The Pricing Problem of Rainbow Option in Uncertain Financial Market

MINGCHONG LIAO, YUANGUO ZHU

School of Mathematics and Statistics

Nanjing University of Science and Technology

Nanjing 210094, Jiangsu

CHINA

Abstract: - In this paper we mainly investigate pricing problems of rainbow option under uncertain financial

market. The price of the underlying asset is assumed to obey an uncertain process. Uncertain differential equa-

tions are used to build a price model. Furthermore, the differential equations under the uncertain mean-reverting

model are solved to deduce the pricing formulas of several rainbow options. Additionally, in order to verify the

reasonableness of our pricing formulas, some numerical experiments are designed to show the prices of these

options.

Key-Words: - Uncertainty theory; Option pricing; Rainbow option; Uncertain differential equation.

Received: August 26, 2021. Revised: March 21, 2022. Accepted: April 25, 2022. Published: July 6, 2022.

1 Introduction

An option is a contract in which the rights and obli-

gations of both parties are not equal, and the buyer

pays an option fee in exchange for the right to buy or

sell an asset at an agreed price on or before the expi-

ration date. Although options have been around since

the late 18th century, they are not widely used due to

the pricing problem of option fees. Until 1973, the fa-

mous Black-Scholes formula on the basis of stochas-

tic differential equations proposed by Black and Sc-

holes [2] made great progress in option pricing theory,

and option prices were expressed by various differen-

tial equations. After more in-depth research and fur-

ther expansion of the B-S formula, it has been applied

to many other financial derivatives pricing models.

With the development of option pricing theory, op-

tions have become the most dynamic derivative finan-

cial products, which have been rapidly developed and

widely used, and the types of options have also in-

creased very quickly. Among them, options involving

two or more risky assets are often referred to as rain-

bow options. In 2015, the price of two path-dependent

derivatives was determined using the disturbance the-

ory in a two-dimensional asset model with random

correlation and volatility by Marcos et al. [6]. Wang

et al. studied the pricing problem of fragile Euro-

pean options under Markov modulation jump diffu-

sion process in 2017 [21]. In 2020, Edeki et al. use the

separated variable transformation method (HSVTM)

to give a exact (closed form) solution of the classical

Black Scholes option pricing model with time fraction

[7]. Later in 2021, Aimi and Guardasoni [1], relying

on the collocated Boundary Element method, extend-

ed a semi-analytical approach to the barrier option

technique to barrier option pricing with earnings de-

pendent on multiple assets. Under the Merton jump-

diffusion model, Ghosh and Mishra [9] studied the

fast, parallel, and numerically accurate pricing of two-

asset American options in 2022.

The traditional option pricing theory have often s-

tarted from the perspective of probability theory, re-

garding the underlying asset price as a Wiener pro-

cess, and construct price models based on this, then

conducts further derivation. However, the traditional

option pricing theory requires the sample size is large

enough to find a distribution function that is close e-

nough to the frequency, but in many cases, for various

reasons, we often cannot obtain enough samples or

even no sample to find the available probability distri-

butions. In these cases, we need to rely on the expert's

belief degree in each uncertain event.

In 2007, for describing the belief degree, Liu [10]

proposed an uncertainty theory, which is based on the

axiomatic system of regularity, duality, subadditivity

and product measure, to solve the mathematical prob-

lem that cannot be solved by probability theory due to

insufficient sample data or the absence of sample, and

further refined the theory in 2010 [12]. At the same

year, Chen and Liu [4] studied the existence and u-

niqueness theorem for uncertain differential equation-

s. In 2013, Liu [14] utilized some paradoxes to prove

that the actual price of a stock does not follow any

of Ito's stochastic differential equations, which over-

threw the view of traditional stochastic financial the-

ory. Therefore, it is reasonable to express asset prices

with uncertain differential equations.

