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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.


The obvious difference between the pricing and hedging problem in complete markets 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 matrixAm×n we can find pseudoinverse matrix notified by A+ that fulfill four conditions below

a. AA+A=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.


c. (AA+)′ =AA+,

d. (A+A)


Pseudoinverse of matrixA can be obtained by using algebra theorem below:

Theorem 1.1. Suppose we have a matrix Am×n and fulfill P AQ =

B 0

0 0


B is a r × r non singular matrix, the pseudoinverse of matrix Am×n is A+n×m=Q

B−1 U


P, with

U =−B

−1 P1P


2 , V =−Q

+ 2Q1B


, W =Q+2Q1B

−1 P1P2,

Pm×m= ✔

P1 P2

, Qn×n= ✂

Q1 Q2

, P2+=P ′ 2

P2P ′ 2

✑−1 , Q+2 =


✑−1 Q′2,

whereP andQis an elementary row (column) matrix operation of matrix Am×n,

r is rank of matrixAm×n.


Here stock prices are assumed to follow trinomial model: the initial stock price which is denoted by S0. At each time step the stock price move by factors

of ρ1 has three possibilities value (1 +a1), (1 +a2), (1 +a3) with probabilities

p1, p2, p3 respectively with p1+p2+p3 = 1. The bank account process is

non-random and is given by B0 = 1 and B1 = (1 +r), r is an interest rate positive

constant. The movement of the stock change can be represented with the following diagram:

S0 ≺

t= 0

(1 +a1)S0, probp1

(1 +a2)S0, probp2

(1 +a3)S0, probp3

t= 1

First we start with one period away. Let C be the current of the call, and C1, C2, C3 be its value at the end of the period if the stock price goes

to (1 +a1)S0, (1 +a2)S0, and (1 +a3)S0 respectively. Then we have payoffs

function at the end of period for the option


andK is contract value. Therefore,

In the reason for seeking rational option price, from the value of callC we form a portfolio that containingγshares of stock and the dollar amountβin riskless bonds. So we have the value of this portfolio is γS0+β. At the end period the

value of this portfolio will have three possibilities

(γS0+β) ≺

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:


and 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:

This assumption is realistic, due to bothρ1andrcome from one term, return. For

above assumption we unfortunately have two equation for three unknown.

Gauss elimination method is not working here to find a unique solution forp1, p2, p3.


Here U and W are null matrix because from B−1 and V are enough to get

pseudoinverse of A.Further we obtain pseudoinverse matrixA:


Finally the new probabilities are obtained:


called pseudoprobabilities. From Gauss elimination method we have relationship betweenai andr

So we must take takec≥r. Finally the condition below is right

a <0≤r≤c < b⇔a3<0≤r≤a2< a1

Using these probabilities the option price in equation (2) can be simplified as an expectation of the payoffs function in the pseudoprobability world.

C = (1 +r)−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 functionsCi= max{0, S0(1 +ai)−K}the fair price is determined as follows


Resetting the notation of ai, the pseudoprobability in equation (3) can be represented as general formula



P i=1


i −r


P i=1

ai+ 3raj−aj


P i=1




P i=1


i − P3



2 , j= 1,2,3. (4)

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 proposition

Proposition 2.1. Risk neutral probability in CRR binomial model [2] can be generalized as a pseudoprobability formula

For the binomial model, takeρ1={a1, a2}, a2<0≤r < a1andras positive

interest rate. We know that in binomial risk neutral probabilityp=p1 = r −a2


1−p=p2= a1 −r a1−a2.

Next both of them will be proved follow pseudoprobability formula

p1 = r−a2




= a





= P


i −r P

ai+ 2ra1−a1Pai 2Pa2

i −( P

ai)2 , i= 1,2

p2 = a1−r

a1−a2 ×



= a





= P


i −r P

ai+ 2ra2−a2Pai 2Pa2

i −(



After finishing one period option pricing trinomial model, let see the two periods expiration time. The stock can take on six possible values after two periods:

(1 +a1)S0 ≺

S0 ≺ (1 +a2)S0 ≺

(1 +a3)S0 ≺

t= 0 t= 1

(1 +a1)2S0,

(1 +a1) (1 +a2)S0

(1 +a1) (1 +a3)S0

(1 +a2) (1 +a1)S0

(1 +a2)2S0,

(1 +a2) (1 +a3)S0

(1 +a1) (1 +a3)S0

(1 +a2) (1 +a3)S0

(1 +a3)2S0

t= 2.

Similarly, the payoff function for the call after two periods:

C1 ≺

C ≺ C2 ≺

C3 ≺

t= 0 t= 1

C11= max{0,(1 +a1)2S0−K}

C12= max{0,(1 +a1) (1 +a2)S0−K}

C13= max{0,(1 +a1) (1 +a3)S0−K}

C21= max{0,(1 +a2) (1 +a1)S0−K}

C22= max{0,(1 +a2)2S0−K}

C23= max{0,(1 +a2) (1 +a3)S0−K}

C31= max{0,(1 +a3) (1 +a1)S0−K}

C32= max{0,(1 +a3) (1 +a2)S0−K}

C33= max{0,(1 +a3)2S0−K}

t= 2

Cij stands for the value of a call two periods from the current time if the stock move by factor (1 +ai)×(1 +aj) ;i, j= 1,2,3.From the previous analysis we have the relations between one period payoff function and two period payoff function

C1= (1 +r) −1

(C11×(p1) +C12×(p2) +C13×(p3)) (5)

C2= (1 +r) −1

(C21×(p1) +C22×(p2) +C23×(p3))

C3= (1 +r) −1

(C31×(p1) +C32×(p2) +C33×(p3))


equation (5) into equation (2) we obtain

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) withnperiod and give:


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 (Sn−K,0), and n periods time, the fair price is determined by the formula below :

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 S0 = 1$ = 8000 rp. Stock change price


With least square error strategy we have γ∗

1 = 0.5 andβ ∗

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 Xa1 fa1 Xa2 fa2 Xa3 fa3 SSE

0 266.6 266.7 800 266.7 0 266.7 0 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 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


Using pseudoinverse matrix to solve incomplete market problem we obtain unique 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.


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

kstate variables”,Management Science, 37(1991), 21-31.


Abdurakhman: Department of Mathematics, Universitas Gadjah Mada, Yogyakarta 52281, Indonesia.


Subanar: Department of Mathematics, Universitas Gadjah Mada, Yogyakarta 55281, Indonesia.


S. Guritno: Department of Mathematics, Universitas Gadjah Mada , Yogyakarta 52281, Indonesia.