An Approach of Price Process, Risk Measures and European Option Pricing Taking into Account the Rating

In this paper, by taking into account the rating in a new concept of economic space, we propose a model of the dynamics of an economic particle, a model of price process, an extension of risk measures, and a new approach of option pricing with associated hedging portfolio.


Introduction
Risk is inherent to all human activities. In all times, the matter is how to avoid it or how to reduce its impact. In Finance, risk can be classified in two groups: quantifiable risks and unquantifiable risks. Main quantifiable risks are market risk and credit risk. For unquantifiable risks, we can mention, among others, operational risk and legal risk. In this paper, we focus on quantifiable risks. For these risks, many approaches of modeling exist in literature. For these risks, many approaches of modeling exist in literature. Market risks models can be classified in two main groups: dispersion risk measures and capital requirement risk measures. For the first group, main contributors are Markowitz in 1952 [1], Sharpe in 1963 [2], Konno and Yamazaki in 1991 [3] and Hamza and Janssen in 1995 [4]. Capital requirement risk measures have been studied amongst others by Artzner et al. in 1999 [5], Föllmer and Shied in 2002 [6], Rockafellar et al. in 2002 [7] and Detlefsen and Scandalo in 2006 [8]. In the field of credit risk models, there are two main approaches. Structural models in which the default is an endogenous process linked to the value of firm have been studied amongst others by Merton in 1974 [9], Black and Cox in 1976 [10], Longstaff and Schwartz (1995) [11] and Hsu et al. (2004) [12]. Reduced form models where the default can occur at any moment as it is defined as the first jump of some stochastic process have been studied amongst others by Jarrow et al. (1992) [13], Jarrow et al. (1995) [14] and Duffie et al. in 1999 [15]. After the step of modeling, the problem is to control or to mitigate the impact of risk using appropriate financial instruments. The interested reader can consult [16] to have more information on these instruments. In this paper, we focus on market risk mitigation. Let's mention that portfolio selection problem constitutes an important aspect of market risk mitigation and is intensively studied. Recent developments in this field have been made by Bo et al. (2015) [17] and Lin et al. (2017) [18]. To the best of our knowledge, there is up to now no work on market risk models and option pricing problem which takes explicitly into account the rating of the underlying's issuer. For instance, if the same asset is issued by two economic agents with different ratings, it is not fair to consider that these assets must have the same behavior, because the way each economic agent manages its firm should have significant impact on the evolution of its asset price's process. The rating of an economic agent acts as a measure of its health. It expresses the future capacity to respect its engagements. It was firstly used in 1860 by Henry Varnum Poor in USA to measure the quality of debts issued by railways companies. First agencies of rating appear in 1909 for Moody's and 1910 for Standard Statistics and Poor's. In 1941, Standard Statistic merges with Poor's to form Standard and Poor's. For more information about rating, we refer to the paper of Lawrence [19]. Ratings are available for banks, states, municipalities, corporations, investment funds, pension funds and insurance companies. These ratings act as decision aid for investors who have excess funds needed by these entities. A rating is a string of letters, for example those given by Standard and Poor's have the forms AAA, AA, A and BBB for investment grades; BB, B and CCC which are speculative grades for entities with low capacity of development in the future; CC and C which are ratings for entities near to default and D for entities in default. The question is how to incorporate the rating explicitly in the price's process? This notion is not new in literature. For instance, in the model of credit risk (Credit-Metrics) built by Bhatia et al. (1997) [20] for the bank J.P Morgan, the authors took into account the rating of the issuer of bonds in the valuation of its forward price. Hackbarth et al. (2004) [21] proposed a model of credit risk where the default threshold is rating dependent. However, in these approaches, the matrix of transition giving the probability of moving from one rating to another is calibrated most often for one year. This means that the possibility of instantaneous change of rating is excluded. Another drawback is that they are used only for debts instruments. Shapiro (2015) [22] and Grzegorz (2016) [23] used the notion of risk profile and the quality of credit, but not in the direction of rating approach. Victor Olkhov (2016) [24] [25] proposed a model of asset price's process incorporating effectively the issuer's rating. He supposed that rating methodolo-gies can be extended in order to take values in  . He also supposed that if an economic participant is subject to n risks, its rating should be an n -dimensional vector. He so defined the notion of economic agent and economic space with dimension the number