Liu [11] first introduced uncertainty theory into the

stock model in 2009, he used the geometric Liu pro-

cess to construct uncertain stock models and derived

the pricing formula for European call options. Fur-

thermore, based on uncertain differential equations,

Zhu [25] studied the application of uncertain optimal

control problem in portfolio selection models. After

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that, Chen [3] deduced the pricing formula for Amer-

ican options in 2011, Sun and Chen [17] derived the

pricing formula for Asian options in 2015. In addi-

tion, in 2011, Peng and Yao [16] considered an op-

tion pricing model with mean reversion process and

derived the pricing formulas for European and Amer-

ican options under the model. Chen et al. [5] pro-

posed an uncertain stock model with periodic divi-

dends. Additionally, Yao [23] proposed an uncertain

floating-rate stock model, in which both stock prices

and interest rates follow uncertain differential equa-

tions. Sun and Su [18] proposed a mean-reverting

stock model under floating interest rates. Yang et

al. [22] derived a pricing formula for Asian bar-

rier options in 2019. In the same year, Gao et al.

[22]deduced a pricing formula of American barrier

options. Lu et al. [15] proposed an uncertain stock

model based on fractional differential equations, and

discussed the pricing of European-style options un-

der this model. Tian et al. [19] determined a barrier

option pricing problem under the mean-reverting s-

tock model. In 2021, Wang and Ralescu[20] studied

pricing formulas of lookback option for the uncertain

Heston volatility model. Gao [8] studied the pricing

problem of Asian rainbow options based on uncertain

stock models.

In this paper, by leveraging knowledge of uncer-

tainty theory, the price of the underlying asset is treat-

ed as an uncertain process, based on uncertain differ-

ential equations, while taking into account the mean

reversion characteristics of asset prices. After that,

the pricing formulas of several types of rainbow op-

tions are deduced. This paper is organized as follows:

The second part of this paper briefly states some the-

orems and definitions related to this paper. The third

part derives the pricing of put 2 and call 1 type rain-

bow options. The fourth section studies rainbow call

on max option and rainbow call on min option. The

fifth part deduces the pricing formula of rainbow put

options, including rainbow put on max option and

rainbow put on min option. Section sixth of this paper

gives some brief conclusions.

2 Preliminaries

Definition 1 (Liu [11] [13]) Providing that (Γ,L)is

a measurable space. A set function M:L → [0,1]

is called an uncertain measure if it satisfies the fol-

lowing conditions: (i) (normality axiom) M{Γ}= 1

for the universal set Γ; (ii) (duality axiom) M{Λ}+

M{Λc}= 1 for any event Λ; (iii) (subadditivity ax-

iom) M∞



i=1

Λi≤

∞



i=1 M{Λi}for every countable

sequence of events Λ1,Λ2,···.

Definition 2 (Liu [11]) The uncertain measure on the

product σ-algebra Lis called product uncertain mea-

sure defined by the following product axiom: (Prod-

uct Axiom) Let (Γk,Lk,Mk) (k= 1,2,···)repre-

sent uncertainty spaces. The product uncertain mea-

sure Mis an uncertain measure satisfying

M∞



k=1

Λk=

∞



k=1 Mk{Λk}.

Definition 3 (Liu [10]) An uncertain variable is a

function ξfrom an uncertainty space (Γ,L,M)to the

set of real numbers such that {ξ∈B}is an event for

any Borel set Bof real numbers.

Definition 4 (Liu [10]) The uncertainty distribution

Φof an uncertain variable ξis defined by

Φ(x) = M{ξ≤x}

for any real number x.

Theorem 1 (Liu [12]) For any events Λ1and Λ2with

Λ1⊂Λ2, we have

M{Λ1} ≤ M{Λ2}.

Theorem 2 (Liu [14]) A function Φ−1: (0,1) → ℜ

is the inverse uncertainty distribution of an uncertain

variable ξif and only if it is continuous and

Mξ≤Φ−1(α)=α

for all α∈(0,1).

Definition 5 (Liu [10]) Let ξbe an uncertain vari-

able. Then the expected value of ξis defined by

E[ξ] = +∞

0M{ξ≥x}dx −0

−∞ M{ξ≤x}dx

Theorem 3 (Liu [10]) Assume that there is uncer-

tainty distribution ϕfor an uncertain variable ξ. If

E[ξ]exists,then

E[ξ] = +∞

−∞

xdϕ(x).

Furthermore if ϕis regular, then we also have

E[ξ] = 1

ϕ−1(α)dα.

