Two Approaches for Option Pricing under Illiquidity

The paper focuses on option pricing under unusual behaviour of the market, when the price may not be changed for some time what is quite a common situation on the modern (cid:28)nancial markets. There are some patterns that can cause permanent price gaps to form and lead to illiquidity. For example, global changes that have a negative impact on (cid:28)nancial activity, or a small number of market participants, or the market is quite young and is just in the process of developing, etc. In the paper discrete and continuous time approaches for modelling market with illiquidity and evaluation option pricing were considered. Trinomial discrete time model improves upon the binomial model by allowing a stock price not only to move up, down but stay the same with certain probabilities, what is a desirable feature for the illiquid modelling. In the paper parameters for real (cid:28)nancial data were identi(cid:28)ed and the backward induction algorithm for building call option price trinomial tree


Intoduction
Analysis of dierent nancial markets shows that during global crises that have a negative impact on nancial activity we can observe some kinds of risky assets which have the periods in their dynamic without change.Such behavior is typical for emerging markets with low number of transactions, for interest rate markets and for commodity markets.So for these markets the problem of evaluating fair price of derivative instruments on stocks have become extremely important.
The classical diusion models for continuous time like Black-Scholes-Merton (B-S) and its discrete variant -binomial tree model of Cox-Ross-Rubinstein (C-R-R) [2] are incapable of adequately modelling illiquidity for real-life asset dynamic and evaluate derivatives.This happens because classical binomial C-R-R model allows a stock price only to move up or down and do not take into account the stagnation periods.In benchmark B-S model Brownian motions is perpetually moving and we can not use it for modeling periods with motionless stock returns too.
In order to overcome this diculty for discretetime approach was considered the trinomial tree model.This model improves upon the binomial model by allowing a stock price not only to move up or down, but stay the same with certain probabilities, what is a desirable properties for the illiquid modelling.
For continuous-time approach one can notice, that the constant periods of stagnation in nancial processes are analogous in nature to the trapping events of the subdiusive particle.Therefore, the physical models of subdiusion can be successfully applied to describe nancial data.See for example paper [6], where option pricing was proposed in fractional jumpâdiusion model, papers [7] for Black-Scholes formula and [8], [14] for Bachelier 1 xtliy hhestyuk knowledges (nnil support from the projet ortfolio mngement for illiquid mrkets @hnrX PHPPHHWWA funded y the unowledge poundtionF model in subdiusive regime.
The aim of the work was to consider two different approaches for modelling market with stagnation periods: to apply trinomial tree model and propose IG prosess as a subordinator for subdiusive model.
The paper is organized as follows.In the next section we remind what is trinomial tree model and how we can apply it to nd fair option price for real historical data.This section is based on the papers [5], where dierent types of trinomial tree models are presented.We show how model parameters for real nancial data can be identied and the backward induction algorithm for building call option price trinomial tree can be applied.
In the third section we consider IG process as subordinator of subdiusive GBM and its properties.The simulation of the trajectories for subordinator, inverse subordinator and subdiusive GBM were performed.Also we describe Monte Carlo option pricing techniques for this case.We choose the parameters u, d, m to match the volatility σ of the stock price.The step is of length ∆t.According to the assumption from [3]: . (1) Also one can match the rst two moments of our models distribution according to the no arbitrage condition.In a risk-neutral world, the expected return on all assets is equal to the risk-free interest rate (this means that all expected gains are discounted at the rate) and the variance can be expressed as follow [3]: var(S t ) = S 2 0 e 2r∆t (e σ 2 ∆t − 1) (3) We equate two values for mathematical expectation (2) and variation (3) to form two equations of the system.Also, using the property that the sum of the probabilities equal to unity, we write down the third equation.So, we got a system of three equations and three unknown variables: From this system the probability values for the trinomial model are: The above setting (1) for parameters and (5) for probabilities we use in the next sections for option pricing numerical result.
Option pricing for trinomial model .The methodology when pricing options using a trinomial tree is exactly the same as when using a binomial tree.To determine the option price f based on the trinomial tree, the following algorithm is used: 1. Declare and initialize S( 0 for the put option We apply this algorithm for option pricing for getting numerical results for real nancial data with stagnation's periods.
Numerical results for trinomial model.
We consider Airbnb company spot price S 0 = = 103.51for June 24, 2022.The strike price is K = 100 for call options with maturity T is given for ten dierent dates.The yearly volatility for returns of the underlying asset is computed as σ = = 0.5758, the yearly riskless interest rate is set as r = 0.16.
For these input parameters we compute jump sizes and the transition probabilities  See Graph for T = 5.The fair price for this call option is C = 6.1957.
The results for dierent times of maturity are demonstrated in the gure 4. Subdiusion processes with IG subordinator and its simulation.For modelling of illiquidity in continuous case it is useful to apply the subdiusion process, which is used in statistical physics for describing the trapping events of the subdiusive particle.In physics, this process usually is described by Fokker-Planck fractal equations.
Equivalent description of subdiusion there exists in terms of subordination, where the standard diusion process is time-changed by the socalled inverse subordinator.In this section we consider B-S model and the standard diusion process GBM for describing underlying risky asset in subdiusive regime.For it we replace the calendar time t in classical GBM [7] with some stochastic process H(t) and obtain sub- In formula ( 13) H(t) is called the inverse subordinator and it dended as The inverse subordinator H(t) is also called a "hitting time" and is interpreted as the time of rst reaching a certain price, which may not change for some time.By construction, the inverted process may be constant.Therefore, any process subordinated by H(t) exhibits motionless periods.
The dinition (3.1) of the inverse subordinator is based on the use of some other random process [14]).
In this paper we propose to take the Inverse Gaussian process IG as a subordinator for the subdiusive modelling.The G(t) process is a nondecreasing Levy process (i.e., process with stationary independent increments), where the increments G(t + s) − G(s) follow the inverse Gaussian G(δt, γ) distribution with probabilities density function (PDF) with parameters γ and δ (see For the standard IG distribution, where γ = δ = 1 the PDF will be Then for any moment t we have E(G(t)) = t, var(G(t)) = t.
The tail probability for G(δt, γ) is studied in [9] and equals The q-th order moments of the G(δt, γ) are given by where K q (ω) is the modied Bessel function of the third kind with index q, dened in [9].
The simulation of the trajectory G(t) is demonstrated below on Figure 6. the inverse to the IG process.The IIG process was studied in [9], where were found as q-th order moments of the IIG(δt, γ)) as its tail behaviour.
In order to simulate the approximate trajectory inverse subordinator H(t), we dene H(∆t) with the step length ∆ as follows [9]: where ∆ is the step length and G(∆n) is the value of the Inverse Gaussian process G(t) evaluated at n.
The simulation of the trajectory H(t) is demonstrated on Figure 7.
For simulation of the trajectory subdiusive GBM X(t) we remind that the Ito equation allows modeling the time dynamics of an arbitrary stochastic process by means of the iterative scheme[15]: In paper [16] were considered iterative schemes for fractal activity time processes with inverted gamma subordinator.
For modeling stochastic subdiusive GBM we propose the next iterative scheme where ε is white noise with normal standard distribution, ∆H(t) have IIG distribution.The simulation of the trajectory X(t) according ( 16) is demonstrated on the Figure 6.12) is given by:  First we simulate N trajectories of subordinator G(t), that is a process of independent stationary increments having IG distribution.
After that we simulate N values of the inverse IG subordinator H(T ) for every given time to maturity T and calculate N option price values, using Black-Scholes option pricing formula (17).
Then nd the fair price as a mean for N scenarios, obtained in the previous step according ( 21).
The results are presented in the graphic shape in Fig. 8.For more detail we need to compute and compare the estimation errors.