of main risks present in the market. In his paper, he assumed that the evolution of an economic agent or economic particle in economic space is modeled by a Brownian motion. In this paper, we also assume as Victor Olkhov that the rating methodologies can be extended in such a way that they take their value in  . Once this is done, we suppose that • Each financial variable issued by an economic agent possesses a rating which is the rating of the issuer in the corresponding market. An economic agent is an entity (individual, firm, local governments or state) which has something to exchange in the market. Financial variables are bonds, obligations, goods and services, loans, debts or even work. For example a firm producing two types of goods 1 G and 2 G and issuing an obligation B possesses three rating. For a given good, rating is related on the degree of fulfilment of standard required for the class of product this good belongs to. For debt instruments, rating is related to the degree of creditworthiness of the issuer.
• Rating assumes values in ( ) { } 0, ∞ ∞  with 0 as the best one, acknowledged in the present rating setting as AAA.
• The axis ( ) 0, ∞ is oriented in the sense of poorest quality of the rating. • If an economic agent is absent in a market of a specific financial variable, its rating for this variable is ∞ .
Since the number of different financial variables in the market is infinite, the dimension of economic space is thus infinite and for a given economic agent, its coordinates possess just a finite number of rating different from ∞ . In the new setting adopted, our first contribution in this paper is the proposition of a new approach of economic particle in economic space combined with a process modeling its dynamics. Our second contribution is to propose a new model of price process taking into account the rating. In our third contribution, we propose new measures of risk which are extensions of capital requirement risk measures, deviation risk measures, quantile-based risk measures and utility-based-risk measures incorporating the rating. The fourth contribution is the derivation of a PDE associated to European call option pricing based on a single asset that incorporates the rating. Using the Feynman-Kac formula, we obtain a closed form solution and we also deduce the hedging portfolio. Our approach of option pricing differs from which proposed by Black and Scholes (1973) in their seminal work [26] by the fact that the payoff is not a function of the underlying's price only, but also of the underlying's rating.
The remainder of the paper is organized as follows. In section 2, we present the notion of economic space and dynamics of economic particle. In Section 3, given an asset, we propose a model of price process that incorporates its rating.
In Section 4, by taking into account the notion of rating, we propose extensions of capital requirement risk measures, deviation risk measures, quantile-based  [24] after the assumption that rating methodologies can be extended in such a way that rating can assume values in  , defined for the first time an economic space whose dimension is the number of main risks assessed in the market. The author claimed that an economic agent also called economic particle moves in economic space by following a drifted Brownian motion. In this paper, we also assume as Victor Olkhov [24] that the rating methodologies can be extended in such a way that they take their values in  . Furthermore, we claim that each financial variable issued by an economic agent possesses a rating which is the rating of the issuer in the corresponding market. Financial variables are bonds, obligations, goods and services, loans, debts or even work. For example a firm producing two types of goods 1 G and 2 G and issuing an obligation B possesses three rating.
For a given good, rating is related on the degree of fulfilment of standard required for the class of product this good belongs to. For debt instruments, rating is related to the degree of creditworthiness of the issuer. For contracts and derivatives, the rating is that of the underlying's issuer in the corresponding underlying's market. Thus the number of ratings (coordinates) for an economic particle depends on the number of financial variables issued. Since the number of different financial variables existing in the market is infinite, the dimension of economic space is infinite. Thus every economic particle moves just in the subspace of the entire economic space. The dimension of this subspace is equal to the number of financial variables issued. We will set the number 0 as the best rating, acknowledged in the present rating setting as AAA. That is in our setting the range of rating is ( ) { } 0, ∞ ∞  and the axis is oriented in the sense of poorest quality of the given financial variable. If it is debt instrument, great ratings mean bad quality of issuer's creditworthiness. But if it is a good or a service, great rating traduce the weakness of the degree of fulfilment to standard required for the class of products to which belongs this good or service. If an economic agent does'nt issue a given financial variable, then its rating for this financial variable is ∞ . Otherwise, it is a positive real number.