Theorem 4 (Yao and Chen [24]) Suppose that there

are the solution Xtand α-path Xα

tfor an uncertain

differential equation

dXt=f(t, Xt)dt +g(t, Xt)dCt.(1)

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Then, the time integral s

Y(Xt)dt possesses an in-

verse uncertainty distribution

ψ−1

s(α) = s

Y(Xα

t)dt, s > 0

where Y(x)is a function owning strictly increasing

feature.

Theorem 5 (Liu [12]) Let ξ1,ξ2,···,ξnbe indepen-

dent uncertain variables with regular uncertainty dis-

tributions ϕ1,ϕ2,···,ϕn, respectively. If f(ξ1,ξ2,···,

ξn) is strictly increasing with respect to ξ1,ξ2,···,ξm

and strictly decreasing with respect to ξm+1,ξm+2,

···,ξn, then f(ξ1,ξ2,···,ξn) has an inverse uncer-

tainty distribution

ψ−1

s(α) = f(ϕ−1

1(α),···, ϕ−1

m(α),

ϕ−1

m+1(1 −α),···, ϕ−1

n(1 −α)).

Definition 6 (Yao and Chen [24]) Let αbe a number

between 0and 1. An uncertain differential equation

dXt=f(t, Xt)dt +g(t, Xt)dCt

is said to have an α-path Xα

tif it solves the corre-

sponding ordinary differential equation

dXα

t=f(t, Xα

t)dt +|g(t, Xα

t)|Φ−1(α)dt

where Φ−1(α)is the inverse standard normal uncer-

tainty distribution, i.e.,

Φ−1(α) = √3

πln α

1−α.

Theorem 6 (Yao and Chen [24]) Let Xtand Xα

tbe

the solution and α-path of the uncertain differential

equation

dXt=f(t, Xt)dt +g(t, Xt)dCt,

respectively. Then

M{Xt≤Xα

t,∀t}=α,

M{Xt> Xα

t,∀t}= 1 −α.

where Xtpossesses an inverse uncertainty distribu-

tion

Ψ−1

t(α) = Xα

3 Put 2 and Call 1 option

A put 2 and call 1 option(PCO) include two assets,

among which the expected price of asset 1 rises and

the expected price of asset 2 falls, the owner of the P-

CO have rights to replace the bearish Asset 2 with the

bullish Asset 1 on the expiration date. It means that,

upon expiration, if the two assets fall and rise as ex-

pected, and their difference is greater than the option

fee, the greater the difference. The higher the return

of the holder. The yield of the option depends on the

the difference between the two assets at maturity. We

will get the pricing formula of PCO through rigorous

derivation in this section. In addition, we will obtain

the option price through a numerical algorithm.

We assume that the underlying asset prices of op-

tion with a maturity time Tobey different uncertain

differential equations. At the same time, considering

that theoretically the stock price cannot always rise

or fall, its mean-reversion characteristic is inevitable.

We propose the following model:











dZt=rZtdt

dSt=u1(m1−a1St)dt +σ1StdC1t

dVt=u2(m2−a2Vt)dt +σ2VtdC2t

(2)

where Zton behalf of the bond price, Stand Vtrep-

resent, respectively, the price of Asset 1 and Asset 2,

C1tand C2tare independent Liu processes, σ1and u1

are respectively, the log-diffusion and log-drift of St,

σ2and u2are respectively, the log-diffusion and log-

drift of Vt. Moreover, ris the riskless interest rate and

mi/airepresents the mean reversion speed.

Because C1tand C2tare independent Liu process,

so Stand Vtare independent of each other. From Def-

inition 6 we know that the α-path Sα

tand Vα

tof Stand

Vt, respectively, satisfy

dSα

t=u1(m1−a1Sα

t)dt +|σ1Sα

t|Φ−1(α)dt, (3)

and

dV α

t=u2(m2−a2Vα

t)dt+|σ2Vα

t|Φ−1(α)dt. (4)

The solutions of (3) and (4), respectively, are

Sα

t=S0

u1m1−exp −u1a1+ Φ−1(α)σ1t

u1a1−Φ−1(α)σ1

,(5)

and

Vα

t=V0

u2m2−exp −u2a2+ Φ−1(α)σ2t

u2a2−Φ−1(α)σ2

.(6)

From Theorem 6, we know Sα

tand Vα

tare also in-

verse uncertain distributions of Stand Vt, respective-

ly.

