Cva and Vulnerable Options in Stochastic Volatility Models

Abstract

This work aims to provide an efficient method to evaluate the Credit Value Adjustment (CVA) for a vulnerable European option, which is an option subject to some default event concerning the issuer solvability. Financial options traded in OTC markets are of this type. In particular, we compute the CVA in some popular stochastic volatility models such as SABR, Hull et al., which have proven to fit quite well market derivatives prices, admitting correlation with the default event. This choice covers the relevant case of Wrong Way Risk (WWR) when a credit deterioration determines an increase in the claim value. Contrary to the structural modeling adopted in [G. Wang, X. Wang & K. Zhu (2017) Pricing vulnerable options with stochastic volatility, Physica A 485, 91–103; C. Ma, S. Yue & Y. Ma (2020) Pricing vulnerable options with Stochastic volatility and Stochastic interest rate, Computational Economics 56, 391–429], we use the reduced-form intensity-based approach to provide an explicit representation formula for the vulnerable option price and related CVA. Later, we specialize the evaluation formula and construct its approximation for the three models mentioned above. Assuming a CIR model for the default intensity process, we run a numerical study to test our approximation, comparing it with Monte Carlo simulations. The results show that for moderate values of the correlation and maturities not exceeding one year, the approximation is very satisfactory as of accuracy and computational time.

CVA in fractional and rough volatility models

“…All the computational procedures were implemented by using MatLab R2021b on a Intel(R) Core(TM) i7-4700MQ CPU @ 2.40GHz. The results of the approximations are reported in Tables ( 2)- (7) and shown in the Figures (1)-( 6) for an ATM call option with S 0 = 100, K = 100 and maturities T = 0.25, 0.5, 1 years, respectively. The time to compute the set of 45 adjustments with the Monte Carlo algorithm was approximately one hour, while about 30 secs are needed for the computation of our approximation, which requires the numerical evaluation of multiple integrals.…”

“…Finally, we considered also the sensitivity of our approximation to the Hurst parameter H. Figure (7) reports the errors between Monte Carlo estimate and the first order approximation as a function of the correlation ρ and the Hurst parameter H. Even in this case, the dependence of the error from H is almost constant and depends mainly on the correlation parameter, however remaning of order 10…”

“…Finally, the factor E t σ s can be computed by using a log-normal (moment-matching) approximation as in [7].…”

“…Therefore, extensive work was done to develop representation formulas that might lead to handier expressions and to finding alternative computational methods, as in [1], and more recently in [8] and [7].…”

Pricing vulnerable fader options under stochastic volatility models

“…Actually, vulnerable options have been studied in stochastic volatility models. 1 As for default risk, there are two typical methods: structural models and reducedform models. In the structural model, vulnerable European options are investigated in different stochastic volatility models (e.g., Yang et al [40], Lee et al [22], Wang et al [31] and Ma et al [25]).…”