A Mellin Transform Approach to the Pricing of Options with Default Risk.

The stochastic elasticity of variance model introduced by Kim et al. (Appl Stoch Models Bus Ind 30(6):753-765, 2014) is a useful model for forecasting extraordinary volatility behavior which would take place in a financial crisis and high volatility of a market could be linked to default risk of option contracts. So, it is natural to study the pricing of options with default risk under the stochastic elasticity of variance. Based on a framework with two separate scales that could minimize the number of necessary parameters for calibration but reflect the essential characteristics of the underlying asset and the firm value of the option writer, we obtain a closed form approximation formula for the option price via double Mellin transform with singular perturbation. Our formula is explicitly expressed as the Black-Scholes formula plus correction terms. The correction terms are given by the simple derivatives of the Black-Scholes solution so that the model calibration can be done very fast and effectively.

Introduction

In general, derivatives traded on stock exchanges are considered to be no default risk. However, several derivatives such as currency options, options on precious metals, interest rate swaps, or credit default swaps, still have been traded in over-the-counter(OTC) market where there is no the exchange or clearing corporation. Therefore, many buyers of the related derivatives are vulnerable to the default risk. In other words, the option holder is always apt to have counterparty credit risks since the option writer of the counterparty may not fulfil the appointed contracts at the maturity. So, we need to take account of the default risk of the contingent claim for pricing the options on the defaultable instruments in such OTC market. the option is called a ‘vulnerable option’ when the option buyer is exposed to the option writer’s default risk which is an example of a counterparty risk. Financial crises such as the Global Financial Crisis, Eurozone crisis and COVID-19 pandemic have brought the issue of counterparty risk to academic researchers and practitioners. We also note that counterparty risks are an important risk factor recognized by Basel III. Especially, the recent economic crisis caused by the COVID-19 pandemic has increased the importance of the default risk on the derivatives. According to BIS report,1 the gross market value of OTC derivatives rose from $11.6 trillion to $15.5 trillion during the first half of 2020. Correspondingly, gross credit exposures reached $3.2 trillion at end-June 2020, which is the largest increase since 2009. The largest growth indicates that many holders of the option contracts should be seriously aware of the default possibilities.

The pricing of European vulnerable options has been already proposed by Johnson and Stulz (1987). Since then, several works have been reported by researchers on the pricing of vulnerable options. For example, Klein (1996) derived a formula for European vulnerable option when a proportion of nominal claim is paid out in default depending on the amount of other liabilities as well as the assets of the counterparty. In that study, the value of the asset of the counterparty is assumed to follow a geometric Brownian motion. Hung and Liu (2005) obtained a vulnerable option pricing formula under an incomplete market condition. The advantage of the formula is that it can evaluate nontraded assets underlying vulnerable options. Yang et al. (2014) assumed that volatility of the asset value follows a fast mean-reverting Ornstein-Ulenbeck (OU) process and derived an analytic formula for European vulnerable option. Yoon and Kim (2015) derived an explicit closed form formula of European vulnerable option with stochastic interest rates following the Hull-White interest rate process.

In this study, we evaluate the price of vulnerable options based on the stochastic of elasticity of variance (SEV) model introduced by Kim et al. (2014). The SEV model forecasts market’s volatility more accurately than the constant elasticity of variance (CEV) model so that investors can employ a dynamic investment strategy reducing the risk more effectively. In particular, it is a useful model for extraordinary volatility behavior which would take place in a financial crisis. Refer to Kim et al. (2015). The model consists of two separate scales in such a way that there are the following important merits. The first one is that the number of necessary parameters for calibration is minimized while the model well reflects the essential characteristics of the underlying asset and the firm value of the option writer. The second one is that singular perturbation technique can be implemented to extract essential information for the properties of the option price. The SEV model has been used to the pricing of other types of derivatives. For example, Yoon and Park (2016) and Yoon et al. (2013) evaluated turbo warrant options and perpetual American options, respectively. It could be hybridized with a pure stochastic volatility model such as the Heston model (1993) as done by Choi et al. (2016).