Firstly, from the holder's point of view, the payoff

at expiration date Tis

(ST−VT)+.

Suppose the option fee the holder paid at time 0 is fpc.

Then the net return of the holder at the initial moment

−fpc +exp(−rT ) (ST−VT)+.

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Secondly, assuming bank is the seller of the option,

then the payoff of bank at time Tis

−(ST−VT)+.

Thus at the initial moment, the bank owns net return

fpc −exp(−rT ) (ST−VT)+.

For fairness, the expected returns of buyers and sellers

should be equal, so we have

−fpc +exp(−rT )E(ST−VT)+

=fpc −exp(−rT )E(ST−VT)+.(7)

Obviously, the PCO price is

fpc =exp(−rT )E(ST−VT)+.

Theorem 7 Suppose a PCO option for Model (2) has

a maturity time T. Then, the price of option is

fpc =exp(−rT )1

(S0h1(α)

−V0h2(1 −α)) dα, (8)

where for i= 1,2,

hi(α) = uimi−exp −uiai+ Φ−1(α)σiT

uiai−Φ−1(α)σi

.(9)

Proof. First of all, we can get Sα

Tand −V1−α

T, which

are, respectively, inverse uncertain distribution of ST

and −VT. Because STand −VTare independent of

each other, so the uncertain variable

(ST−VT)+

has an inverse uncertain distribution

Sα

T−V1−α

T+

(10)

by Theorem 5, where

Sα

T=S0

u1m1−exp −u1a1+ Φ−1(α)σ1T

u1a1−Φ−1(α)σ1

V1−α

T=V0

u2m2−exp −u2a2+ Φ−1(1 −α)σ2T

u2a2−Φ−1(1 −α)σ2

Finally, we can obtain the pricing formula of PCO

fpc =exp(−rT )E(ST−VT)+

=exp(−rT )1

0Sα

T−V1−α

T+

dα (11)

by Theorem 3. Instituting the expressions of Sα

Tand

V1−α

Tinto (11) gets the conclusion (8). The proof is

completed.

To calculate the option price, according to the def-

inition of integral, we divide the integrating interval

into 100 sub intervals to calculate the integral sum for

approximating the integral. On this basis, we design

the following algorithm:

Algorithm 1 (Option price for model (2))

Step 1. Input the values of parameters: S0,V0,m1,

m2,a1,a2,σ1,σ2,u1,u2and T.

Step 2. Let αstart at α0= 0.01 and grow to 0.99 at

a step of 0.01 to obtain

αj=αj−1+ 0.01, j = 1,2,···,99.

Step 3. Calculate hi(αj)by (9) for i= 1,2,j= 1,

2,···,99.

Step 4. Calculate

fpc =exp(−rT )×0.01 ×



j=1

(S0h1(αj)

−V0h2(1 −αj)).

Example 1 Set S0= 5,V0= 4,m1=m2= 4,

a1=a2= 1,σ1= 0.01,σ2= 0.01,u1= 0.06,

u2=−0.04, and T= 1. Then the Put 2 and Call

1 option price for model (2) is obtained to be fpc =

9.6367 by Algorithm 1.

4 Rainbow call on option

In this section we will study rainbow call on options

(RCO), which include rainbow call on max option-

s (RCMAO) and rainbow call on min options (R-

CMIO). By solving the asset-price model and further

derivation, we obtain the pricing formulas of RCO.

Moreover, some numerical experiments are designed

to calculate the option prices.

We assume that the underlying asset prices of op-

tion with a maturity time Tobey different uncertain

differential equations. At the same time, considering

that theoretically the stock price cannot always rise or

fall, the following model is proposed:











dZt=rZtdt

dS1t=u1(m1−a1S1t)dt +σ1S1tdC1t

dS2t=u2(m2−a2S2t)dt +σ2S2tdC2t

···

dSnt =un(mn−anSnt)dt +σnSntdCnt

(12)

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where Ztis a bond price, ris a riskless interest rate,

Sit represents the price of Asset i,Cit represents in-

dependent Liu processes, uiand σiare, respectively,

the log-drift and log-diffusion of Sit (i= 1,2, . . . , n),

mi/airepresents the mean reversion speed.

