A Jump Diffusion Model with Fast Mean-Reverting Stochastic Volatility for Pricing Vulnerable Options

Te Black–Scholes–Merton option pricing model is a classical approach that assumes that the underlying asset prices follow a normal distribution with constant volatility. However, this assumption is often violated in real-world fnancial markets, resulting in mispricing and inaccurate hedging strategies for options. Such discrepancies may result into fnancial losses for investors and other related market inefciencies. To address this issue, this study proposes a jump difusion model with fast mean-reverting stochastic volatility to capture the impact of market price jumps on vulnerable options. Te performance of the proposed model was compared under three diferent error distributions: normal, Student-t , and skewed Student-t , and under diferent market scenarios that consist of bullish, bearish, and neutral markets. In a simulation study, the results show that our model under skewed Student-t distribution performs better in pricing vulnerable options than the rest under diferent market scenarios. Our proposed model was ftted to S&P 500 Index by maximum likelihood estimation for the mean and volatility processes and Gillespie algorithm for the jump process. Te best model was selected based on AIC and BIC. Samples of the simulated values were compared with the S&P 500 values and MSE computed at various sample sizes. Values of MSE at diferent sample sizes indicate signifcant decrease to actual MSE values demonstrating that it provides the best ft for modeling vulnerable options.


Introduction
Over-the-counter (OTC) markets have grown signifcantly in recent years, raising concerns about default risk, particularly following the 2008 global fnancial crisis.Since OTC options are not subject to regular market reconciliation and margin replenishment, they expose option holders to increased credit risks.Tese options, which are vulnerable to counterparty credit risk, are known as vulnerable options.Furthermore, OTC markets are dominated by trend traders, whose tendency to follow market trends often leads to market congestion, resulting in the whipsaw efect infuencing the price discovery of options that are traded in this market.During such periods, increased price volatility disrupts pricing inputs like implied volatility, making it challenging to accurately determine option prices.In response to these challenges and with potential applications in controlling counterparty credit risk and facilitating smoother negotiation processes for these options during market congestion, research for improved pricing models has emerged as an increasingly vital topic in fnance.Tese eforts aim to mitigate pricing inefciencies and enhance risk management for OTC options in today's dynamic fnancial landscape.
Black and Scholes [1] pioneered the concept of option pricing models in their seminal paper by assuming that the underlying asset price follows a geometric Brownian motion with a constant mean and volatility and that the asset return series follows a normal distribution.However, this model has been exposed to criticism and limitations due to these assumptions.Te assumption that the underlying asset follows a geometric Brownian motion with a constant mean and volatility means that the price of the underlying asset moves continuously and follows a smooth and predictable path over time.However, empirical studies have shown that the underlying asset's price curve is not smooth but has jumps.Te other assumption that the underlying asset return series follows a normal distribution contradicts the empirical studies because asset returns' distribution has leptokurtic features implying that the distribution of asset returns has a higher peak and asymmetric heavier tails than those of the normal distribution.While these assumptions make the model more tractable, they create limitations when applied to real-world fnancial markets.Such limitations include option mispricing during extreme events such as market crashes, inadequacy in capturing market realities such as volatility skew, inaccurate prediction of discontinuous payofs, and signifcant losses to fnancial institutions that rely heavily on this model for risk management.
Ki et al. [2] proposed a closed pricing formula for European options the case where the return of the underlying asset follows extended normal distribution for diferent degrees of skewness and kurtosis relative to the normal distribution.Numerical experiments and a comparison of the empirical performance of the proposed model with the Black-Scholes model were done through the estimation of implied parameters such as standard deviation, skewness, and kurtosis of the return on the underlying asset from the market prices of the KOSPI 200 Index options.Te results demonstrated that the actual density of the underlying asset depicts skewness to the left with high peaks.Similarly, Burger and Kliaras [3] empirically investigated the Black--Scholes model and the Merton model both constructed regarding normal distribution and the double exponential jump difusion model which does not assume a normal distribution of the stock returns but a distribution that has got a higher peak and two heavier tails and also considers the empirical abnormity called volatility smile.Te empirical results showed that the double exponential jump difusion model ftted the stock data better as compared to the other two models.Gatheral et al. [4] noted that the Black-Scholes approach underestimates the probability of extreme events in asset returns' distribution due to its constant volatility assumption.Moreover, Liu et al. [5] utilized a model of stochastic volatility featuring jumps within the price of the underlying asset and the counterparty asset value to derive solutions for the option price.Teir fndings indicated that the stochastic volatility model with jumps can provide more accurate pricing compared to the Black-Scholes model.Similarly, Zhou et al. [6] considered an improved model for pricing vulnerable options by incorporating the dynamics of the underlying asset and counterparty asset as a class of jump difusion processes.Tese fndings challenge the model's validity in real-world scenarios and emphasize the need for more sophisticated approaches to option pricing and risk management.Besides the jump difusion model employed in this study, other difusion processes have been explored in existing literature, such as self-difusion and cross-difusion, as evident in recent studies (see Chen et al. [7], Chen and Wu [8], Chen and Srivastava [9], and Zhu et al. [10]).
Tis paper proposes to study a jump difusion model under the Student-t and skewed Student-t distributions instead of the Gaussian distribution.Te dynamics of the underlying asset price, option writer's asset value, and stochastic volatility are derived, and the pricing formula of the vulnerable options is obtained.Te application of the proposed model is demonstrated by considering three different error distributions (normal, Student-t, and skewed Student-t) and three market trends (bullish, bearish, and neutral) and performing simulations of the model under the diferent market trends and error distributions.Empirical results are obtained by ftting the proposed model using the S&P 500 Index prices under the three diferent error distributions.Performance of the proposed model was tested by computing the AIC and BIC and the mean square error under diferent sample sizes.By addressing these aspects, this study aims to present a more accurate and robust approach to pricing vulnerable options, ofering valuable insights into risk management and option pricing strategies for fnancial practitioners.
Te rest of this paper is organized as follows.In Section 2, we describe the methodology of pricing vulnerable options using the jump difusion model with fast meanreverting stochastic volatility.Section 3 discusses the simulation results obtained from the application of this methodology.Section 4 presents the empirical results.Finally, in Section 5, we provide the conclusions and suggestions for further research.