A Mellin transform is an integral transform that can be thought of as the multiplicative version of the two-sided Laplace transform. The Mellin transform can be a very useful tool for the transformation of partial differential equations (PDEs) as described in Brychkov (1992). In general, the Mellin transform technique, if it is plausible for the pricing of a given option, would not require the complexity of the calculation as appeared in the probabilistic approaches. The Mellin transform technique has been used for pricing financial derivatives in Panini and Srivastav (2004), Panini and Srivastav (2005), Brychkov (1992), Frontczak and Schöbel (2010), Frontczak (2013), Jeon et al. (2016), Jeon et al. (2017) and Guardasoni et al. (2020). In this study, we first employ singular perturbation technique to obtain PDEs for the vulnerable option price and then exploit the double Mellin transform introduced by Krapivsky and Ben-Naim (1994) to derive an analytic solution for each PDE. The double Mellin transform provides an explicit closed form formula.

Model calibration in financial engineering is as important as the model itself. Calibration is equivalent to identifying the parameter values so that the model can reproduce market prices as accurately as possible. The speed as well as the accuracy of calibration is crucial because practitioners use the calibrated parameters to price complicated derivatives and to work out trading strategies. A main contribution of this paper is that our option price formula is given in a relatively simple closed form explicitly. It is the Black–Scholes price plus correction terms that can be calculated by simply taking derivatives of the Black–Scholes formula. In this sense, our pricing formula is better than the well-known formula under the Heston model introduced by Heston (1993), one of the most popular stochastic volatility models for pricing derivatives, which is given analytically but requires a numerical method to compute the complex integral. Actually, the semi-analytical pricing formula for the European vanilla options by Heston model obtained by the use of Fouier transform techniques comes up with the instability and a great deal of computing time in obtaining the solutions, including the problem of the model calibrations. So, the existence of the closed solution for the option pricing is very important because it directly affects the computing time in implementing the option’s data fitting (or the calibration of option’s model) and the numerical stability related to the pricing accuracy. This paper shows that the SEV model with the Mellin transform leads to a better analytical solution for pricing derivatives than the Heston model with the Fourier transform in terms of the stability (or the accuracy) and the computing time.

The structure of the paper is organized as follows. Section 2 gives a brief review of the SEV model. In Sect. 3, we apply multiscale asymptotic analysis to constructing an explicit approximation of the European vulnerable option price. The explicit analytic solution is given by the Melling transform technique in Sect. 4. Section 5 is committed to investigating the sensitivity of the vulnerable option value with regard to the model parameters and verifying the accuracy and efficiency of the approximate formula. Section 6 provides concluding remarks.

Model Formulation

We first set out to write the SEV model given by Kim et al. (2014). In a market measure, the underlying asset price satisfieswhere and m are constants and and are small positive parameters and . The elasticity parameter is fluctuated by financial circumstance and the magnitude of its fluctuation is shown to be small as in Kim et al. (2015).

Next, let be the market value (firm value) of the assets of the option writer. Its dynamics is given bywhere and are constants and and . We note here the comment from Hoffmann (2018) that the value of firms should not be based on just shareholders but should include all groups of stakeholders.

By using the changes of Brownian motionswe can transform the market measure to an equivalent martingale measure under which the dynamics of the processes and becomeswhere r is a risk-free interest rate and is the market price of risk. The correlation structure of the Brownian motions and is given byWe need to give a restriction on f in order to guarantee the regularity condition. It is assumed that f is bounded and is small () and positive enough to satisfy the inequality apart from the absorbing boundary condition at . Refer to Delbaen and Shirakawa (2002).

In this study, we consider a European vulnerable call option with its payoff given byHere, K is the strike price of the option. , and are related with the credit loss event in the vulnerable option. is the critical value such that default occurs if the value of the option writer’s assets, , is less than . In practice, this term is stochastic rather than constant but it is assumed to be a constant in this paper. D is the value of total liabilities. can be less than D because of an additional liability owing to the possibility of a counterparty keeping operation. represents the deadweight costs related with bankruptcy expressed as a percentage of the value of the assets of the option writer. If is larger than or equal to the default boundary , no credit loss occurs and the payoff becomes full payment . If not, the proportion of the nominal claim is paid out.

