Lookback option pricing problem of uncertain mean-reverting currency model

A lookback option is a maturity option that pays off based on the maximum or minimum stock price over the life of the option. This paper investigates the problem of pricing a lookback option based on the uncertain mean-reverting currency model and designs the algorithms to calculate the formulations. Furthermore, discussions about parameters and results are drawn in the paper.


Introduction
The option pricing problem which was proposed by Bachelier (1900) in 1900 was constantly popular. But it has not been solved scientifically until stochastic process differential equation was founded in 1951 by Ito (1951). Following that, Black and Scholes (1973) investigated the famous Black-Scholes option pricing formula in 1973, which assumed the stock price followed a geometric Wiener process. Considering the jump phenomena of stock price, Merton (1973) reconstructed the model. After that, amount of stochastic differential equations had been used to describe the stock price and various option pricing formulas were derived.
In the application of Black-Scholes option pricing formula, investors pointed out the strict restriction of the prerequisite to use the formula. In addition, two paradoxes were found by Liu (2013) showing that the stock price in real financial market is impossible to follow stochastic differential equations. To deal with the conflicts between real mark and probability theory, uncertainty theory was built by Liu (2007) in 2007, which was based on normality, duality, subadditivity and product axioms. Then it was applied to amount of fields, such as uncertain programming , uncertain logic Liu (2011), uncertain reliability analysis and uncertain risk analysis . The basic concept uncertain variable in uncertainty theory was defined by Liu (2007) to model the quantity associated with human uncertainty. Meanwhile, uncertainty distribution was given to describe uncertain variables, excepted value was provided to rank uncertain variables.
Uncertain process, a sequence of uncertain variables driven by time, was explored by Liu (2008) in 2008 to model the evolution of uncertain phenomena. As a special and important type of uncertain process, Liu (2009) designed Liu process within the framework of uncertainty theory as a counterpart of Wiener process. Following that,  began the pioneer work of uncertain finance based on the assumption of the stock price's fluctuation follows a geometric Liu process. After that, its European option, American option Chen (2011) and Asian option Sun and Chen (2015) Zhang and Liu (2014) pricing formulas were derived, respectively.
Lookback option is a path-dependent option that offers the payoff depending on the maximum or minimum value of the underlying asset price. The pricing problem of lookback option has been widely discussed since it was first valuated by Goldman et al. (1979). Conze and Viswanathan (1991) used the joint distribution of the spot price and its historical maximum (or minimum) to obtain the closed-form valuation formulas. Quanto lookback option price formulas were derived by Dai et al. (2004). With framework of uncertainty theory, Zhang et al. (2019) firstly investigated the lookback option problem. Based on that, Gao et al. (2018) developed the lookback call and put option pricing formulas following the uncertain exponential Ornstein-Uhlenbeck model.
By viewing the foreign exchange rate as uncertain process and describing an uncertain differential equation as Liu process, an uncertain currency model was proposed by Liu et al. (2015) in 2015. Based on the foundation, Shen and Yao (2016) presented a mean-reverting currency model by considering the exchange rate in the long term in the real global market. Sheng and Shi (2018) investigated the Asian currency option pricing problem following the mean-reverting stock model. Then Wang and Ning (2017) built a uncertain currency model with floating interest rates by considering the uncertain fluctuations in the financial market from time to time.
The existing scholarship in this field, however, tended to overlook the combination of lookback option and uncertain mean-reverting currency model. Following the previous work on option pricing problem within framework of uncertainty theory lead, this paper will combine lookback option and uncertain mean-reverting currency model to derive the lookback option pricing formulas for the uncertain meanreverting currency model. The rest of the paper is organized as follows. In the next section, this paper will review some preliminary knowledge in uncertainty theory. In Sect. 3, the lookback call option will be introduced and its pricing formula will be derived. An algorithm will be designed to calculate the price numerically, a numerical experiment will be given and the sensibility of single variable will be analyzed finally. In Sect. 4, the lookback put option will be discussed in the similar way. In Sect. 5, the conclusion will give some conclusions.

Preliminaries
In this section, some preliminary concepts from uncertainty theory as needed are reviewed for further understanding the paper.

