Modeling credit risk with mixed fractional Brownian motion: An application to barrier options

Javed Hussain; Munawar Ali

doi:10.1515/nleng-2024-0003

Artikel Open Access

Modeling credit risk with mixed fractional Brownian motion: An application to barrier options

Javed Hussain und Munawar Ali

Veröffentlicht/Copyright: 18. April 2024

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen

Aus der Zeitschrift Nonlinear Engineering Band 13 Heft 1

Abstract

This article aims to examine the pricing of debt and equity in the context of credit risk structural models, where the value of a company’s assets is influenced by mixed fractional Brownian motion. Three distinct scenarios are analyzed, including when the assets are trade-able, fixed, and subject to partial recovery of debt. The study culminates with the evaluation of debt pricing under the barrier model, where a bankruptcy threshold is established for the company’s asset value.

Keywords: credit structural model; mixed fractional Brownian motion; barrier option

1 Introduction

Classical Brownian motion (BM) is a widely used tool for modeling randomness in fields such as biology, finance, physics, chemistry, and mathematics. However, it has limited scope in explaining natural phenomena due to its property of independent increments. To address this, fractional Brownian motion (FBM) was introduced, which depends on a Hurst index H , which is a parameter that describes long-range dependence. FBM has limitations, as it is neither a Markov process nor a semi-martingale, making traditional Itô calculus tools unsuitable for studying its dynamics. In addition, FBM can result in arbitrage opportunities. To overcome these limitations, mixed fractional Brownian motion (MFBM) was introduced, which is a combination of FBM and independent BM.

Credit risk refers to the possibility of a borrower declaring bankruptcy and being unable to fulfill their debt obligations. This is relevant to companies that are partially funded by debt. The challenge lies in determining the value of the debt or the interest rate set by debt holders to compensate for the potential loss they face in case of bankruptcy by the company. Credit risk derivatives serve the purpose of informing a lending organization about the likelihood of default and under what conditions it may occur. Traditional credit risk models, based on the assumption of a company’s asset value following standard or FBM, have been found to be flawed. These models often result in credit spreads that are higher than actual spreads and inaccurately predict corporate bond prices. Huang and Huang [1] have shown that these models frequently mis-value corporate debt. The contingent claims that approach, widely used in the field, has also been proven to be ineffective in explaining corporate debt prices despite its long history of recognition as problematic. The diffusion approach, commonly used in the field, fails to consider a firm’s immediate default probability and does not match actual credit spread curves.

Recent financial empirical studies (such as by Ding et al. [2]) have discovered self-similarity and long-range dependence in financial assets, contradicting the classical Black–Scholes model.

There are two methods to model credit risk: structural model approach and reduced form model approach [3]. Merton introduced these methods [4,5] by assuming that a company’s value follows classical BM, represented by the equation:

d V ( t ) = μ V ( t ) d t + σ V ( t ) d B P ( t ) ,

where μ is the drift coefficient, σ is the volatility, and B P ( t ) is the BM under the probability measure P .

Despite being the most widely accepted model for credit risk pricing, Merton’s model has theoretical limitations and may not accurately fit real-life data. This is because financial return series often exhibit nonlinearity and long-range dependence, which are not incorporated in the Merton model that relies on classical BM [6]. To address these limitations, there is a need for Merton fractional models and even more advanced Merton mixed fractional models [6–8], which are considered improvements over the existing models [9–12].

The MFBM is commonly used to model natural phenomena and price financial derivatives, particularly credit risk derivatives. There is a wealth of research on FBM in various sources, including refs. [13–15]. These works extend financial derivatives driven by FBM, while the study by Cheridito et al. [16] contains important results on MFBM. The pricing of European call options driven by MFBM is covered in the study by Murwaningtyas et al. [17]. Leccadito [8] extensively discusses credit risk derivatives driven by FBM. Some of the more interesting and relevant are presented in refs. [18–22].

In this article, we propose an alternative model, the MFBM model, which has shown improved results. The credit structural model is developed under the assumption that the value process of the company ( V ( t ) ) t ≥ 0 follows MFBM, described as follows:

(1.1) d V ( t ) = μ V ( t ) d t + σ V ( t ) ( d B ˆ ( t ) + d B ˆ H ( t ) ) ,

where B ˆ H ( t ) is an FBM and B ˆ ( t ) is a BM with respect to a probability measure P ˆ H . In particular, we price the equity and debt when the Merton model is driven by MFBM. Also, equity and debt in case of partial recovery and fixed assets have been priced. We have also presented a key theorem that gives the relation between credit spread and quasi-debt ratio.

2 MFBM preliminaries and some important results

Andrey Nikolaevich Kolmogorov (1903–1987), a Soviet Russian mathematician, was the first person who wrote an article about FBM in 1940. Kolomogorov introduced certain processes with time continuity, stationary increments, and with self-similarity property and named those processes as “Weiner Spirals.” Moreover, it was Benoît B. Mandelbrot, a French mathematician, who formulated an integral representation for FBM through BM and named such processes “FBM.” It became a topic for conversation again in the 1900s because of its long-range dependence. Since an FBM is neither a Markov process and nor a semi-martingale, so classical Itô’s calculus fell apart. Different attempts were made to generalize the existing stochastic analysis. Due to some of those efforts, people got remarkable results.

2.1 FBM

An FBM is a centered Gaussian process ( B H ( t ) ) t ≥ 0 , where H ∈ ( 0 , 1 ) , such that

(2.1) E [ ( B H ( t ) ) ( B H ( s ) ) ] = 1 2 ( ∣ t ∣ 2 H + ∣ s ∣ 2 H − ∣ t − s ∣ 2 H ) ,

for all t , s ≥ 0 . Some properties of FBM are as follows:

E [ ( B H ( t ) ) ] = 0 .
E [ ( B H ( t ) ) 2 ] = t 2 H .
The increments of an FBM are stationary, i.e., B H ( t + s ) − B H ( s ) = B H ( t ) , for all t , s ≥ 0 .
An FBM is a self-similar process i.e., B H ( a t ) = a H B H ( t ) , for all t ≥ 0 .
Almost all trajectories of FBM are continuous.

An FBM is similar to a standard BM for H = 1 2 . The Hurst index H determines the correlation between the increments of FBM. Increments are negatively correlated for H ∈ 0 , 1 2 and positively correlated for H ∈ 1 2 , 1 [14]. For H ≠ 1 2 , an FBM is neither semi-martingale nor a Markov process and the model depends on arbitrage opportunities [16]. To overcome this problem, an MFBM was introduced [17].

2.2 MFBM

An MFBM is a stochastic process ( M H ( t ) ) t ≥ 0 defined as follows:

(2.2) M H ( t ) = a B H ( t ) + b B ( t ) ,

where a and b are constants, B H ( t ) is an FBM with respect to P H , here Hurst index H ∈ ( 0 , 1 ) , and B ( t ) is an independent BM.