The changes of Sit are independent of each oth-

er because Cit (i= 1,2, . . . , n) are independent Liu

process. The α-pathes of systems in Model (12) are

Zα

t=Z0exp(rt)

Sα

it =Si0hi(α)(13)

where

hi(α) = uimi−exp −uiai+ Φ−1(α)σit

µiai−Φ−1(α)σi

(14)

for i= 1,2,···, n.

We assume Ψ−1

1t(α),Ψ−1

2t(α),···,Ψ−1

nt (α)are, re-

spectively, inverse uncertain distributions of S1t,S2t,

···,Snt. Then we have

Ψ−1

it (α) = Sα

it =Si0hi(α)(15)

by Theorem 6 for i= 1,2,···, n.

4.1 Rainbow call on max option

The holder of the RCMAO can buy the highest priced

asset contained in the option at the strike price Kat

time T. It means at time T, when the price of the

highest priced asset is greater than K, the higher price

of the highest priced asset in the rainbow option, the

higher the yield for the option holder. The payoff of

the option depends on the price of the highest priced

asset in the rainbow option on the expiration date.

Firstly, assuming we are the holder of the option,

then our payoff at expiration date is

max

1≤i≤nSiT −K+

Suppose the option fee we paid at time 0 is f1c. The

net return we own at time 0 is

−f1c+exp(−rT )max

1≤i≤nSiT −K+

Secondly, assuming bank is the seller of the option,

then the payoff of bank at time Tis

−max

1≤i≤nSiT −K+

At time 0, the bank charged an option fee, hence the

bank's net return is

f1c−exp(−rT )max

1≤i≤nSiT −K+

From the consideration of fairness, the expected re-

turns of buyers and sellers should be equal, it means

−f1c+exp(−rT )max

1≤i≤nSiT −K+

=f1c−exp(−rT )max

1≤i≤nSiT −K+

Thus we can conclude that the price of RCMAO is

f1c=exp(−rT )Emax

1≤i≤nSiT −K+

Theorem 8 Suppose that a RCMAO for Model (12)

has a maturity time Tand an exercise price K. Then,

its option price is

f1c=exp(−rT )1

0max

1≤i≤nSi0hi(α)−K+

dα

where hi(α)is shown as (14) for i= 1,2,···, n.

Proof. By Theorems 5 and 6, the uncertain variable

max

1≤i≤nSiT −K+

has an inverse uncertainty distribution

max

1≤i≤nSα

iT −K+

Finally, we can obtain the pricing formula of RCMAO

f1c=exp(−rT )Emax

1≤i≤nSiT −K+

=exp(−rT )1

0max

1≤i≤nSα

iT −K+

dα

=exp(−rT )1

0max

1≤i≤nSi0hi(α)−K+

dα.

The proof is completed.

4.2 Rainbow call on min option

The buyer of RCMIO pays an option premium in ex-

change for the right to buy the lowest priced asset in

the rainbow option at the strike price Kon the expiry

date T. Therefore the buyer's benefit depends on the

lowest price of each asset in the rainbow option on the

expiration date.

Firstly, assuming we are the holder of the option,

then our payoff at expiration date Tis

min

1≤i≤nSiT −K+

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Suppose the option fee we paid at time 0 is f2c. Then

our net return at time 0 is

−f2c+exp(−rT )min

1≤i≤nSiT −K+

Secondly, assuming bank is the seller of the option,

then the bank's payoff at time Tis

−min

1≤i≤nSiT −K+

At time 0, the bank charged an option fee, hence the

bank's net return is

f2c−exp(−rT )min

1≤i≤nSiT −K+

From the consideration of fairness, the expected re-

turns of buyers and sellers should be equal, it means

−f2c+exp(−rT )min

1≤i≤nSiT −K+

=f1c−exp(−rT )min

1≤i≤nSiT −K+

Thus we can conclude that the price of RCMIO is

f2c=exp(−rT )Emin

1≤i≤nSiT −K+.

Theorem 9 Suppose that a RCMIO for Model (12)

has a maturity time T and an exercise price K. Then,

its price is

f1c=exp(−rT )1

0min

1≤i≤nSi0hi(α)−K+

dα

where hi(α)is shown as (14) for i= 1,2,···, n.

Proof. The proof is similar to that of Theorem 8.