Methodology
In fnancial markets, the jumps in fnancial asset prices are normally triggered by policy changes, catastrophic events, and big news events.In this section, the jump difusion model with mean-reverting stochastic volatility is derived.
Assume that T � [0, T] is the time level and (Ω, F, F t   t∈T,P ) is the complete probability space where P is the physical probability measure.Te dynamics of the underlying asset price S t and the option writer's asset value V t are assumed to be described by the following stochastic diferential equations (SDEs), respectively: where μ 1 and μ 2 are drift components of underlying asset price and option writer's value, respectively, σ 1 is the volatility of the underlying asset price, σ 2 is the volatility of the option writer's value, and B 1 t and B 2 t are the standard Brownian motions of the underlying asset price and option writer's value, respectively.
To account for jumps in the underlying asset price and option writer's asset value, a jump process is introduced to the right hand side of equation (1).Terefore, the dynamics of S t and V t follow a jump difusion process given by 2 Discrete Dynamics in Nature and Society where S t − and V t − represent the value of S t and V t just before a possible jump of sizes e ξ and e v at time t, respectively.N 1t and N 2t denote the counts of the observed number of jump times prior to time t with intensities λ 1 and λ 2 , respectively.i�1 (e ξ i − 1)) − λ 1 β 1 represent the conditional mean, the stochastic volatility process, and the jump process of the underlying asset, respectively.For model estimation, the conditional mean is ftted by estimating the parameters by the MLE method and then residuals are obtained from the ftted model, and volatility process is also ftted by estimating parameters from the residuals by the MLE method.Simultaneously, we use the Gillespie Algorithm to estimate the jump process.. Te percentage increase in the price of the underlying asset and option writer's asset value given the price changes from S t− to e ξ S t and V t− to e v V t , respectively, is given by where ∆S t � ∆V t ⟶ dt as ∆t ⟶ 0 is the infnitesimal limit dt.Terefore, the total number of jumps is given by To ensure that S t and V t are martingales, the jump components in equation ( 2) are compensated by λ 1 β 1 dt and λ 2 β 2 dt, respectively.
Let Y t denote the stochastic volatility process assumed to follow the Ornstein-Uhlenbeck process given by where ε denotes the inverse of the mean reversion rate and To obtain an arbitrage-free price of the vulnerable option, a risk-neutral measure P * from Johannes and Polson [11] is introduced in equations ( 2) and ( 5) to get where and Λ(Y t ) are defned under P * .According to Klein [12], the payof of a vulnerable European call option at time T is given by where  D is the default boundary, D is the total claim value, and α is the dead-weight cost of fnancial distress expressed as a percentage of option writer's asset value.
Terefore, the price of a vulnerable call option at time t ≤ T is given by Discrete Dynamics in Nature and Society where E P * [•] is the conditional expectation under the riskneutral measure P * .As a consequence, the solution of P(t, s, v, y) is given by a PDE in Oksendal [13] with the terminal condition Given the terminal condition in equation ( 9), obtaining an analytical solution for this partial diferential equation (PDE) is not feasible due to its complexity.As a result, numerical simulations of the proposed model were performed.Te Gillespie algorithm was used to simulate the price jumps.We defned three distinct states representing the bullish (BU), bearish (BE), and neutral (NE) market trends, each associated with specifc rates governing the occurrence of market price jumps.Tese rates were calibrated to refect the intensity of jumps in each market state.We denote the state vector x � (x 1 , x 2 , x 3 ) where x 1 , x 2 , x 3 represents the counts of BE, BU, NE, respectively.Table 1 lists the interactions between diferent states, propensity functions of reactions, and the net vectors.From Table 1, if the market experiences an event that leads to (1) a downward jump in stock prices, the economy moves from BU to BE and is modeled in reaction R 1 , (2) an upward jump in stock prices, the economy moves from BE to BU and is modeled in reaction R 2 , (3) and no jump in stock prices, the economy in either BU or BE moves to NE and is modeled in reactions R 3 , R 4 , R 5 , and R 6 .
To implement the Gillespie algorithm, we adapted the steps outlined in Altıntan et al. [14].