Under the risk neutral measure , the price of a European vulnerable option is determined by the formulawhere r is the risk free rate of interest over [t, T].

Using the Feyman–Kac formula (cf. Oksendal 2013) , solves the partial differential equation (PDE)where .

We note that the PDE problem (2.8) has no analytic solution for general f. However, for small parameters and , one can proceed with asymptotic analysis to obtain an explicit approximate solution, which will be carried out in the next section.

Multiscale Analysis

Now, we apply multiscale asymptotic analysis developed by Fouque et al. (2011) to derive an approximate formula for satisfying (2.8). We denote by the averaging with respect to the invariant distribution of the process .

First, we define the differential operator asAs the operators and depend on , we use Taylor expansion with respect to to express the operators and in powers of as follows:Notice that is exactly the operator corresponding to the Black–Scholes model with constant volatility for vulnerable European options. We denote it by from now on.

We are interested in the solution of the PDE (2.8) of the form Substituting this into (2.8) yields the following hierarchy of PDEs:Then we plug the expansionsinto (3.13) and obtainassuming that and satisfy the order condition .

We obtain the following Propositions 3.1 and 3.2.

Proposition 3.1

Let and for . Assume that the partial derivative of with respect to y does not grow as much as for when y goes to infinity. Then for and it is independent of y and satisfies the PDEwhose solution is given by

Proof

Plugging the expansion into the PDE (3.13), we obtain the recursive relationwhere is defined for consistency.

If , we have the ordinary differential equation (ODE) . By solving this ODE, one can findfor some functions and independent of y. From the assumed growth condition, must disappear. So, is independent of the y variable, that is, and thus . Then (3.18) for yields the ODE , meaning that is independent of y, that is, and thus . Then (3.18) with leads to the PDESince (3.20) is a Poisson equation for in terms of , one can apply the solvability condition on the source term from the Fredholm alternative. Therefore, we obtain (3.16) and the solution of it is given by (3.17) from Theorem 1 in Yoon and Kim (2015).

Now, from (3.16) and (3.20), the ODE is satisfied so that , which yields the PDE . Then (3.18) for leads to , which is a Poisson equation of y in terms of . Again, the solvability condition leads to the PDE . The solution of this equation with the final condition is . Similarly, for , (3.18) becomes , which leads to . Then, from the final condition , the solution is exactly . Then, by mathematical induction, one can have the result for .

Proposition 3.2

Assume that the partial derivatives of and with respect to y do not grow as much as for as y goes to infinity. Then and are the independents of y and satisfies the PDEand satisfies the PDEwhere , , , , , , , , and is the solution of the ODE and it is described by the indefinite integralwhere is the usual standard normal distribution function.

From the PDE (3.14), we haveFrom the order term of (3.23), we have . From the assumed growth condition, is y-independent. Since all the terms in and contain a partial derivative with respect to y, we have from the order terms of (3.23) and thus is y-independent. This yields . Next, from the order 1 terms of (3.23), we have the PDESince this is a Poisson’s equation in y in terms of , the Fredholm alternative leads toIf we subtract (3.25) from , we obtainBy utilizing the definition of , we have the desired PDE described by (3.21).

From the order terms of (3.23), we have . Noticing that this is a Poisson’s equation with respect to , we obtainMeanwhile, from (3.24) and (3.26), we have in the following formfor some function c(t, x, v) which is independent of y. Then, by substituting (3.28) into (3.27) and using the definition of the operator , we obtain the desired result given by (3.22).

By the same technique as above, satisfies the PDEwhere and .

From Propositions 3.1 and 3.2, and the above result for , the option price is approximated byThe accuracy of this approximation is theoretically given in Sect. 4. The closed form solution of is given by (3.17) and the closed form solutions of , and will be derived in Sect. 4. Mellin transform is a powerful tool to derive the analytic solutions.