Comparison of the two models
In this section we compare numerical results for AIRBNB company for two proposed models.It is a trinomial tree model and subdiusive B-S model with IG subordinator.
Our aim is not only to compare these two models each with other, but also to show that both models adequately describe the illiquid market.
In Fig. 9 we compare the subdiusive B-S formula for European call options with the classical one and with option pricing using trinomial tree model.We estimated the values of subdiusive B-S formula using Monte Carlo methods based on the above described simulation procedure.

Conclusion
In the paper two dierent approaches for modelling market with stagnation periods were considered.We apply well-known trinomial tree model in discrete time case and propose subdiusive model with IG subordinator in continuous time case.
For the option pricing the backward induction algorithm trinomial tree model was used.In the continuous time case Monte-Carlo method was proposed.
The programmed model can be used to valuate option price by several dierent methods and it can help to make decision.
To compare numerical results we used absolute relative percentage (ARPE) and root mean squared errors (RMSE).
In the framework of the paper we compared option prising results in situation when strike price K was xed (in the money), while time to maturity In the future we are going to examine the ARP pricing errors of the proposed option pricing models in more detail (see paper [17] ) and consider the pricing errors as a regression on the time to maturity T (in years), the moneyness of the option, and a binary variable that is set to unity, if the option is a call and to zero in the case of a put.This can indicate a level of explanatory value of moneyness, maturity and the put-call dummy in the model.
Figure below.