To calculate the option price, we design the fol-

lowing algorithm according to the definition of defi-

nite integral:

Algorithm 2 (Option price for model (12))

Step 1. Input the values of parameters: S10,S20,

...,Sn0,m1,m2,. . .,mn,a1,a2,. . .,an,

T,σ1,σ2,...,σn,u1,u2,...,unand K.

Step 2. Let αstart at α0= 0.01 and grow to 0.99 at

a step of 0.01 to obtain

αj=αj−1+ 0.01, j = 1,2,···,99.

Step 3. Calculate hi(αj)by (14) for i= 1,2,···,n,

j= 1,2,···,99.

Step 4. Calculate

f1c=exp(−rT )×0.01



j=1 max

1≤i≤nSi0hi(αj)−K+

and

f2c=exp(−rT )×0.01



j=1 min

1≤i≤nSi0hi(αj)−K+

Example 2 Set n= 5,S10 = 5,S20 = 4,S30 = 3,

S40 = 2,S50 = 1,m1=m2=m3= 1,m4=m5=

2,a1=a2=a3= 0.1,a4=a5= 0.5,σ1=σ2=

... =σ5= 0.5,u1= 0.05,u2= 0.04,u3= 0.03,

u4= 0.02,u5= 0.01, and T= 1,K= 10. Then the

RCMAO price is obtained to be f1c= 32.2840 and

the RCMIO price is f2c= 2.5957 by Algorithm 2.

5 Rainbow put on option

In this section, we will reveal the pricing formula of

rainbow put on option (RPC), which include rainbow

put on max option (RPMAO) and rainbow put on min

option (RPMIO). After further derivation, the pricing

formula of options is obtained. As in the previous sec-

tion, we will also use some numerical experiments to

verify the rationality of the formula.

5.1 Rainbow put on max option

The holder of RPMAO has the right to sell the

highest-priced asset contained in the option at the

strike price Kon time T. It means on the expiration

date, when the strike price Kis higher than the price

of the highest priced asset in RPMAO, the lower price

of the highest priced asset in the option, the higher the

yield for the option holder. So the return of the op-

tion depends on the highest price among all asset in

the RPMAO on the expiration date.

Assume that RPMAO possesses a maturity time T

and an exercise price Kfor Model (12).

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Firstly, assuming we are the holder of the option,

then our payoff at expiration date Tis

K−max

1≤i≤nSiT +

Suppose the option fee we paid at time 0 is f3c, then

at time 0, the net return we have is

−f3c+exp(−rT )K−max

1≤i≤nSiT +

Secondly, form the perspective of seller, the payoff of

the option seller on expiration date Tis

−K−max

1≤i≤nSiT +

At the initial moment, the seller of option charged an

option fee, hence the seller's net return is

f3c−exp(−rT )K−max

1≤i≤nSiT +

From the consideration of fairness, the expected re-

turns of buyers and sellers should be equal. It means

−f3c+exp(−rT )K−max

1≤i<n SiT +

=f3c−exp(−rT )K−max

1≤i≤nSiT +

Thus we can conclude that the price of RPMAO is

f3c=exp(−rT )EK−max

1≤i≤nSiT +.

Theorem 10 Suppose that a RPMAO option for

Model (12) has a maturity time T and an exercise

price K. Then, its price is

f3c=exp(−rT )1

−max

1≤i≤nSi0hi(1 −α)+

dα,

where hi(α)is shown as (14) for i= 1,2,···, n.

Proof. By Theorems 5 and 6, the uncertain variable

K−max

1≤i≤nSiT +

has an inverse uncertainty distribution

K−max

1≤i≤nS1−α

iT +

Finally, we can obtain the pricing formula of RPMAO

f3c=exp(−rT )EK−max

1≤i<n SiT +

=exp(−rT )1

0K−max

1≤i≤nS1−α

iT +

dα

=exp(−rT )1

−max

1≤i≤nSi0hi(1 −α)+

dα.

The proof is completed.

5.2 Rainbow put on min option

The holder of RPMIO has the right to sell the lowest-

priced asset contained in the option at the strike price

Kon the expiration date T, it means on the expira-

tion date, when the strike price Kis higher than the

price of the lowest priced asset contain in RPMIO, the

higher price of the lowest priced asset in the option,

the lower the yield for the option holder. So the return

of the option depends on the lowest price among all

asset in the RPMIO on the expiration date.