The Dynamics of Economic Particle in the Economic Space
In this paper, we assume that economic particles move randomly in the economic space. That is, for a given economic particle, its ratings change randomly.
If an economic particle is present in n ( n ∈  ) different markets, its rating at time t is a vector ( ) ( ) 1 2 , , , , , , , where for each i ( 1, , the number of financial variables issued in a specific market.
For example, a given firm can issue more than one type of debt instruments.
From now on, we write We assume that these regulations and standards can be extended to all financial variables. The set of regulations or standards required at time t for a given financial variable is ( ) e R t . These regulations and standards are exogenous to economic agents, that is regulators do not have something to exchange in the market and do not intervene directly in market processes. For a given financial variable we suppose that amongst the objectives of its issuer (economic particle), there is the improvement of its rating in the corresponding market.
We make the following assumptions about the variation t t t t X X X +∆ ∆ = − of rating between t and t t + ∆ 1) There is a depreciation of the rating due, amongst others, to the evolution of regulations and standards and to the degradation of production's factors.
Since regulations and standards are exogenous, this depreciation is measured by the quantity ( ) 2) The internal effort to ameliorate the rating is proportional to the current one. It is measured by the term 3) The movement of the issuer in economic space is subject to randomness measured by the term Brownian motion and f is a function on 2 +  .
Considering the above assumptions, we have , F t x is the total value issued by economic particles with rating x at time t.
2) In the case of portfolio P containing n ( 3) For financial instruments such as forwards, futures and options, the rating is that of the underlying issuer.

Motivations
In the pioneer work of Louis Bachelier in 1900 [27], he proposed to model price process of a given asset as a Brownian motion, but his model has a drawback because price can assume negative values. Samuelson [28] proposed a model where the logarithm of price is a Brownian motion and by this way price can only assume positive values, which is more realistic. It has been shown by Mandelbrot in 1966 [29] that the hypothesis of log-normality of price does not fit with experimental data. Other classes of model have been proposed like the Constant Elasticity of Variance Model and models incorporating jumps [30]. Due to the tractability, in the majority of problems in finance, the log-normal approach is used as a first approximation of price process dynamics. The common feature of all these models is that the rating of the asset's issuer is not taking into account. This is not too realistic because for a given financial variable, the specific risk of its issuer is not taken into account in the price's process. This omission can create important damage in the fu- ture. An investor who enters in a contract with one of the issuers can suffer a lost due to its rating deterioration. Victor Olkhov [24] proposed a model of asset's price process that incorporates the rating of the issuer. In this paper, we propose a model in which the negative impact of bad (high) rating on the asset price is taken into account.

The Model
We are given a triplet is a filtration and  is the historical probability. For a given financial variable, with price t S at time t, Black and Scholes (1973) [26] assumed that t S follows a log-normal process. If the market is driven by one source of randomness modeled by a standard Brow- where k is positive real number and t X is the rating of the issuer. From rela-

Graphical Illustration
In Figure 1, we illustrate the evolutions of rating and those of the same asset where one is adjusted to rating and the other is not. We can observe that the evolutions can be truly different. At every time t, the ratio of these prices is proportional to e t kX . This emphasizes the fact that neglecting the rating in the process of price can create a disagreement.
We have the following important lemma To prove Lemma 3.1, we need the following Lemma known as reflection principle of the Brownian motion.
be a standard Brownian motion and w a positive real number. Then We now give the proof of Lemma 3.1.
Proof. From relation (4) From squeeze Theorem, we obtain the result.  Remark 3.1. From Lemma 3.1 we deduce that, if the issuer's rating of a financial variable becomes too large or deteriorates considerably, this financial variable is worthless. No economic agent is willing to buy it.
Up to now in risk measures treatment, to the best of our knowledge, there exists no research work that takes explicitly into account the rating of the issuer of a financial variable. This omission of the specific uncertainty coming from the issuer can entails undervaluation of the actual degree of riskiness associated to a financial instrument and consequently can explain the fact that all risk measures proposed up to now turn out to be insufficient to protect the investors in period of financial crisis against excessive losses. The approach that we propose in the next section, incorporate the rating of the issuer in the valuation of the degree of riskiness. The properties of known risk measures have been preserved in our approach. This can allow to have another approach of portfolio allocation problem.