Uncertain variable
Definition 1 Liu (2007) Let be a nonempty set, and L be a σ -algebra over . A set function M : L → [0, 1] is called an uncertain measure if it satisfies the following axioms: Axiom 1 (Normality Axiom) M{ } = 1 for the universal set .
Axiom 3 (Subadditivity Axiom) For every countable sequence of events 1 , 2 , . . ., we have Note that the triplet ( , L, M) is called an uncertainty space.
Besides, the product uncertain measure on the product σalgebra was defined by Liu (2009) as follows.
Axiom 4 (Product Axiom) Let ( k , L k , M k ) be uncertainty spaces for k = 1, 2, . . . , n. The product uncertain measure M is an uncertain measure satisfying where k are arbitrarily chosen events from L k for k = 1, 2, . . . , n, respectively. (2007) An uncertain variable ξ is a measurable function from an uncertainty space ( , L, M) to the set of real numbers, i.e., for any Borel set B, the set {γ ∈ |ξ(γ ) ∈ B} is an event in L. (2007)) The uncertainty distribution of an uncertain variable ξ is defined by

Definition 3 (Liu
for any real number x. Liu and Lio (2020) proposed that a real-valued function (x) on is an uncertainty distribution if and only if it is a monotone increasing function satisfying Definition 4 Liu (2010) Let ξ be an uncertain variable with regular uncertainty distribution (x). Then the inverse function −1 (α) is called the inverse uncertainty distribution of ξ .

Example 1 An uncertain variable ξ is called linear if it has a linear uncertainty distribution
Example 2 An uncertain variable ξ is called normal if it has a normal uncertainty distribution , where e and σ are real numbers with σ > 0. The inverse uncertainty distribution of normal uncer- Definition 5 Liu (2007) Let ξ be an uncertain variable. Then provided that at least one of the two integrals is finite.

Uncertain differential equation
An uncertain process is a sequence of uncertain variables indexed by time. A formal definition of uncertain process is stated as follows.
Definition 6 Liu (2008) Let T be a totally ordered set (e.g., time) and let ( , L, M) be an uncertainty space. An uncertain process is a function X t (γ ) from T × ( , L, M) to the set of real numbers such that {X t ∈ B} is an event for any Borel set B of real numbers at each time t.
Definition 7 Liu (2014) Uncertain processes X 1t , X 2t , . . . , X nt are said to be independent if for any positive integer k and any times t 1 , t 2 , . . . , t k , the uncertain vectorş are independent, i.e., for any Borel sets B 1 , B 2 , . . . , B n of k-dimensional real vectors, we have An uncertain process X t is said to have independent increments if are independent uncertain variables where t 0 is the initial time and t 1 , t 2 , . . . , t k are any times with t 0 < t 1 < . . . < t k . An uncertain process X t is said to have stationary increments if, for any given t > 0, the increments X s+t − X s are identically distributed uncertain variables for all s > 0.
Definition 8 ) An uncertain process C t is said to be a Liu process if (i) C 0 = 0 and almost all sample paths are Lipschitz continuous, (ii) C t has stationary and independent increments, (iii) every increment C t+s − C s is a normal uncertain variable with an uncertainty distribution Definition 9 Liu (2009) Let X t be an uncertain process and let C t be a Liu process. For any partition of closed interval provided that the limit exists almost surely and is finite. In this case, the uncertain process X t is said to be integrable.

Theorem 2.2 Yao and Chen (2013) Let Z t be an independent sample-continuous increment process. Then Z t has a regular uncertainty distribution t (x). Let J (x) be a real value strictly increasing function. It shows that the time integral
has an inverse uncertainty distribution −1 Definition 10 Liu (2008) Let C t be a Liu process, assume f and g are continuous functions. Then is called an uncertain differential equation. A solution is an uncertain process X t that satisfies the equation identically in t.
Definition 11 Yao and Chen (2013) The α-path (0 < α < 1) of an uncertain differential equation with an initial value X 0 is a deterministic function X α t with respect to t that solves the corresponding ordinary differential equation where −1 (α) is the inverse uncertainty distribution of standard normal uncertain variable,i.e., (2013) Let X t and X α t be the solution and α-path of the uncertain differential equation

Theorem 2.3 Yao and Chen
respectively. Then, This formula is well known as Yao-Chen formula.
Theorem 2.4 Yao and Chen (2013) Let X t and X α t be the solution and α-path of the uncertain differential equation has an inverse uncertainty distribution Theorem 2.5 Yao (2013) Let X t and X α t be the solution and α-path of the uncertain differential equation respectively. Then, for any time s > 0 and strictly increasing function J (x), the supremum has an inverse uncertainty distribution −1