Some key properties of MFBM are as follows:

M H ( t ) is a centered Gaussian process.
M H ( 0 ) = 0 a.s.
For all s , t ∈ R the covariance function of M H ( t ) is
(2.3) cov ( M H ( t ) , M H ( s ) ) = a 2 min { t , s } + b 2 2 ( ∣ t ∣ 2 H + ∣ s ∣ 2 H − ∣ t − s ∣ 2 H ) .
The increments of M H ( t ) are stationary, i.e., M H ( t + s ) − M H ( s ) ∼ M H ( t ) , for all t , s ≥ 0 .
M H ( t ) is a self-similar process.
The increments of M H ( t ) are uncorrelated for H = 1 2 , positively correlated for H ∈ ( 1 2 , 1 ) , and negatively correlated for H ∈ ( 0 , 1 2 ) .
The increments of M H ( t ) exhibit long-range dependence for H = 1 2 .

Cheridito et al. [16] proved that an MFBM is equivalent to a BM for H ∈ ( 3 4 , 1 ) . The next section is devoted to some results on quasi-conditional expectation without proof, which were introduced by the study by Necula [23]. The basics of these results in the case of FBM can be found in the stduy by Hu and Øksendal [13]. Here, we are only quoting those theorems, which will be helpful in the preceding sections. Their proofs can be found in the study by Xiao et al. [24].

2.3 Quasi-conditional expectation and related results

This section is devoted to some results on quasi-conditional expectation, which were introduced by the study by Necula [23]. The basics of these results in the case of FBM can be found in the stduy by Hu and Øksendal [13]. Here, we are only quoting some theorems that will be helpful in the preceding sections. Their proofs can be found in the study by Xiao et al. [24].

Let ( Ω , ℱ H , P H ) be the probability space such that B H ( t ) is FBM with respect to P H and B ( t ) is an independent BM.

Theorem 2.1

For t ∈ ( 0 , T ) and ξ , η ∈ C

E ˜ [ exp ( ξ η B H ( t ) + ξ B ( t ) ) ∣ ℱ H ( t ) ] = exp ξ η B H ( t ) + ξ B ( t ) + 1 2 ξ 2 η 2 × ( T 2 H − t 2 H ) + 1 2 ξ 2 ( T − t ) ,

where ℱ H ( t ) is σ -algebra generated by ( B H ( t ) , 0 ≤ s ≤ t ) and E ˜ [ . ∣ ℱ H ( t ) ] on left-hand side denotes quasi-conditional expectation with respect to ℱ H ( t ) under the probability measure P H .

Using the above theorem, the quasi-conditional expectation of a function of an MFBM can be easily determined, which is given in the next theorem.

Theorem 2.2

For a function f that satisfies E ˜ [ f ( B H ( T ) , B ( T ) ) ] < ∞ , then for every t ∈ ( 0 , T ) and ξ , η ∈ C , we have the formula

E ˜ [ f ( ξ η B H ( T ) + ξ B ( T ) ) ∣ ℱ H ( t ) ] = ∫ R exp − ( y − ξ η B H ( t ) − ξ B ( t ) ) 2 2 ( ξ 2 η 2 ( T 2 H − t 2 H ) + ξ 2 ( T − t ) ) 2 π ( ξ 2 η 2 ( T 2 H − t 2 H ) + ξ 2 ( T − t ) ) d y .

We can easily obtain the next corollary by setting f ( y ) = 1 A ( y ) .

Corollary 2.1

If A ∈ ℬ ( R ) , then

E ˜ [ 1 A ( ξ η B H ( T ) + ξ B ( T ) ) ∣ ℱ H ( t ) ] = ∫ A exp − ( y − ξ η B H ( t ) − ξ B ( t ) ) 2 2 ( ξ 2 η 2 ( T 2 H − t 2 H ) + ξ 2 ( T − t ) ) 2 π ( ξ 2 η 2 ( T 2 H − t 2 H ) + ξ 2 ( T − t ) ) d y .

Consider the process

α B H * ( t ) + β B * ( t ) = α B H ( t ) + α 2 t 2 H + β B ( t ) + β 2 t ,

for all α , β ∈ R , then by fractional Grisanov theorem, α B H * ( t ) + β B * ( t ) is a new MFBM under probability measure P H * .

Let us define another process

X ( t ) = exp − α B H ( t ) − 1 2 α 2 t 2 H − β B ( t ) − 1 2 β 2 t ,

for all t ∈ [ 0 , T ] .

Theorem 2.3

For a function f that satisfies E ˜ [ f ( B H ( T ) , B ( T ) ) ] < ∞ , then for every t ∈ ( 0 , T ) and ξ , η ∈ C , we have the

E ˜ * [ f ( α B H ( T ) + β B ( T ) ) ∣ ℱ H ( t ) ] = 1 X ( t ) E ˜ [ f ( α B H ( T ) + β B ( T ) ) X ( T ) ∣ ℱ H ( t ) ] .

The aforementioned theorem gives a relationship between a quasi-conditional expectation E ˜ [ . ∣ ℱ H ( t ) ] with respect to P H and a quasi-conditional expectation E ˜ * [ ⋅ ∣ ℱ H ( t ) ] with respect to P H * . By using this theorem, one can easily find the discounted expectation of a function of MFBM with the help of the next given theorem, which can be understood as the Feynman-Kac formula (or fundamental theorem of asset pricing) in the mixed fractional environment.

Theorem 2.4

The value of a bounded ℱ H ( t ) -measurable claim V ∈ L 2 ( P H ) at time t ∈ [ 0 , T ] is given by

(2.4) V t = e − r ( T − t ) E ˜ [ V T ∣ ℱ H ( t ) ] ,

where r denotes risk-less interest rate.

3 Debt pricing in credit risk structural models driven by MFBM

We start by establishing some terminology. Let’s consider a company that was established at the time 0, when assets were acquired for V ( 0 ) . The company receives funding from two sources. Financial investors provide E ( 0 ) , referred to as equity. The rest of the total, D ( 0 ) = V ( 0 ) − E ( 0 ) , called debt, is either obtained from a bank or raised by issuing bonds. We assume that the company will repay its debt at time T without any additional cash flows between 0 and T . The interest rate agreed upon with the bank is denoted by k D . There is always a risk of default by the debt holder, so this interest rate must be higher than the risk-free rate r , as stated in Proposition 1.2 from the study by Marek Capiński [3]. The amount that the debt holder needs to repay at time T is F = e k D T D ( 0 ) .

Definition 3.1

[3] The quasi-debt ratio is defined as follows:

l = F e − r T V ( 0 ) ,

and the credit spread is defined as follows:

s = k D − r ,

where k D satisfies

F = D ( 0 ) e k D T .