An Extension of Risk Measures
In this section we propose an extension of risk measure in static case. Trading is C. Tadmon adjusted-to-rating) future wealth or future payoff of a financial instrument. For a given financial variable issued by an economic particle with initial rating 0 X in the corresponding market. We denote by ( ) Y X its adjusted-to-rating future payoff; where X is its random future rating in the corresponding market. We also denote by Y the future payoff of the same financial variable if it was issued by an economic particle with rating 0. If t S is the price at time t of this financial variable, from relation (4) we have S is the price of the same financial variable if it was issued by an economic particle with rating 0. We have the relation where 0 R and R are respectively the total return of investment on the financial variable issued by economic particle with rating 0 and initial rating 0 X . By taking the logarithm in relation (8), we have The relation (9) [31].
In this way, the adjusted-to-rating future payoff is less than Y if 0 X X > and greater than Y if 0 X X < . In the extreme case where X is too large, the future payoff can be equal to zero an even less than zero.
Definition 4.1. The adjusted-to-rating future payoff of a financial variable issued by an economic particle with initial rating 0 X in the corresponding market, is given by the relation where 1 , , p χ  and 2  are defined as above.
This can be interpreted by the fact that in this case, there is no uncertainty coming from the rating of the issuer. We recover by this way the case where the rating is not taken into account. For the investor, the difference is just on the amount invested at the beginning of trade.

Scenario-Based Risk Measures
Before giving an extension of risk measure in scenario-based risk measures, let's recall the definition of coherent risk measure given by Artzner et al. [5].
Here ( ) L ∞ Ω is the set of essentially bounded real value random variables defined on Ω . Given a financial variable issued by an economic particle with initial rating 0 X , we defined the associated adjusted-to-rating future wealth ( ) Y X by the relation (10). In the case of capital requirement risk measures or scenario-based risk measures, a good candidate for is a premium to be put aside or invested in a safe way together with the given financial variable in order to rule out possible loss coming from the risk bearing by the rating of the issuer. At the end, if there is an improvement in the rating of the issuer, this amount will be an additional gain, otherwise it will be a buffer reducing potential losses. Given a scenario-based risk measure ρ and a financial variable with adjusted-to-rating future wealth ( ) Y X , we defined the level of riskiness of ( ) Y X by applying ρ to (10).
This means that bad ratings have positive impact on the level of riskiness. 2) This means that good ratings have negative impact on the level of riskiness. 3) Definition 4.2. For a risk measure ρ , the acceptance set associated is defined by 2) The investment manager who gives his portfolio to a trader.
3) The exchange's clearing firm that has to secure transactions between all parties. Theorem 4.1. Given a risk measure and its acceptance set  , let ( ) Y X be the adjusted-to-rating future wealth associated to a financial variable issued by an economic particle with initial rating 0 X and Y the future wealth if it is issued by an economic particle with rating 0. Let's suppose that Y ∈  .
Proof. Since risk measure as defined in [5] is static, the premium Using the monotonicity of ρ , we conclude that 2) If 0 X X > almost surely, then from the definitions of p and ( )  [36] proved that under mild continuity assumptions, they can be represented as worst expected loss with respect to a given set of probability models. In our setting, we have the following representation result.  is the expectation under Q. This characterization generalizes the earliest one given by Artzner et al. [5] in the case of finite sample spaces.
Proof. Replacing X by ( ) Y X in [36], Theorem 2, we obtain the expected result.
The law invariance property is also preserved because the motion of an economic particle in economic space follows an Ornstein-Ulhenbeck process with specific characteristics known at every time.
The definition we gave above is for static risk measures. It doesn't take into account additional information that can be granted during the period of trading.
Evaluation is done only one time; it's the reason why they are also called single period risk assessment. Cvitanić and Karatzas [37] proposed an approach of multi-period risk assessment that follows the first one proposed by Hakansson [38] as an improvement of Harry Markowitz's model. Other approaches were provided by several authors such as Detlefsen and Scandalo [8], Föllmer and Penner [39] and Bion-Nadal Jocelyne [40]. These authors expressed risk as capital requirement. Detlefsen and Scandalo [8] proposed a suitable way to assess periodically the riskiness of a given financial variable incorporating available ad- For any X and Y in ( ) We obtain the conditional measure of risk adjusted-to-rating by applying ρ to ( ) Y X . In our setting, the representation provided by Detlefsen and Scandalo [8] is expressed as follows.
| is absolutly continuous with respect to and in Q Q P Q P = =    , α is a (random) penalty function.
Proof. Replacing X by ( ) Y X in the proof given by Detlefsen and Scandalo [8], Theorem 1, the expected result is obtained. 

