In recent years, the default risks faced by financial institutions have been increasing, and the corresponding capital requirements have also been increasing. The financial market is full of risks. There are numerous examples of losses or company failures caused by poor risk management in the market. The Association of Interbank Money Market Dealers issued guidelines on credit risk mitigation tools in October 2010. This marks the official launch of the Chinese version of the CDS credit derivatives market. This favorably promotes the release of credit risk. The pricing of credit default swaps is a kind of financial credit derivatives. This allows the credit risk of risky assets to be separated from asset ownership. The market-oriented arrangement for transferring credit risk gives risky assets liquidity characteristics . It has played an important role in bank credit portfolio management and the financial risk control of listed companies. Some scholars have studied the pricing of corporate bonds under the reduced model where the interest rate obeys the Vasicek model, and the default intensity is a linear function of the interest rate. Some scholars focus on the random recovery rate model. They treat the recovery rate as an exponential function of default strength. We established a pricing model and used the technique of solving partial differential equations to give an analytical solution to corporate bonds. This way, the impact of recovery parameters and default strength parameters on credit spreads is analyzed . The above documents assume that the company's asset value Vt meets the standard Brownian motion-driven stochastic differential equation in a given probability space. However, the standard Brownian motion cannot describe the fractal characteristics of financial markets, such as self-similarity and long memory. Fractional Brownian motion has become the easiest way to compensate for the shortcomings of the Brownian motion model because fractional Brownian motion is a Gaussian process.
Its properties mainly include additive invariance, self-similarity, thick tail, discontinuity, and long-term correlation. This makes the fractional Brownian motion neither a Markov process nor a semimartingale. This has brought huge difficulties to avoid arbitrage opportunities in the financial market under fractional Brownian motion . Some scholars have proposed using the fractional Brownian jump-diffusion model to avoid the existence of arbitrage, and the results are still not ideal.
Therefore, this article attempts to study the capital valuation adjustment of the value of the derivatives transaction contract. We use the semi-copy method to design a reasonable hedging portfolio strategy . The partial differential equation of valuation adjustment is derived based on the partial differential equation of the value of the derivative transaction contract and the risk-free value of the contract . At the same time, we apply the Feynman-Kac formula to derive a valuation adjustment expression closer to the market from separating the capital valuation adjustment. As mentioned earlier, the calculation of the capital valuation adjustment further revised the price of the derivative contract value.
We use the replication method to establish a reasonable hedging portfolio strategy. This enables hedging of derivatives . Using this method to hedge against the counterparty's default risk has become an important means of bank risk management. At the same time, banks are also using scientific and reasonable calculation methods and appropriate models to measure the default risk of trading contracts. This will strengthen the management of capital valuation adjustments and reduce the cost of capital. We set the price of the underlying asset stock as a Brownian motion
This model assumes that the random process followed by the price of the underlying asset stock is geometric Brownian motion. The model does not consider the jump behavior of stock prices . This article sets the price process of the underlying asset as a jump-diffusion process in the subsequent model asset assumptions. The purpose is to enable the model to cover this risk factor. At the same time, we have added stock options to the hedging portfolio. The purpose is to hedge against the volatility risk of stock prices.
Assume that the value of the transaction contract between the bank and its counterparty is
When the bank or counterparty defaults, the liquidation or claim of the trading contract position is usually determined by the Mark-to-Market
When the counterparty defaults, there are:
When the value of the transaction contract between the bank and its counterparty is
Any hedging portfolio strategy should include at least different assets such as the subject matter, bonds issued by both parties to the transaction, collateral, capital, and cash accounts. Therefore, the risk should include the stock of
The prices issued by the bank are
The basis difference between bonds of different grades is not considered for the time being. We use bonds with different recovery rates to offset the balance sheet impact of derivatives.
The repurchase account
In addition, the asset constraints of the hedging portfolio strategy are set to:
Since the hedging investment strategy Π satisfies
We derive the partial differential equation of the total valuation adjustment from the differential form of the hedging portfolio strategy containing stock options and the cost of capital. Then the Feynman-Kac formula derives the expression of the total valuation adjustment . At the same time, we separate the capital valuation adjustment calculation expression. According to the Ito lemma of the jump-diffusion process, we find that the differential form of the contract value
Over time, the differential form of hedging investment strategy Π is:
According to the assumptions, we assume
Reorganize the above formula 1 to get:
In the formula
Banks take into account the interests of shareholders, so they hope that the hedging strategy in the above formula can be transformed into a form of self-financing. The drift term in the above formula is required to be zero, then the partial differential equation about the transaction contract value
The risk-free value
From equation (2), the total valuation adjustment
According to the Feynman-Kac formula, it can be found that the partial differential equation is expressed as
When the boundary condition is
In the formula,
Credit valuation adjustment
Financing valuation adjustment
Margin valuation adjustment
Collateral valuation adjustment
The expression of capital valuation adjustment
The price of bonds issued by fund companies depends on many factors. Mainly include asset volatility, bond maturity date, the weight of short-term debt and medium and long-term debt in the default boundary, the debt recovery rate at the time of default, market risk-free interest rate, financing leverage, etc. When drawing up capital structure plans, fund companies usually use debt ratios and debt financing ratios to assess bankruptcy risks. Assume that D1/S0=0.05 and D2/S0=0.1. We use Matlab software to program equation (16) and iteratively solve equation (17). When ρ = 0.15, 0.2, 0.25, 0.3, the influence of the simulated Hurst parameter H on the default probability of short-term bonds is shown in Figure 1 and Figure 2.
It can be seen that the default probability of the short-term debt of the fund company gradually increases with the increase of the Hurst parameter H of the mixed fraction jump-diffusion fraction Brownian motion. When the risk factor becomes larger, the probability of default also rises. This is consistent with empirical evidence.
This article studies the cost of capital by including capital items in the hedging portfolio strategy. First, a hedging portfolio is designed from the perspective of an incomplete hedging strategy. The portfolio includes stocks, bonds with different recovery rates, stock options, capital, and different cash accounts. This article assumes that the stock price of the underlying asset obeys the jump-diffusion process to increase the rationality of asset price changes. This article also considers the risk of fluctuations in the underlying asset price. Based on this, stock options are added to the hedging portfolio to reduce the risk of stock price fluctuations.
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