During the time period from 0 to T , the assets are utilized to generate returns, which are then distributed between two groups of investors at a time T . As a priority, debt is repaid with added interest, and the remaining funds are given to equity holders [3]. It is clear that the funds are generated by selling the company’s assets at the time T . We make the assumption that the company’s assets are trad-able.

3.1 Debt pricing for company with tradeable assets

With the assumption of a liquid market, let V ( t ) , for t ∈ [ 0 , T ] , be the value of the company. At a time T , we sell the returns and dissolve the business to assess the financial standing of the company, with V ( T ) representing the value of the company at that time. In this scenario, we may encounter two situations:

V ( T ) < F : In this case, the value of the company is less than the debt repayment amount, and the debt holders receive V ( T ) , while the equity holders receive nothing.
V ( T ) ≥ F : In this case, the debt holders receive the full amount F , and the equity holders receive the rest of the funds.

From these two conditions, we can conclude that the value of the equity is equivalent to a call option with the value of the assets as the underlying security and F as the strike price, i.e.,

(3.1) E ( T ) = ( V ( T ) − F ) + ,

and

(3.2) D ( T ) = min ( F , V ( T ) ) = F − max ( F − V ( T ) , 0 ) = F − ( F − V ( T ) ) + .

In the following theorem, we calculate the debt pricing formula in the mixed fractional Merton model.

Theorem 3.1

Assume that the assets of the company are tradeable and follow the mixed fractional Black–Scholes model (cf. [4]), i.e., satisfy the stochastic differential equation

(3.3) d V ( t ) = μ V ( t ) d t + σ V ( t ) ( d B P ( t ) + d B P , H ( t ) ) ,

where B P , H is an FBM, B P is a standard BM with respect to probability measure P H , and constants μ and σ denote the average and volatility, respectively, of the value V ( t ) of company. Then at any time t ∈ [ 0 , T ] , the debt that the company need to repay is,

(3.4) D ( V ( t ) , σ , r , T , F ) = V ( t ) N ( − d 1 ) + e − r ( T − t ) F N ( d 2 ) ,

where

d 1 = ln V ( t ) F + r ( T − t ) + 1 2 σ 2 ( T 2 H − t 2 H + T − t ) σ 2 ( T 2 H − t 2 H + T − t ) , d 2 = ln V ( t ) F + r ( T − t ) − 1 2 σ 2 ( T 2 H − t 2 H + T − t ) σ 2 ( T 2 H − t 2 H + T − t ) ,

and N is a distribution function of the standard normal random variable.

Proof

By following standard no-abitrage argument and invoking the fractional Girsanov theorem, we can obtain the risk neutral measure P ˆ H equivalent to real-world probability measure P H such that V ( t ) satisfies

d V ( t ) = r V ( t ) d t + σ V ( t ) ( d B ˆ P ˆ H ( t ) + d B ˆ P ˆ H , H ( t ) ) .

Then the debt D ( t ) at any time t ∈ [ 0 , T ] can be computed from the Feynman–Kac formula (Theorem 2.4),

D ( t ) = e − r ( T − t ) E P ˆ H [ D ( T ) ∣ ℱ H ( t ) ] , = e − r ( T − t ) E P ˆ H [ F − ( F − V ( T ) ) + ∣ ℱ H ( t ) ] , = e − r ( T − t ) F − e − r ( T − t ) E P ˆ H [ ( F − V ( T ) ) + ∣ ℱ H ( t ) ] , = e − r ( T − t ) F − P BS ( V ( t ) , σ , r , T , F ) ,

where P BS ( V ( t ) , σ , r , T , F ) = F e − r ( T − t ) N ( − d 2 ) − V ( t ) N ( − d 1 ) can be treated as Black–Scholes price of put option on value of company V with strike F . Thus, substituting the value of P BS in the last equation gives the required expression for debt D ( t ) (Figure 1).

$Figure 1 Fair price of equity and debt vs the Hurst parameter and debt at time T. It is clear that equity decreases as H → 1 H\to 1 and D ( T ) D\left(T) increases. Similarly, D ( 0 ) D\left(0) increases as H → 1 H\to 1 and D ( T ) D\left(T) increases.$

Figure 1

Fair price of equity and debt vs the Hurst parameter and debt at time T. It is clear that equity decreases as H → 1 and D ( T ) increases. Similarly, D ( 0 ) increases as H → 1 and D ( T ) increases.

Keeping in view Definition 3.1, the following theorem gives the expression for credit spread in the fractional framework.

Theorem 3.2

In the framework of Theorem 3.1, the credit spread can be given as follows:

s = − 1 T ln N ( d 2 ) + N ( − d 1 ) l ,

where

d 1 = − ln l + 1 2 σ 2 ( T 2 H + T ) σ T 2 H + T , d 2 = − ln l − 1 2 σ 2 ( T 2 H + T ) σ T 2 H + T ,

and quasi-debt ratio l = F e − r T V ( 0 ) .

Proof

Recall that s = k D − r , and F = D ( 0 ) e k D T . Therefore,

s = 1 T ln F D ( 0 ) − r = − 1 T ln D ( 0 ) F e − r T .

By using D ( 0 ) from Eq. (3.4), we infer that

s = − 1 T ln F e − r T − P ( 0 ) F e − r T , = − 1 T ln 1 − P ( 0 ) F e − r T , = − 1 T ln 1 − F e − r T N ( − d 2 ) − V ( 0 ) N ( − d 1 ) F e − r T , = − 1 T ln 1 − N ( − d 2 ) + V ( 0 ) N ( − d 1 ) F e − r T , = − 1 T ln N ( d 2 ) + N ( − d 1 ) l .

Finally, by using expression for d 1 from Theorem 3.1, we obtain

d 1 = ln V ( 0 ) F + r T + 1 2 σ 2 ( T 2 H + T ) σ 2 ( T 2 H + T ) , = ln V ( 0 ) F e − r T + 1 2 σ 2 ( T 2 H + T ) σ 2 ( T 2 H + T ) , = − ln l + 1 2 σ 2 ( T 2 H + T ) σ 2 ( T 2 H + T ) .

Similarly, we can compute,

□ d 2 = − ln l − 1 2 σ 2 ( T 2 H + T ) σ 2 ( T 2 H + T ) .

Theorem 3.3

For

s ( T ) = − 1 T ln N ( d 2 ( T ) ) + N ( − d 1 ( T ) ) l ,

with

d 1 ( T ) = − ln l + 1 2 σ 2 ( T 2 H + T ) σ 2 ( T 2 H + T ) ,

and

d 2 ( T ) = − ln l − 1 2 σ 2 ( T 2 H + T ) σ 2 ( T 2 H + T ) .