Simulation Results
In this section, we illustrate the application of the jump difusion model with the Gillespie algorithm, as presented in equation ( 2).Te Gillespie algorithm was selected for its unique features, including its ability to simulate individual reactions rather than the system as a whole, which results in a more precise and accurate simulation of the underlying stochastic processes.Moreover, its fexibility in handling complex models with multiple stochastic variables and events, combined with its computational efciency and suitability for large datasets, makes it ideal for real-time simulations.Tese features make the Gillespie algorithm a powerful tool for simulating complex systems, such as the jump process in the proposed model.
Figure 1 illustrates the comparison of the adapted model and the ftted model under three diferent error distributions.
Table 2 presents a set of parameters adapted from Turchyn [15] and Altıntan et al. [14] based on real market data.Te adapted set of parameter estimates in Table 2 was used to ft the proposed jump difusion model with fast mean-reverting stochastic volatility.
Te parameters of the proposed model were estimated using the maximum likelihood estimation method.Table 3 presents the parameter estimates of the ftted model using three distinct residual distributions, namely, normal, Student-t, and skewed Student-t, under three market trends (bearish, bullish, and neutral).Te parameter estimates were found to be independent of the market trends (bearish, bullish, and neutral).Tis implies that the proposed model is appropriate to price vulnerable options as it will provide the jump process of the underlying asset regardless of whether the market is bullish, bearish, or neutral.

Empirical Results
Tis section presents the empirical results and discussions of the results obtained from ftting the proposed model to the dataset consisting of 4410 daily average closing prices of the S&P 500 Index, covering the period from 1 st January 2005 to 31 st July 2022.Te data exclude weekends and holidays downloaded from https://www.investing.com.In Figure 2, the time series plot of the daily prices depicts the trend of the prices over time, while the log returns plot highlights the stochastic volatility dynamics in the daily prices.Signifcant price jumps are observed during the 2007-2008 global

Reaction list
Reaction propensity 4 Discrete Dynamics in Nature and Society fnancial crisis period and COVID-19 pandemic period.Te market crashes caused the returns to exhibit extreme high asymmetric volatility and scattered jumps.Te log returns also appear to fuctuate around the mean level, exhibiting volatility clustering, where large changes are followed by large changes, and small changes are followed by small changes.Tis implies that the log return exhibits conditional heteroskedasticity that can be modeled using conditional heteroskedastic models.Table 4 presents descriptive summary statistics and statistical tests of the index prices, log returns, and squared log returns of the S&P 500 Index prices.Te minimum and maximum values provide a range of the observed values in the data.Te standard deviations are all positive giving an indication of the volatility of the underlying asset prices.Te negative skewness of the log returns indicates that the distribution is negatively skewed, with the tail on the left side longer than the right side.Furthermore, the kurtosis of the log returns is greater than 3, indicating that the distribution is heavy tailed, and extreme values occur more frequently than in a normal distribution.Tese characteristics are consistent with the presence of volatility clustering and fat tails in the distribution of log returns.Te Jarque-Bera (JB) test for normality confrms that the log returns are not normally distributed.ARCH-LM test for the residuals of the log returns confrms the presence of heteroskedasticity.Terefore, the jump difusion model with fast meanreverting stochastic volatility could be useful in describing the dynamics of vulnerable options using the S&P 500 Index as the underlying asset.Te proposed jump difusion model was ftted to the S&P 500 Index prices under the three error distributions.Table 5 presents the parameter estimates of the ftted model under diferent error distributions.Te parameter estimates for both the conditional mean and volatility equations are confrmed to be statistically signifcant.In addition, the shape parameter and skewness parameter for both the Student-t and skewed Student-t distributions are statistically signifcant.Tus, the use of heavy-tailed innovation distribution seems justifed to account for skewness and excess kurtosis in the asset returns.
To evaluate the relative goodness of ft of the proposed model under the diferent distributions to S&P 500 Index, the AIC and BIC of the proposed model were computed under various sample sizes.Table 6 presents the AIC and BIC values under diferent sample size values.Te results demonstrate that as the sample size increases, skewed Student-t has the smallest AIC and BIC.Tis demonstrates that the proposed model with skewed Student-t distribution fts the return series more appropriately compared to the other two error distributions.
In order to test the performance of the proposed model under the three error distributions (normal, Student-t, and   skewed Student-t), the mean square error (MSE) was computed for both the ftted model and the S&P 500 Index prices.Table 7 presents the MSE values under diferent sample size values.Te results demonstrate that as the sample size increases, the MSE of the proposed model decreases for all three distributions.Tis means that as the sample size increases, the suggested model's forecast of S&P 500 Index prices becomes increasingly accurate, and thus the model's prediction accuracy grows.Tese fndings imply that when pricing vulnerable options using the proposed method, the distribution for the error component is an important factor in attaining accurate results.Overall, skewed Student-t distribution showed the lowest MSE value among the three distributions for all sample sizes.Tis demonstrates that the proposed model with skewed Studentt distribution fts the return series more appropriately compared to the other two error distributions.Tese fndings support prior fndings that, under the assumption   .