Similarly, assume that RPMIO price obey Model

(12).

Firstly, the payoff of the option holder on expira-

tion date Tis

K−min

1≤i≤nSiT +

Suppose the option fees we paid at time 0 is f4c. Then

at time 0 we have the net return

−f4c+exp(−rT )K−min

1≤i≤nSiT +

Secondly, from the perspective of seller, the payoff of

the option seller on expiration date Tis

−K−min

1≤i≤nSiT +

At the initial moment, the seller of option charged an

option fee, hence the seller's net return is

f4c−exp(−rT )K−min

1≤i≤nSiT +

From the consideration of fairness, the expected re-

turns of buyers and sellers should be equal, it means

−f4c+exp(−rT )K−min

1≤i≤nSiT +

=f4c−exp(−rT )K−min

1≤i≤nSiT +

.(16)

WSEAS TRANSACTIONS on BUSINESS and ECONOMICS

DOI: 10.37394/23207.2022.19.103

Mingchong Liao, Yuanguo Zhu

E-ISSN: 2224-2899

1186

Volume 19, 2022

In summary, we can conclude that the price of RPMIO

f4c=exp(−rT )EK−min

1≤i≤nSiT +.

Theorem 11 Suppose that a RPMIO for Model (12)

has a maturity time Tand an exercise price K. Then,

its price is

f4c=exp(−rT )1

−min

1≤i≤nSi0hi(1 −α)+

dα,

where hi(α)is shown as (14) for i= 1,2,···, n.

Proof. The proof is similar to that of Theorem 10.

To calculate the option price, we design an algo-

rithm as follows:

Algorithm 3 (Option price for model (12))

Step 1. Input the values of parameters: S10,S20,

. . .,Sn0,m1,m2,...,mn,a1,a2,. . .,an,

T,σ1,σ2,. . .,σn,u1,u2,. . .,unand K.

Step 2. Let αstart at α0= 0.01 and grow to 0.99 at

a step of 0.01 to obtain

αj=αj−1+ 0.01, j = 1,2,···,99.

Step 3. Calculate hi(αj)by (14) for i= 1,2,···,n,

j= 1,2,···,99.

Step 4. Calculate

f3c=exp(−rT )×0.01



j=1 K−max

1≤i≤nSi0hi(1 −αj)+

and

f4c=exp(−rT )×0.01



j=1 K−min

1≤i≤nSi0hi(1 −αj)+

Example 3 Set n= 5,S10 = 5,S20 = 4,S30 = 3,

S40 = 2,S50 = 1,m1=m2=m3= 1,m4=

m5= 2,a1=a2=a3= 0.1,a4=a5= 0.5,

σ1=σ2=... =σ5= 0.5,u1= 0.05,u2= 0.04,

u3= 0.03,u4= 0.02,u5= 0.01, and T= 1,K=

10. Then the RPMAO price is f3c= 9.5073 and the

RPMIO price is f4c= 29.6979 by Algorithm 3.

6 Conclusion

In this paper, uncertain differential equations are used

to describe the price of underlying assets. From the

perspective of uncertainty theory, combined with the

stock price mean regression model, five types of rain-

bow options are studied. After that, through stric-

t derivation, we obtain the pricing formulas of five

types of rainbow options. Finally, we verified the ra-

tionality of the option pricing formula through some

numerical experiments.

The research results of this paper are to build the

asset price model through uncertain differential equa-

tions. Considering the fact that fractional order differ-

ential equations have achieved many successful prac-

tices in economics and other fields in recent years, we

will further consider introducing uncertain fraction-

al order differential equations to build the asset price

model, and study the pricing of options on this basis.

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Contribution of individual authors to

the creation of a scientific article

(ghostwriting policy)

Mingchong Liao completed the draft writing.

Yuanguo Zhu supervised the writing and put forward

suggestions for revisions.

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WSEAS TRANSACTIONS on BUSINESS and ECONOMICS

DOI: 10.37394/23207.2022.19.103

Mingchong Liao, Yuanguo Zhu

E-ISSN: 2224-2899

1188

Volume 19, 2022