We have

lim T → 0 s ( T ) = 0 , l < 1 , ∞ , l ≥ 1 .

Proof

Case: 1

For l < 1 , we have

lim T → 0 d 1 ( T ) = lim T → 0 − ln l + 1 2 σ 2 ( T 2 H + T ) σ 2 ( T 2 H + T ) = ∞ .

Similarly,

lim T → 0 d 2 ( T ) = lim T → 0 − ln l − 1 2 σ 2 ( T 2 H + T ) σ 2 ( T 2 H + T ) = ∞ .

Consequently,

lim T → 0 N ( d 2 ( T ) ) + N ( − d 1 ( T ) ) l = 1

and

lim T → 0 ln N ( d 2 ( T ) ) + N ( − d 1 ( T ) ) l = 0 .

Now using L’Hopital’s rule

lim T → 0 s ( T ) = − lim T → 0 d d T ln N ( d 2 ) + N ( − d 1 ) l d d T T , = − lim T → 0 d d T N ( d 2 ) + N ( − d 1 ) l N ( d 2 ) + N ( − d 1 ) l , = − lim T → 0 d d T N ( d 2 ( T ) ) − 1 l lim T → 0 d d T N ( − d 1 ( T ) ) , = − 1 2 π lim T → 0 e − 1 2 d 2 ( T ) 2 d d T ( d 2 ( T ) ) − 1 l 2 π lim T → 0 e − 1 2 d 1 ( T ) 2 d d T ( − d 1 ( T ) ) , = 0 .

The aforementioned result is zero because the exponential e − 1 2 d 1 ( T ) 2 and e − 1 2 d 2 ( T ) 2 converge to zero rapidly than the derivatives

d d T N ( d 1 ( T ) ) = α σ 2 ln l + σ 2 ( T 2 H + T ) + r α T 2 H + T

and

d d T N ( d 2 ( T ) ) = α σ 2 ln l + σ 2 ( T 2 H + T ) − r α T 2 H + T

converge to ∞ and − ∞ as T → 0 , where

α = 2 H T 2 H − 1 + 1 4 ( T 2 H + T ) 3 2 .

Case: 2

First consider l = 1 , then

d 1 ( T ) = − ln l + 1 2 σ 2 ( T 2 H + T ) σ 2 ( T 2 H + T ) , d 2 ( T ) = − ln l − 1 2 σ 2 ( T 2 H + T ) σ 2 ( T 2 H + T )

become

d 1 ( T ) = σ T 2 H + T 2 , d 2 ( T ) = − σ T 2 H + T 2 .

As a result,

lim T → 0 d 1 ( T ) = lim T → 0 d 2 ( T ) = 0 .

Finally,

lim T → 0 ( N ( − d 1 ( T ) ) + N ( d 2 ( T ) ) ) = N ( 0 ) + N ( 0 ) = 1 2 + 1 2 = 1

and

lim T → 0 ln ( N ( − d 1 ( T ) ) + N ( d 2 ( T ) ) ) = 1 .

Now using l’Hopital’s rule

lim T → 0 s ( T ) = − lim T → 0 d d T ln ( N ( d 2 ) + N ( − d 1 ) ) d d T T , = − lim T → 0 d d T ( N ( d 2 ) + N ( − d 1 ) ) N ( d 2 ) + N ( − d 1 ) , = − lim T → 0 d d T N ( d 2 ( T ) ) − lim T → 0 d d T N ( − d 1 ( T ) ) , = − 1 2 π lim T → 0 e − 1 2 d 2 ( T ) 2 d d T ( d 2 ( T ) ) + 1 2 π lim T → 0 e − 1 2 d 1 ( T ) 2 d d T ( d 1 ( T ) ) .

lim T → 0 e − 1 2 d 1 ( T ) 2 = 1 , lim T → 0 e − 1 2 d 2 ( T ) 2 = 1

and

lim T → 0 d d T ( d 1 ( T ) ) = lim T → 0 σ ( 2 H T 2 H − 1 + 1 ) 4 T 2 H + T = ∞ ,

also

lim T → 0 d d T ( d 2 ( T ) ) = lim T → 0 − σ ( 2 H T 2 H − 1 + 1 ) 4 T 2 H + T = − ∞ .

So, with this, we have

lim T → 0 s ( T ) = ∞ .

Finally, considering l > 1 ,

lim T → 0 d 1 ( T ) = lim T → 0 − ln l + 1 2 σ 2 ( T 2 H + T ) σ 2 ( T 2 H + T ) = − ∞ .

Similarly,

lim T → 0 d 2 ( T ) = lim T → 0 − ln l − 1 2 σ 2 ( T 2 H + T ) σ 2 ( T 2 H + T ) = − ∞ .

Consequently,

lim T → 0 ln N ( d 2 ( T ) ) + N ( − d 1 ( T ) ) l = ln N ( − ∞ ) + N ( ∞ ) l = ln 1 l = − ln l .

To conclude (Figure 2),

□ lim T → 0 s ( T ) = − lim T → 0 ln N ( d 2 ( T ) ) + N ( − d 1 ( T ) ) l T = − − ln l lim T → 0 T = ∞ .

$Figure 2 Credit spread corresponding to l ≥ 1 l\ge 1 and l < 1 l\lt 1 .$

Figure 2

Credit spread corresponding to l ≥ 1 and l < 1 .

3.2 Partial recovery

In case of default or bankruptcy, i.e., when V ( T ) < F , full debt repayment of F may not be possible due to expenses incurred for legal and bankruptcy processes. In the event of default, the company’s assets are sold, and debt holders receive α V ( T ) , where α ∈ [ 0 , 1 ] is known as the recovery rate. The debt holders do not receive the full value of V ( T ) as a result of the costs associated with bankruptcy procedures, such as fees for bailiffs and legal service providers [3]. On the other hand, if V ( T ) ≥ F , debt holders receive the full amount. We use F α to represent the debt that would be recovered in the event of default. Hence, the debt payoff becomes

D ( T ) = F α 1 { V ( T ) ≥ F α } + α V ( T ) 1 { V ( T ) < F α } .

Theorem 3.4

(Partial recovery of debt) Assume that the assets of the company are trade-able and follow mixed fractional Black–Scholes model (cf. [4]), i.e., satisfy the stochastic differential equation

(3.5) d V ( t ) = r V ( t ) d t + σ V ( t ) ( d B ˆ P ˆ H ( t ) + d B ˆ P ˆ H , H ( t ) ) ,

where B ˆ P ˆ H , H is an FBM, B ˆ P ˆ H is a standard BM with respect to probability measure P H , and constants r and σ denote the risk-free interest rate and volatility, respectively, of the value V ( t ) of company. Then at any time t ∈ [ 0 , T ] ,

D ( t ) = F α e − r ( T − t ) N ( d 2 ) + α V ( t ) N ( − d 1 ) ,

where

d 1 = ln V ( t ) F α + r ( T − t ) + 1 2 σ 2 ( T 2 H − t 2 H + T − t ) σ 2 ( T 2 H − t 2 H + T − t ) , d 2 = ln V ( t ) F α + r ( T − t ) − 1 2 σ 2 ( T 2 H − t 2 H + T − t ) σ 2 ( T 2 H − t 2 H + T − t ) ,

and N is a distribution function of the standard normal random variable.