Conclusions and Recommendation
Tis paper presents a jump difusion model with fast meanreverting stochastic volatility for pricing vulnerable options.Te proposed model's application is illustrated by ftting the model using three distinct residual distributions, namely, normal, Student-t, and skewed Student-t, under three market trends (bearish, bullish, and neutral).From empirical results, the distribution of the log return series of S&P 500 Index was found to be negatively skewed and heavy tailed and this confrms the existing research that log returns of underlying asset of vulnerable options are not normally distributed.In addition, the proposed model with skewed Student-t distribution fts the return series of the S&P 500 Index more appropriately compared to normal distribution.
To validate these results, future research can expand the analysis to diferent historical options datasets, explore other types of options, and compare this with other option pricing models.

Figure 1 :
Figure 1: Comparison of the adapted model to the ftted model under skewed Student-t, Student-t, and normal distributions.

Figure 3
Figure 3 illustrates the comparison between the ftted model under various error distributions and the S&P 500 Index.Te ftted model's plot under the skewed Student-t distribution closely resembles the plot of the S&P 500 Index prices.Tese fndings show that the proposed model under the skewed Student-t distribution could be a useful model for pricing vulnerable options.Te proposed jump difusion model was ftted to the S&P 500 Index prices under the three error distributions.Table5presents the parameter estimates of the ftted model under diferent error distributions.Te parameter estimates for both the conditional mean and volatility equations are confrmed to be statistically signifcant.In addition, the shape parameter and skewness parameter for both the Student-t and skewed Student-t distributions are statistically signifcant.Tus, the use of heavy-tailed innovation distribution seems justifed to account for skewness and excess kurtosis in the asset returns.To evaluate the relative goodness of ft of the proposed model under the diferent distributions to S&P 500 Index, the AIC and BIC of the proposed model were computed under various sample sizes.Table6presents the AIC and BIC values under diferent sample size values.Te results demonstrate that as the sample size increases, skewed Student-t has the smallest AIC and BIC.Tis demonstrates that the proposed model with skewed Student-t distribution fts the return series more appropriately compared to the other two error distributions.In order to test the performance of the proposed model under the three error distributions (normal, Student-t, and

Figure 2 :
Figure 2: Times series, log return, and squared log return series of the S&P 500 Index for the period starting from 1st January 2005 to 31st July 2022.

Figure 3 :
Figure 3: Comparison of the ftted model under skewed Student-t, Student-t, and normal distributions to the S&P 500 Index prices from April 2011 to October 2011.
the expected jump size conditional on information available at time t.Within this model represented by equation (2), μ, Y t , and (

Table 1 :
Reaction list and reaction propensities of the jump process.

Table 3 :
Parameter estimates of the ftted model using three diferent error distributions (normal, Student-t, and skewed Student-t) under three diferent market trends (bearish, bullish, and neutral).

Table 4 :
Summary statistics of prices, log returns, and squared log returns of S&P 500 Index.

Table 5 :
Parameter estimates of the proposed model ftted to the S&P 500 Index under the three error distributions.

Table 6 :
AIC and BIC of the proposed model under normal, Student-t, and skewed Student-t distributions for various sample sizes.

Table 7 :
Mean square error of the proposed model under normal, Student-t, and skewed Student-t distributions for various sample sizes.