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

: 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 in ﬂ uenced by mixed fractional Brownian motion. Three distinct scenarios are analyzed, including when the assets are trade-able, ﬁ xed, 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.


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 misvalue 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: where μ is the drift coefficient, σ is the volatility, and ( ) B t P is the BM under the probability measure .
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][7][8], which are considered improvements over the existing models [9][10][11][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][14][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][19][20][21][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: where ( ) B t ˆH is an FBM and ( ) B t ˆis a BM with respect to a probability measure ˆH.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 quasidebt ratio.

MFBM preliminaries and some important results
Andrey Nikolaevich Kolmogorov (1903Kolmogorov ( -1987)) 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].

MFBM
An MFBM is a stochastic process ( ( )) ≥ M t H t 0 defined as follows: Cheridito et al. [16] proved that an MFBM is equivalent to a BM for . 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].

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  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.
and ∈ ξ η , , we have the formula We can easily obtain the next corollary by setting Consider the process and ∈ ξ η , , we have the

H H H H
The aforementioned theorem gives a relationship between a quasi-conditional expectation with respect to H and a quasi-conditional expectation [ | ( )] ͠ ⋅ t * H with respect to 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 is given by where r denotes risk-less interest rate.

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 V E 0 0 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 The quasi-debt ratio is defined as follows: rT and the credit spread is defined as follows: where k D satisfies 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.

Debt pricing for company with tradeable assets
With the assumption of a liquid market, let ( ) V t , for [ ] ∈ t T 0, , 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: 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., ( ) ( ( ) and 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 where B H , is an FBM, B is a standard BM with respect to probability measure H , and constants μ and σ denote the average and volatility, respectively, of the value ( ) V t of company.Then at any time [ ] ∈ t T 0, , the debt that the company need to repay is, , , , , 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 ˆH equivalent to real-world probability measure H such that ( ) Then the debt ( ) 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 expres- sion for debt ( ) D t (Figure 1).
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: and quasi-debt ratio rT By using ( ) D 0 from Eq. (3.4), we infer that Finally, by using expression for d 1 from Theorem 3.1, we obtain rT Similarly, we can compute, ( ) We have T 0 Proof.Case: 1 For < l 1, we have Similarly, Consequently, Now using L'Hopital's rule The aforementioned result is zero because the exponential and converge to ∞ and −∞ as → T 0, where As a result, Finally, and T 0 1 2 Now using l'Hopital's rule So, with this, we have Similarly, Consequently, To conclude (Figure 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 ( ) αV 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 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 where and N is a distribution function of the standard normal random variable.
Proof.We will begin by using Feynman-Kac formula (Theorem 2.4).Credit risk derivatives driven by MFBM  7

r T t H r T t α V T F V T F H α r T t V T F H r T t V T F H
Since  Then by using By using above change of variables, we obtain .
So, we have  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, which represents the value of equity with fixed assets.Finally, for debt payoff, we have where d 1 and d 2 are same as earlier (Figure 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:

B
To determine the debt amount to be paid, we define two random variables:  Credit risk derivatives driven by MFBM  9

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 and ( ) D T increases.Similarly, ( ) D 0 increases as → H 1 and ( ) D T increases.
r T t , which is the com- pounded amount of the Barrier.