Proof

We will begin by using Feynman-Kac formula (Theorem 2.4). For any time 0 ≤ t ≤ T ,

D ( t ) = e − r ( T − t ) E ˜ [ D ( T ) ∣ ℱ H ( t ) ] , = e − r ( T − t ) E ˜ [ F α 1 V ( T ) ≥ F α + α V ( T ) 1 V ( T ) < F α ∣ ℱ H ( t ) ] , = F α e − r ( T − t ) E ˜ [ 1 V ( T ) ≥ F α ∣ ℱ H ( t ) ] + α e − r ( T − t ) E ˜ [ V ( T ) 1 V ( T ) < F α ∣ ℱ H ( t ) ] .

Since

E ˜ [ 1 V ( T ) ≥ F α ∣ ℱ H ( t ) ] = N ( d 2 ) ,

where

d 2 = ln V ( t ) F α + r ( T − t ) − 1 2 σ 2 ( T 2 H − t 2 H + T − t ) σ 2 ( T 2 H − t 2 H + T − t ) .

Now for the second part

E ˜ [ V ( T ) 1 V ( T ) < F α ∣ ℱ H ( t ) ] ,

consider the process

(3.6) σ B H * ( t ) + σ B * ( t ) = σ B H ( t ) − σ 2 t 2 H + σ B ( t ) − σ 2 t ,

for all σ ∈ R , then by fractional Girsanov theorem [13], σ B H * ( t ) + σ B * ( t ) is a new mfBm under probability measure P H * . Let us define another process

(3.7) Z ( t ) = exp σ B H ( t ) − 1 2 σ 2 t 2 H + σ B ( t ) − 1 2 σ 2 t ,

for all t ∈ [ 0 , T ] , then

E ˜ [ V ( T ) 1 V ( T ) < F α ∣ ℱ H ( t ) ] = E ˜ V ( 0 ) e r T exp σ B H ( t ) − 1 2 σ 2 t 2 H + σ B ( t ) − 1 2 σ 2 t 1 V ( T ) < F α ∣ ℱ H ( t ) , = V ( 0 ) e r T E ˜ [ Z T 1 V ( T ) < F ∣ ℱ H ( t ) ] , = V ( 0 ) e r T E ˜ [ Z T 1 y < d 1 * ∣ ℱ H ( t ) ] , = V ( 0 ) e r T Z t E ˜ * [ 1 y < d 1 * ∣ ℱ H ( t ) ] , = V ( 0 ) e r T Z t E ˜ * [ 1 V ( T ) < F ∣ ℱ H ( t ) ] .

Since

V ( t ) = V ( 0 ) exp σ B H ( t ) − 1 2 σ 2 t 2 H + σ B ( t ) − 1 2 σ 2 t + r t .

Then by using

σ B H * ( t ) + σ B * ( t ) = σ B H ( t ) − σ 2 t 2 H + σ B ( t ) − σ 2 t ,

we have

V ( t ) = V ( 0 ) exp σ B H * ( t ) + 1 2 σ 2 t 2 H + σ B * ( t ) + 1 2 σ 2 t + r t .

Now for V ( T ) < F α , we have

σ B H * ( t ) + σ B * ( t ) < ln F α V ( 0 ) − 1 2 σ 2 ( T 2 H + T ) − r T ≕ d 1 * .

It implies

E ˜ * [ 1 V ( T ) < F α ∣ ℱ H ( t ) ] = E ˜ * [ 1 V ( T ) < F α ( σ B H * ( t ) + σ B * ( t ) ) ∣ ℱ H ( t ) ] , = ∫ − ∞ d 1 * exp − ( y − ξ η B H * ( t ) − ξ B * ( t ) ) 2 2 ( ξ 2 η 2 ( T 2 H − t 2 H ) + ξ 2 ( T − t ) ) 2 π ( ξ 2 η 2 ( T 2 H − t 2 H ) + ξ 2 ( T − t ) ) f ( y ) d y .

Let

z = y − σ B H * ( t ) − σ B * ( t ) σ 2 ( T 2 H − t 2 H ) + σ 2 ( T − t ) ,

then we have

d z = d y σ 2 ( T 2 H − t 2 H ) + σ 2 ( T − t ) .

By using above change of variables, we obtain

E ˜ * [ 1 V ( T ) < F α ∣ ℱ H ( t ) ] = ∫ − ∞ d 1 * − σ B H * ( t ) − σ B * ( t ) σ 2 ( T 2 H − t 2 H ) + σ 2 ( T − t ) 1 2 π e − z 2 2 d z , = ∫ − ∞ d 1 1 2 π e − z 2 2 d z , = N ( − d 1 ) .

So, we have

E ˜ [ V ( T ) 1 V ( T ) < F α ∣ ℱ H ( t ) ] = V ( 0 ) e r T Z t N ( − d 1 ) , = V ( 0 ) e r T exp σ B H ( t ) − 1 2 σ 2 t 2 H + σ B ( t ) − 1 2 σ 2 t N ( − d 1 ) , = e r ( T − t ) V ( t ) N ( − d 1 ) .

Finally, the debt payoff for partial recovery at any t ∈ [ 0 , T ] becomes (Figure 3)

D ( t ) = F α e − r ( T − t ) N ( d 2 ) + α V ( t ) N ( − d 1 ) .□

$Figure 3 Fair price of equity and debt vs the Hurst parameter and recovery rate. Here, V ( 0 ) = 100 V\left(0)=100 , r = 0.05 r=0.05 , σ = 0.3 \sigma =0.3 , F α = 90 {F}_{\alpha }=90 , H ∈ ( 0 , 1 ) H\in \left(0,1) , α ∈ ( 0 , 1 ) \alpha \in \left(0,1) , and T = 5 T=5 for the sake of getting a significant change in E ( 0 ) E\left(0) and D ( 0 ) D\left(0) . It is clear that equity and debt increase as recovery rate α → 1 \alpha \to 1 .$

Figure 3

Fair price of equity and debt vs the Hurst parameter and recovery rate. Here, V ( 0 ) = 100 , r = 0.05 , σ = 0.3 , F α = 90 , H ∈ ( 0 , 1 ) , α ∈ ( 0 , 1 ) , and T = 5 for the sake of getting a significant change in E ( 0 ) and D ( 0 ) . It is clear that equity and debt increase as recovery rate α → 1 .