Deviation Risk Measures
In If ρ is a given deviation risk measure, the idea here is the same as before, to apply ρ to adjusted-to-rating future return.

Quantile-Based Risk Measures
In this class of risk measures, we are concerned with the distribution of future losses. Given a financial variable, issued by an economic particle with initial rating 0 X , its adjusted-to-rating future loss ( ) Y X is given by the relation and conditional value at risk (CVaR). In the existing literature, these risk measures don't take explicitly into account the rating of the issuer. Our idea here is also to apply each of these risk measures to adjusted-to-rating future loss. This allows to incorporate the rating of the issuer in the assessment of the degree of riskiness of the issued financial instrument.
The example of VaR stands as follows. Given a financial instrument issued by an economic agent with initial rating 0 X , by applying the VaR on the adjusted-to-rating future loss ( ) Y X , we obtain the relation is the cumulative distribution function of ( ) Y X .

Utility-Based Risk Measures
In this category of risk measures, used mainly in the insurance industry, p has the same interpretation as in scenario-based risk measures. Namely p is a premium to be put aside or invested in a safe way together with the given financial instrument in order to rule out possible loss coming from the adverse change in the rating of the issuer. At the end of period, if there is an improvement in the rating of the issuer, this amount will be an additional gain. Otherwise it will be a buffer reducing potential losses. The idea here is also to apply to adjusted-to-rating future wealth the utility function of the regulator.
For example, given a financial variable issued by an economic agent with initial rating 0 X , its adjusted-to-rating future payoff ( )

European Option Pricing in Economic Space
In Section 2, we introduced the notion of economic space and economic particle.
In Section 3, we proposed a model of price process taken into account the rating of the underlying issuer. In what follows, we propose a new approach of European option pricing in economic space. An option is a contract in which the writer of the option grants to the buyer of the option the right, but not the obligation, to purchase from or sell to the writer something at a specified price within a specified period of time (or at a specified date). The writer, also referred to as the seller, grants the right to the buyer in exchange for a certain sum of money which is called the option price or option premium. The price at which the asset may be bought or sold is called the exercise price or strike price. The date after which an option is void is called the expiration date. When an option grants the buyer the right to purchase the designed instrument from the seller, it is referred to as a call option, or call. When the option buyer has the right to sell the designed instrument to the writer, the option is called a put option, or put.
Buying calls or selling puts allows the investor to gain if the price of the underlying asset rises. Selling calls or buying puts allows the investor to gain if the price of the underlying asset falls. An option is also categorized according to when the option buyer may exercise the option. The option that may be exercised at any time before (including) the expiration date, is referred to as American option whereas the option that may be exercised only at the expiration date is called European option. The formula proposed by Black and Scholes [26] as the price or premium of an European call (or put) didn't take into account the rating of the issuer of the underlying asset. To the traditional characteristics of an option, we add the exercised rating κ that determines the desired quality of the underlying at expiration date. In the case of exercise of the option, the seller must buy or sell to the buyer the underlying financial variable having κ for the specified price K. Hence, for a given financial variable issued by a given economic agent, the option contract should specify the strike, the period or expiration date and the exercise rating. At the maturity, there are four possibilities: • The quality of the underlying (rating of the issuer) can deteriorate in such a way that even its price is below the strike. • The quality can be improved, but the underlying's price is less than the strike.
• The quality can be deteriorated but the underlying's price is greater than the strike.
• The quality can be improved and the underlying's price is greater than the strike.  [9]; namely the first structural model of credit risk. But this work didn't take into account the rating of the issuer of the underlying asset. To derive the Black Scholes and Merton's PDE for an European option in economic space, we make the following assumptions 1) Trading is made on a fixed period of time T.