Mellin Transform for Option Pricing

To derive analytic solutions for , and , we use the double Mellin transform method in this section. A brief review of it is as follows. For a locally Lebesgue integrable function , the double Mellin transform is defined byand if and and are given such that , then the inverse of the double Mellin transform is expressed asThe leading term is the European vulnerable option price under the Black–Scholes model with constant volatility. The existence of the double Mellin transform of was discussed in Yoon and Kim (2015) and it was explicitly derived by Yoon et al. (2013). Let , , and be the double Mellin transforms of , , and , respectively. Then, from the properties of the double Mellin transform, the PDEs (3.21), (3.22) and (3.29) are transformed into ODEs given byandrespectively.

Next, we solve these ODEs. We first prove the following lemma first.

Lemma 4.1

The solution of the ODEis given bywhere is the double Mellin transform of P(t, x, v) and and are some functions of and .

We seek a solution of (4.9) in the formPlugging this into (4.9) yieldsWe can easily solve these equations whose solutions are given byTherefore, we obtain the solution of given by (4.10).

By making use of Lemma 4.1 and (4.1) and (4.4), we obtainwhere and are given by (4.3), (4.5) and (4.6), respectively. So, we obtain the following results.

Proposition 4.1

The correction terms and are given byrespectively, where and are defined by (3.21) and (3.22), respectively.

From the definition of in (4.2) , we haveApplying the inverse double Mellin transform on in (4.12) and making use of the properties of the inverse of double Mellin transform yieldwhere and are defined as (4.16), (4.17) and (3.21), respectively.

Similarly, to obtain the explicit expression for , we apply the inverse double Mellin transform to (4.13) which giveswhere and are defined as (4.16), (4.18) and (3.22), respectively.

The following lemma gives the solution of (4.7).

Lemma 4.2

The solution of the ODE (4.7), i.e.,is given bywhere

We seek a solution of (4.20) in the formSubstituting this into (4.20) yields the following ODEs for , and .One can easily solve these equations and they are given by (4.22), (4.23) and (4.24), respectively.

Using Lemma 4.2, the inverse double Mellin transform and its properties, we obtainwhere and are given by (3.21) and (3.29), respectively. Then, by using explicit expression for given by (4.14), direct computation givesTo sum up, we have derived the corresponding ODEs from the pricing PDEs through the utilization of Mellin transform and then found the solutions of the ODEs based upon Lemmas  4.1 and 4.2. Finally, the pricing correction terms and are obtained by the inverse double Mellin transform and its properties.

Proposition 4.2

Let be the solution of the PDE problem (2.8) and an approximation given by (3.30). When the payoff function h(x, v) is continuous, for any (t, x, y, z) and for any and , we havewhere notation represents terms that are of order for any .

The proof of this proposition is similar to the proof of the Proposition 2.5 in Fouque et al. (2016) and the proof of Theorem 4.1 in Choi et al. (2016). In particular, Fouque et al. (2016) provides the detailed explanation for the second order approximation of the option price so that readers can understand the error accuracy mentioned in (4.28). In addition, compared to Choi et al. (2016), the only difference of our paper is that the payoff function h depends on an additional process in this paper. Since the payoff function of vulnerable option is bounded by the payoff function of European vanilla option, that is, , the proof of accuracy does not deviate from that of Choi et al. (2016). Therefore, combining the above two references, the verification for the error estimation in (4.28) is completed.

We present a numerical evidence of the accuracy (4.28) by computing and comparing the approximation and the exact solution in the following section.

Implications

This section shows the price behavior of a vulnerable call option with respect to the model parameters as well as the price accuracy of our analytic formula. In the previous section, we have shown that the option price is expressed as an explicit closed form by using the Mellin transform techniques. Due to the analytical solution, we are able to analyze the sensitivity of the option price against the model parameters in an effective manner.

Figure 1 exhibits the price behavior of a vulnerable ATM call option against the underlying asset price (x) and the firm value (v) of the option writer (OW) assets for different choices of the volatilities ( and ) of the asset and the firm value. We choose the same values of the two volatilities because we believe that they go along with each other in general. As shown in the figure, when the underlying asset price, the firm value of the option writer, or the volatilities increases, the price of the vulnerable call option rises as it should be the case. A noticeable fact is that the value of the vulnerable call option falls more sharply when the firm value is smaller than the critical value as the volatilities become smaller. It implies that an event of default is more shocking to the call option buyer when the market has been less volatile.