3.3 Fixed assets

It is unrealistic to assume that all assets of a company are traded. So, we relax this assumption by considering that the company’s value depends on both traded and fixed assets, such as office equipment. Let L be the constant value of the fixed assets. Then,

V ( t ) = L + S ( t ) ,

where

d S ( t ) = r S ( t ) d t + σ S ( t ) ( d B ( t ) + d B H ( t ) ) .

The equity and debt payoffs can be divided into three cases:

If S ( T ) ≥ F , then the debt is fully paid, i.e., D ( T ) = F . The fixed assets are safe and equity holders receive E ( T ) = L + S ( T ) − F .
If S ( T ) < F , but L + S ( T ) ≥ F , then the debt is fully paid, i.e., D ( T ) = F . Both traded, and some fixed assets are sold and equity holders receive E ( T ) = L + S ( T ) − F .
If L + S ( T ) < F , then D ( T ) = L + S ( T ) and E ( T ) = 0 , meaning all assets are sold and bankruptcy procedures begin.

Therefore, equity can be viewed as a call option on S ( T ) with strike F − L , similar to before, with V ( T ) replaced by S ( T ) and F replaced by F − L . Hence,

E ( T ) = max ( S ( T ) − ( F − L ) , 0 ) .

With a similar argument as in Theorem 3.1, we have

E ( t ) = C ( S ( t ) , σ , r , T , F − L ) = S ( t ) N ( d 1 ) − e − r ( T − t ) ( F − L ) N ( d 2 ) ,

where

d 1 = ln S ( t ) F − L + r ( T − t ) + 1 2 σ 2 ( T 2 H − t 2 H + T − t ) σ 2 ( T 2 H − t 2 H + T − t )

and

d 2 = ln S ( t ) F − L + r ( T − t ) − 1 2 σ 2 ( T 2 H − t 2 H + T − t ) σ 2 ( T 2 H − t 2 H + T − t ) ,

which represents the value of equity with fixed assets.

Finally, for debt payoff, we have

D ( T ) = min ( F , L + S ( T ) ) = F − max ( ( F − L ) − S ( T ) , 0 ) ,

and by using put call parity, we obtain

D ( t ) = S ( t ) N ( − d 1 ) + e − r ( T − t ) ( F − L ) N ( d 2 ) ,

where d 1 and d 2 are same as earlier (Figure 4).

$Figure 4 Fair price of equity and debt vs the Hurst parameter and fixed assets value. Here, V ( 0 ) = 100 V\left(0)=100 , r = 0.05 r=0.05 , σ = 0.3 \sigma =0.3 , F α = 90 {F}_{\alpha }=90 , H ∈ ( 0 , 1 ) H\in \left(0,1) , and T = 1 T=1 . It is clear that for a fixed value of H H not close to 1, as it will give smoother path of FBM, equity increases and debt decreases as fixed assets value L → V ( t ) L\to V\left(t) , i.e., S ( t ) = 0 S\left(t)=0 .$

Figure 4

Fair price of equity and debt vs the Hurst parameter and fixed assets value. Here, V ( 0 ) = 100 , r = 0.05 , σ = 0.3 , F α = 90 , H ∈ ( 0 , 1 ) , and T = 1 . It is clear that for a fixed value of H not close to 1, as it will give smoother path of FBM, equity increases and debt decreases as fixed assets value L → V ( t ) , i.e., S ( t ) = 0 .

4 Barrier model

For practical purposes, it is important to assess a company’s performance prior to maturity. Poor performance could lead bondholders to sell their bonds early. However, the situation with debt is different. Early repayment can be agreed upon with the bank and the agreement should specify the early default benchmark (i.e., barrier) and final debt amount to be paid to the bank.

To establish a standard for default, we first assume the value of the company is V ( t ) . Bankruptcy is declared when this value reaches a threshold B at a random time, τ B defined as follows:

τ B ≔ inf t ∣ V ( t ) = B .

To determine the debt amount to be paid, we define two random variables:

m V ( T ) = inf 0 < t < T V ( t )

and

M V ( T ) = sup 0 < t < T V ( t ) ,

where V ( t ) follows the equation

d V ( t ) = r V ( t ) d t + σ V ( t ) ( d B ( t ) + d B H ( t ) ) .

There are two cases for debt:

If τ B > T , debt is paid D ( T ) = F if V ( T ) ≥ F , or D ( T ) = V ( T ) otherwise.
If τ B ≤ T , debt is paid D ( T ) = B e r ( T − t ) , which is the compounded amount of the Barrier.

Hence,

D ( T ) = F 1 { V ( T ) > F , m V ( T ) > B } + B e r ( T − t ) 1 { τ B ≤ T } ,

and the discounted amount of debt becomes,

D ( t ) = e − r ( T − t ) F E ˜ { 1 { V ( T ) > F , m V ( T ) > B } } + e − r ( T − t ) B e r ( T − t ) E ˜ { 1 { τ B ≤ T } } .

It implies

(4.1) D ( t ) = e − r ( T − t ) F E ˜ { 1 { V ( T ) > F , m V ( T ) > B } } + B E ˜ { 1 { τ B ≤ T } } .

The next thing is to evaluate aforementioned expectations. Hence, the following section is devoted to do this where we provide a framework to price barrier model in case of MFBM.

4.1 Barrier option valuation in the MFBM

To price the barrier option, we will start by introducing some financial derivatives, which will be used later. These derivatives have already been priced in an FBM environment as described in the study by Necula [25]. However, in this case, the pricing of the same derivatives will be done in an MFBM environment. The financial derivatives are as follows:

Binary call with strike F , i.e.,
BC = 1 { V ( T ) > F } .
Binary put with strike F , i.e.,
BP = 1 { V ( T ) < F } .
Gap call with strike F , i.e.,
GC = V ( T ) 1 { V ( T ) > F } .
Gap put with strike F , i.e.,
GP = V ( T ) 1 { V ( T ) < F } .

The discounted value of the aforementioned contingent claims can be found easily by using Lemma 3.1 from the study by Necula [25], we have

BC ( t , V ( t ) ) = e − r ( T − t ) N ( d 2 ) ,
BP ( t , V ( t ) ) = e − r ( T − t ) N ( − d 2 ) ,
GC ( t , V ( t ) ) = V ( t ) N ( d 1 ) ,
GP ( t , V ( t ) ) = V ( t ) N ( − d 1 ) ,

where

d 1 = ln V ( t ) F + r ( T − t ) + 1 2 σ 2 ( T 2 H − t 2 H + T − t ) σ 2 ( T 2 H − t 2 H + T − t )

and

d 2 = ln V ( t ) F + r ( T − t ) − 1 2 σ 2 ( T 2 H − t 2 H + T − t ) σ 2 ( T 2 H − t 2 H + T − t ) .