Figure 1 .
Figure 1.Trinomial tree ) 2. Calculate the jump sizes u, d, m 3. Calculate the transition probabilities p u , p d , p m 4. Build the share price tree 5. Calculate the option payos at maturity time T , i.e node N: for the call option

) 6 .
Apply the following backward induction algorithm, where u represents the time position and j the space position f u,j = e −r∆t (p u f u+1,j+1 + p m f u+1,j+1 + p d f u+1,j+1 ) (8) 7. The fair price f of the European call or put option is p d = 0.4169 p m = 0.1663(11)    and build the share price trinomial tree.The rst 5 steps of this tree is demonstrated in the Graph below.

Figure 3 .
Figure 3. Tree of pay-o function for 5 steps

Figure 4 .
Figure 4. Simulated prices for the binomial and trinomial option pricing models called a subordinator G(t).The subordinator G(t), in its turn, is generally a non-decreasing stochastic process with stationary independent increments with right continuous left limits sample paths.Many types of subordinators such as α-stable, tempered-stable, Gamma, Poisson and other have been already applied for dierent subdiusive models of illiquidity (see for example [6], [7], [8],

Figure 5 .
Figure 5. Simulation of the IG process trajectories

Figure 6 .
Figure 6.Simulation of the inverse to the IG process trajectories

Figure 7 .
Figure 7. Simulation of the subdiused Geometric Brownian motion with inversed IG subordinator Monte Carlo method for option pricing in subdiusion Black -Scholes model.The

0 C
Consider a time-changed version of the B-S model, where the underlying risky assets follow (13).Then, as were shown in [7] the market model is arbitrage-free and incomplete and the corresponding fair price of the European call option in subdiusive regime [7] isC sub (S, K, T, σ) = ⟨C (S, K, H(T ), σ)⟩ = ∞ (S, K, x, σ) g(x, T )dx(20)Here, g(x, T ) is the PDF of H(T ) and C(S, K, T, σ) is given by (17).It is worth to mention, that the proof of formula (20) for fair price is based on the common ideas for changed time models, see for examples proof in [11] for Student model with FAT or for Student-like FAT in [10] and their applications in [13], [12].There are two ways of nding the values of the price C(•).One is to calculate C(•) by approximating the integral in (20).However, this can be performed in cases, where g(x, T ) is known exactly.The other way is to nd C(•) by using the Monte-Carlo method.One simulates the trajectories for the inverse subordinator on the interval [0, T ] by the approximation scheme (14).Then, one obtains the fair price as an estimation of the expected value for simulated prices where the inverse subordinator stands for calendar time T in (20) C sub (S, K, T, r, σ) = ⟨C(S, K, H(T ), where C(S, K, T, σ) is taken from Black-Scholes option pricing formula (17).One can see the applying of the Monte-Carlo method for option pricing in subdiusive models, for example, in the papers [7], [8], [14].Numerical results for subdiusive Black--Scholes model .For the company "Airbnb" the input parameters are: S = 103.51,K = 100, r = = 0.16, σ dif f = 0.5758 for the diusion model (see section 2.3 above).

Figure 8 .
Figure 8. Simulated prices for the diusive and subdiusive B-S models

Figure 9 .
Figure 9.Comparison of the trinomial model and the B-S subdiusive approach for the call option pricing It is worth to mention, in econometrics, the root mean squared error (RMSE) (22) is a key criterion for model selection.The mean squared error indicates the mean squared deviation between the forecast and the outcome.It sums the squared bias and the variance of the estimator.The advantage of the ARPE (23) relatively to the RMSE measure is that it gives a percentage value of the pricing error.Therefore, if we use both these errors it provides more insight into the economic signicance of performance dierences.

T
were changing.If we compare classical B-S model with subdiffusive one, the results show that the diusive option pricing B-S model shows better results on the short-term period, while the subdiusive model is more eective on the long-term perspective.Meanwile RMSE is bigger for proposed subdiusive model then for classical B-S one.Comparing subdiusive B-S model with trinomial one we assume that trinomial model has the smallest RMS error.

Table 1 .
The RMS errors for diusion, subdiusion and trinomial models regarding to the market price