Fig. 1

Value of a vulnerable ATM call option against the firm value of the option writer and the underlying asset price

Next, we investigate the analytical option price accuracy numerically through the Monte Carlo (MC) simulation. Table 1 displays the option price given by (3.30) compared to the MC result denoted by . The MC option price is calculated with 100,000 simulation paths, where the parameter values are . In addition, we find the relative percentage error given by in Table 1. As one can notice from the table, the gap between the analytical option price and the numerical price becomes smaller (more rapidly in than ) as or gets close to zero.

Table 1

Comparison between the analytical solution and the numerical solution , where “Error” means the relative percentage error defined by

	P~^δ,ϵ					P_MC
	δ=1	δ=0.5	δ=0.1	δ=0.05	δ=0.01
ϵ=1	7.8403	7.9429	8.0812	8.1142	8.1584	8.1659
Error(%)	3.9862	2.7305	1.036	0.6327	0.0914	–
ϵ=0.5	7.8538	7.9524	8.0854	8.1172	8.1597	8.1667
Error(%)	3.8310	2.6237	0.9948	0.6059	0.0851	–
ϵ=0.1	7.8718	7.9651	8.0911	8.1212	8.1615	8.1680
Error(%)	3.6266	2.4840	0.9414	0.5729	0.0794	–
ϵ=0.05	7.8761	7.9681	8.0925	8.1222	8.1620	8.1682
Error(%)	3.5768	2.4495	0.9274	0.5637	0.0767	–
ϵ=0.01	7.8817	7.9722	8.0943	8.1234	8.1625	8.1684
Error(%)	3.5091	2.4022	0.9073	0.5502	0.0717	–

We, now, find the stochastic elasticity of variance (SEV) effect on the vulnerable option price by comparing to the Black–Scholes (BS) price for the vulnerable option. Figure 2 shows differences between the BS option price and the SEV option price against the firm value of the option writer for three different choices of the critical level (the critical value of default) and (the deadweight costs). As seen in the figure, the price difference becomes larger when or decreases. It is the case also when the firm value of the option writer or time-to-maturity increases. If the firm value is large enough near the maturity, then the difference becomes a maximum level and independent of , and the firm value.

Fig. 2

Difference between the Black–Scholes and the SEV vulnerable call option prices against the firm value of the option writer for three different time-to-maturities

Figure  3 exhibits the Greek, the first derivative of the option price with respect to the market price of risk , given bywhere , and , and is given by (4.15). As shown in the figure, the slope of the Greek increases when the firm value of the OW decreases. It says that the sensitivity of the Greek is rising as the defaulting probability is increasing. In particular, for shorter time to maturity as in Figure 3-(c), the slope of the Greek increases drastically until the firm value of the OW decreases to . It implies that the change of market price of risk has a more significant influence on the option price in the case of higher probability of default and shorter time to maturity.

Fig. 3

The Greek with for three time-to-maturities

Conclusion

There are two possible ways of developing stochastic volatility frameworks from the Black–Scholes model. One way is to randomize the constant volatility of the Black–Scholes model. The Heston model is the best example of it. The other method is to randomize the constant elasticity of the Black–Scholes or CEV model. The SEV model used in this paper belong to this category.

The Heston model or Choi et al. (2016) provides semi-analytical pricing formulas for European vanilla options. These formulas require the calculation of logarithms with complex arguments during the process of computing the inverse Fourier integral. Subsequently, this leads to a numerical instability for a certain range of time to maturities or fast mean reversion, which may ultimately come up with inaccurate solutions and reduce the performance of the option’s data fitting. In addition, when it comes to solving the solution numerically, it may take much time for us to compute the option price and implement option’s data fitting. In this paper, we use the SEV model to obtain a closed form analytical pricing formula for the European options via the Mellin transform approach. This formula only requires the evaluation of the Black–Scholes formula and its derivatives. So, the SEV framework with the Mellin transform gives rise to a better analytical result for option pricing with respect to the stability (or the accuracy) and the computing time than the Heston framework with the Fourier transform. This is, of course, important to practitioners for speedy calculation of more complicated derivatives and effective designing trading strategies.