With the aforementioned evaluation and the notation already provided, i.e.,

τ B ≔ inf { t ∣ V ( t ) = B } ,

m V ( T ) ≔ inf 0 < t < T V ( t ) ,

and

M V ( T ) ≔ sup 0 < t < T V ( t ) ,

we have the following theorem:

Theorem 4.1

If F > B , V ( T ) > B , and τ B > t , then

(4.2) E ˜ { 1 { V ( T ) > F , m V ( T ) > B } } = N ( b 1 ) − V ( t ) B N ( b 2 ) ,

where

b 1 = ln V ( t ) F + r ( T − t ) − 1 2 σ 2 ( T 2 H − t 2 H + T − t ) σ 2 ( T 2 H − t 2 H + T − t )

and

b 2 = ln B 2 F V ( t ) − r ( T − t ) − 1 2 σ 2 ( T 2 H − t 2 H + T − t ) σ 2 ( T 2 H − t 2 H + T − t ) .

Proof

First, let us consider a down-and-out binary call with strike F , barrier B and maturity T , whose payoff becomes

1 { V ( T ) > F , m V ( T ) > B } .

The price of this derivative becomes

DOBC ( t , V ( t ) ) = 0 , τ B < t , ≠ 0 , τ B > t .

Now consider a portfolio that includes a long position in one binary call with strike F , and one short position in 1 B gap puts with strike B 2 F .

At t = τ B , i.e., V ( t ) = B price of this portfolio becomes zero. If τ B > T , then the value of the portfolio becomes equal to that of option, also when τ B ≤ T both portfolio and option agree on the same payoff.

Since mixed fractional Black–Scholes model does not exhibit arbitrage opportunities, so the payoff of portfolio and the option agree for all the time such that t < min { τ B , T } . Hence,

DOBC ( t , V ( t ) ) = BC ( t , V ( t ) ) − 1 B GP ( t , V ( t ) ) .

Finally, we have the following:

(4.3)□ E ˜ { 1 { V ( T ) > F , m V ( T ) > B } } = N ( b 1 ) − V ( t ) B N ( b 2 ) .

For the proof of the next theorem, we will use the similar approach.

Theorem 4.2

If F > B , V ( T ) > B , and τ B > t , then

(4.4) E ˜ { 1 { τ B > T } } = E ˜ { 1 { m V ( T ) > B } } = N ( c 1 ) − V ( t ) B N ( c 2 ) ,

where

c 1 = ln V ( t ) B + r ( T − t ) − 1 2 σ 2 ( T 2 H − t 2 H + T − t ) σ 2 ( T 2 H − t 2 H + T − t )

and

c 2 = ln B V ( t ) − r ( T − t ) − 1 2 σ 2 ( T 2 H − t 2 H + T − t ) σ 2 ( T 2 H − t 2 H + T − t ) .

Proof

Here, we use the similar approach as we have done previously, just by comparing the option with the portfolio consisting of long position in one binary call with strike B , and one short position in 1 B gap puts with strike B . So, the price of option becomes

E ˜ { 1 { τ B > T } } = E ˜ { 1 { m V ( T ) > B } } = N ( c 1 ) − V ( t ) B N ( c 2 ) ,

where c 1 and c 2 are same as given earlier.□

4.2 Debt and equity in case of Barrier model

From Sections 4.1 and 4.2, we can easily find the amount of debt and equity. Let us put this in the form of a theorem.

Theorem 4.3

Using the notation provided in this section, the value of the debt becomes

D ( t ) = e − r ( T − t ) F N ( b 1 ) − V ( t ) B N ( b 2 ) + B N ( − c 1 ) + V ( t ) B N ( c 2 ) .

Proof

From Eq. (4.5), we have

(4.5) D ( t ) = e − r ( T − t ) F E ˜ { 1 { V ( T ) > F , m V ( T ) > B } } + B E ˜ { 1 { τ B ≤ T } } , = e − r ( T − t ) F E ˜ { 1 { V ( T ) > F , m V ( T ) > B } } + B P ( { τ B ≤ T } ) , = e − r ( T − t ) F E ˜ { 1 { V ( T ) > F , m V ( T ) > B } } + B ( 1 − P ( { τ B > T } ) ) , = e − r ( T − t ) F E ˜ { 1 { V ( T ) > F , m V ( T ) > B } } + B ( 1 − E ˜ { 1 { τ B > T } } ) .

Finally, by using Theorems 4.1 and 4.2, we have

D ( t ) = e − r ( T − t ) F N ( b 1 ) − V ( t ) B N ( b 2 ) + B 1 − N ( c 1 ) − V ( t ) B N ( c 2 ) .

It implies

D ( t ) = e − r ( T − t ) F N ( b 1 ) − V ( t ) B N ( b 2 ) + B N ( − c 1 ) + V ( t ) B N ( c 2 ) ,

where b 1 , b 2 , c 1 , and c 2 are same as mentioned earlier (Figure 5).□

$Figure 5 Fair price of debt vs the Hurst parameter and the value of the barrier. Here, V ( 0 ) = 100 V\left(0)=100 , r = 0.05 r=0.05 , σ = 0.3 \sigma =0.3 , F = 50 F=50 , and 150 , H ∈ ( 0 , 1 ) 150,H\in \left(0,1) , and T = 25 T=25 to obtain significant change in D ( 0 ) D\left(0) . It is clear that as H → 1 H\to 1 and the value of the barrier increases, debt increases. Moreover, when F > V 0 F\gt {V}_{0} and the barrier is small, then D ( 0 ) = 0 D\left(0)=0 , i.e., it would be difficult for the bank to give loan to the company.$

Figure 5

Fair price of debt vs the Hurst parameter and the value of the barrier. Here, V ( 0 ) = 100 , r = 0.05 , σ = 0.3 , F = 50 , and 150 , H ∈ ( 0 , 1 ) , and T = 25 to obtain significant change in D ( 0 ) . It is clear that as H → 1 and the value of the barrier increases, debt increases. Moreover, when F > V 0 and the barrier is small, then D ( 0 ) = 0 , i.e., it would be difficult for the bank to give loan to the company.

5 Conclusion

In this work, we have dealt with credit risk evaluation associated with a company whose underlying assets are driven by the MFBM. Modeling of assets driven by MFBM has the natural advantage of incorporating the memory effect, i.e., assets are random but highly correlated with the previous state’s assets. In particular, we focused on pricing on debt, equity, and credit spread for the cases when the assets of the company (driven by MFBM) are tradable, fixed, and partial recovery is expected in case of the default of the company. Finally, after deriving the pricing formulas for barrier style Gap and binary options, the pricing of the debt and equity has been done in barrier framework. For this purpose, a barrier is set up for the value of assets of the company to declare a default of the company. The simulations and graphs of the valuation formulas have been plotted.

