Universitas Gadjah Mada , Yogyakarta 52281, Indonesia.