An Extension of the Black and Scholes' Equation
2) There is no restriction on selling and buying stocks.
3) There is no dividends, frictions and transactions costs.

4)
There is no arbitrage opportunity.

5)
There exists a risk free savings account (in a bank rated AAA) with constant interest rate r.
6) The option is written on a single financial variable.
Let t S and t X be the price and rating of the underlying respectively at time t. In the original Black Scholes model a risk free self financing portfolio is constructed by using the underlying asset and a derivative which is used to hedge the underlying asset. In the present case, due to the existence of additional risk, namely the risk coming from the deterioration of the rating of the underlying's issuer, to hedge this additional risk, we consider another derivative written on the rating of the underlying's issuer. Hence we construct a portfolio containing at any time t one option ( ) , , underlying's issuer rating. The portfolio's value at time t is given by the relation Under the self financing assumption, the change in the portfolio is expressed as follows We assume further that the functions t V and t U are 1  in time and at least 2  for other variables. If we assume that the variance of the random effect on the variation of underlying's rating is proportional to the current rating, then the function f in relation (5) (5) and (6), we obtain the following relations Substituting (13) and (14) into (12), the change in the portfolio becomes In order for the portfolio to be hedged against movement in financial variable's price and rating of the underlying's issuer, the last two terms of (15) must be zero. This implies that where we suppose that 0 The condition that the portfolio is risk free implies that the change in the portfolio is equal to the change in the risk free saving account; that is Now, equating (15) with (17), and using (16), we obtain the following relation is the price of rating risk. This allows us to obtain the equations to be written as ( ) and ( ) In the case of European call option with strike K, maturity T and exercise rating κ , the goal of the buyer is to buy at time T the underlying asset with quality (rating) κ at a price of K. For an European put option with same characteristics, the goal of the buyer is to sell at time T the underlying with quality κ at a price of K.
, , max e , 0 , , For European put option with same characteristics, the problems to be solved are obtained by replacing the payoff in (22)

Closed-Form Solution of European Call Option
In the sequel, we assume that ( ) The following Theorem gives a solution to the problem (22).
is a solution of the problem (22) 2 0 We consider its associated canonical diffusion Proof. We split problem (22) into problems (25) and (26).
These problems can be written as  Since 1 V and 2 V represent prices, we can reasonably assume that they have We now have ( ) . The proof is completed.  The following Theorem gives the behavior of V when the rating x tends to ∞ .
where * y is the price at time t of the underlying having rating κ .
At time t, the value * y is independent of x. We also have ( )

Greeks and Hedging
We now talk about the Greeks, which are measures of sensitivity of the option price with respect to its variables. They are another way to measure risk associated to an investment. In fact risk associated to a specific variable is canceled if the partial derivative of the price with respect to this variable vanishes for all t.
To obtain Greeks, we need to solve problem (23). We will investigate a solution of the form ( ) ( ) , , , . U t x y yW t x = (38) Using the relation (38), the problem (23) We have the following result.