Funding information: The authors did not receive any external funding to perform this research.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors declare that they have no conflict of interest.
Data availability statement: No data were required to perform this research.

References

[1] Huang JZ, Huang M. How much of the corporate-treasury yield spread is due to credit risk? Rev Asset Pricing Studies. 2012 11;2(2):153–202. 10.1093/rapstu/ras011. Suche in Google Scholar

[2] Ding Z, Granger CWJ, Engle RF. A long memory property of stock market returns and a new model. J Empirical Finance. 1993;1(1):83–106. https://www.sciencedirect.com/science/article/pii/092753989390006D. 10.1016/0927-5398(93)90006-DSuche in Google Scholar

[3] Marek Capiński TZ. Credit risk. 1st ed. Mastering mathematical finance. Cambridge: Cambridge University Press; 2017. Suche in Google Scholar

[4] Merton RC. Theory of rational option pricing. Theory of Valuation. 1973, p. 229–88. 10.1142/9789812701022_0008Suche in Google Scholar

[5] Merton RC. On the pricing of corporate debt: The risk structure of interest rates. J Finance. 1974;29(2):449–70. 10.1111/j.1540-6261.1974.tb03058.xSuche in Google Scholar

[6] Sun X, Yan L. Mixed-fractional models to credit risk pricing. J Stat Econ Methods. 2012;1(3):79–96. Suche in Google Scholar

[7] Biagini F, Fink H, Klüppelberg C. A fractional credit model with long range dependent default rate. Stochastic Process Appl. 2013;123(4):1319–47. 10.1016/j.spa.2012.12.006Suche in Google Scholar

[8] Leccadito A. Fractional models to credit risk pricing [doctoral dissertation]. Universitá degli Studi di Bergamo; 2008. Suche in Google Scholar

[9] Hsieh DA. Chaos and nonlinear dynamics: application to financial markets. J Finance. 1991;46(5):1839–77. 10.1111/j.1540-6261.1991.tb04646.xSuche in Google Scholar

[10] Mariani M, Florescu I, Varela MB, Ncheuguim E. Long correlations and levy models applied to the study of memory effects in high frequency (tick) data. Phys A Stat Mechanics Appl. 2009;388(8):1659–64. 10.1016/j.physa.2008.12.038Suche in Google Scholar

[11] Alvarez-Ramirez J, Alvarez J, Rodriguez E, Fernandez-Anaya G. Time-varying Hurst exponent for US stock markets. Phys A Stat Mech Appl. 2008;387(24):6159–69. 10.1016/j.physa.2008.06.056Suche in Google Scholar

[12] Willinger W, Taqqu MS, Teverovsky V. Stock market prices and long-range dependence. Finance Stochast. 1999;3(1):1–13. 10.1007/s007800050049Suche in Google Scholar

[13] Hu Y, Øksendal B. Fractional white noise calculus and applications to finance. Infin Dimens Anal Quantum Probab Relat Top. 2003;6(1):1–32. 10.1142/S0219025703001110Suche in Google Scholar

[14] Biagini F, Hu Y, Øksendal B, Zhang T. Stochastic calculus for fractional Brownian motion and applications. 1st ed. Probability and its applications. London: Springer-Verlag; 2008. 10.1007/978-1-84628-797-8Suche in Google Scholar

[15] Nualart D. Fractional Brownian motion: stochastic calculus and applications. Proceedings of the International Congress of Mathematicians, European Mathematical Society; 2006. p. 1541–62. 10.4171/022-3/74Suche in Google Scholar

[16] Cheridito P. Mixed fractional Brownian motion. Bernoulli. 2001;7(6):913–34. 10.2307/3318626Suche in Google Scholar

[17] Murwaningtyas EC, Kartiko SH, Suryawan HP. European option pricing by using a mixed fractional Brownian motion. In: Journal of Physics: Conference Series. vol. 1097. IOP Publishing; 2018. p. 012081. 10.1088/1742-6596/1097/1/012081Suche in Google Scholar

[18] Farah EM, Amine S, Ahmad S, Nonlaopon K, Allali K. Theoretical and numerical results of a stochastic model describing resistance and non-resistance strains of influenza. Europ Phys J Plus. 2022 Oct;137(10):1169. 10.1140/epjp/s13360-022-03302-5. Suche in Google Scholar PubMed PubMed Central

[19] Farah EM, Hajri Y, Assiri TA, Amine S, Ahmad S, De la Sen M. A stochastic co-infection model for HIV-1 and HIV-2 epidemic incorporating drug resistance and dual saturated incidence rates. Alexandria Eng J. 2023 Dec;84:24–36. 10.1016/j.aej.2023.10.053. Suche in Google Scholar

[20] Alfwzan W, Yao SW, Allehiany FM, Ahmad S, Saifullah S, Inc M. Analysis of fractional non-linear tsunami shallow-water mathematical model with singular and non singular kernels. Results Phys. 2023 Sep;52:106707. 10.1016/j.rinp.2023.106707. Suche in Google Scholar

[21] Bouzgarrou SM, Znaidia S, Noor A, Ahmad S, Eldin SM. Coupled fixed point and hybrid generalized integral transform approach to analyze fractal fractional nonlinear coupled Burgers equation. Fract Fract. 2023 Jul;7(7):551. 10.3390/fractalfract7070551. Suche in Google Scholar

[22] Ahmad S, Haque S, Khan KA, Mlaiki N. The evolution of COVID-19 transmission with superspreaders class under classical and caputo piecewise operators: real data perspective from India, France, and Italy. Fract Fract. 2023 Jun;7(7):501. 10.3390/fractalfract7070501. Suche in Google Scholar

[23] Necula C. Option pricing in a fractional Brownian motion environment. Bucharest University of Economics, Center for Advanced Research in Finance and Banking—CARFIB; 2008.Suche in Google Scholar

[24] Xiao WL, Zhang WG, Zhang X, Zhang X. Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm. Phys A Stat Mech Appl. 2012;391(24):6418–31. 10.1016/j.physa.2012.07.041Suche in Google Scholar

[25] Necula C. Pricing European and barrier options in the fractional Black–Scholes market. Bucharest University of Economics, Center for Advanced Research in Finance and Banking—CARFIB; 2008.10.2139/ssrn.1289422Suche in Google Scholar

Received: 2023-07-21

Revised: 2024-02-07

Accepted: 2024-03-14

Published Online: 2024-04-18

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/nleng-2024-0003

Schlagwörter für diesen Artikel

credit structural model; mixed fractional Brownian motion; barrier option

Creative Commons

BY 4.0