Uncertain mean-reverting currency model
Assuming that the stock price follows a geometric Liu process,  built the first uncertain stock model, which was applied widely The bond price X t and the stock price Y t are determined by this model, where r is the risk-free interest rate, e is the stock drift, σ is the stock diffusion and C t is a canonical Liu process. Then the model was derived in the international financial market by Liu et al. (2015) ⎧ ⎨ ⎩ d X t = μX t dt dY t = νY t dt d Z t = eZ t dt + σ Z t dt. (2) In this model, X t represents the domestic currency with a domestic interest rate μ, Y t is the foreign currency with a foreign interest rate ν, Z t represents the exchange rate (the domestic currency price of one unit of foreign currency) at time t. It means that one unit of foreign currency equals to Z t unit of domestic currency.
It is found that the change of foreign exchange rate stays around an average level in the long term in real global financial market. Therefore, mean-reverting model was proposed by Shen and Yao (2016) in 2016, in which m a means the average level 3 Lookback call option pricing formula with the fixed strike

The pricing formula of lookback call option
A lookback call option provides the holder the right to sell a certain asset at the highest price during a certain period. Assume that a buyer can buy one unit of foreign lookback option during t ∈ [0, T ] with a fixed striking price K, where T is the expiration time. Then the payoff of the buyer is sup 0≤t≤T where Z t is the currency at time t. Considering the time value of money resulted from the bond, the present value of the payoff is Let f c present the price of the call option, then the expected profit of the buyer is and the expected profit of the bank is According to the fair price principle, the buyer and the bank should gain the same expected profit This pricing formula can be presented as follows:

Theorem 3.1 Assume the lookback call option for the stock model (3) has a strike price K and an expiration time T. Then the price of the lookback call option is
(1 − exp(−at)).
Proof By solving the ordinary differential equation we get the α-path of Z t : By Theorem 2.4, the foreign exchange rate Z t has inverse distribution function And since by Theorem 2.5, the inverse distribution of ( sup The inverse distribution of Therefore, The lookback call option pricing formula is verified.

Numerical experiment
An algorithm is designed to calculate the lookback call option price f c .
Step 6 Calculate the price at time t j If P α i t j > P α i t j−1 and j < M, then return to Step 4. If P α i t j ≥ P α i t j−1 and j = M, then jump to Step 8.
Step 7 Set P α i t j ← P α i t j−1 . If j < M, then return to Step 4.
Step 10 Calculate the lookback call option price Example 3.1 Suppose that the initial exchange rate Z 0 = 5, the domestic interest rate μ = 0.05, the foreign interest rate ν = 0.04, the log-diffusion σ = 0.1, the strike price K=4, the expiration time T=1, m=6, a=1. Then the price of lookback call currency option is f c ≈ 1.4719.

Sensibility analysis
It is obvious that f c is an increasing function of Z 0 . Under the assumption that other parameters except Z 0 are constants, change Z 0 from 0 to 6 with step 0.001, we can draw the figure of lookback call option price f c (Fig. 1).

Since
Under the assumption that other parameters except K are constants, change K from 3 to 6 with step 0.001, we can draw the figure of lookback call option price f c (Fig. 2).
3. Since Under the assumption that other parameters except μ are constants, change μ from 0 to 1 with step 0.001, we can draw the figure of lookback call option price f c (Fig. 3).  4. Since Under the assumption that other parameters except ν are constants, change ν from 0 to 1 with step 0.001, we can draw the figure of lookback call option price f c (Fig. 4) (Tables 1,  2, 3, 4).

The pricing formula of lookback put option
Assume that a foreign lookback put option has a fixed striking price K and an expiration time T. Then the payoff of a buyer is Considering the time value of money resulted from the bond, the present value of the payoff is Let f p present the price of the lookback put option, then the expected profit of the buyer is and the expected profit of the bank is According to the fair price principle, the buyer and the bank should gain the same expected profit This the pricing formula can be presented as follows: (1 − exp(−at)).
Proof We have known that the foreign exchange rate Z t has inverse distribution function the inverse distribution of Therefore, The lookback put option pricing formula is verified.

Numerical experiment
An algorithm is designed to calculate the lookback put option price f p .