脆弱選擇權是在場外交易市場中交易的一種衍生性金融商品，是具有違約風險的選擇權。傳統的文獻大多假設選擇權標的資產服從幾何布朗運動，但金融市場的波動甚大，因此，後續許多學者以跳躍擴散或是群聚跳躍模型代替幾何布朗運動來表示標的資產價值的波動。而在金融市場上，流動性是違約風險重要的因素，因為直接影響資產買賣能力和交易成本。具有良好流動性的市場意味著資產可以迅速、容易地買賣，且交易成本相對較低。這對於投資者和交易者來說非常重要，因為他們可以更容易進出市場，調整投資組合，或者快速執行交易策略。然而，缺乏流動性可能會導致市場價格的劇烈波動，因為即使有小量的交易量也可能對資產價格產生巨大影響。此外，低流動性可能會導致投資者難以及時進行交易，這可能使他們暴露於更大的風險中。因此，在金融市場上，流動性是確保市場有效運作和吸引投資者的關鍵因素之一。交易對手的資產在實際市場中很少被直接交易，這意味著交易對手資產的流動性可能相當有限。而Amihud 的流動性不足指標在實證文獻中非常受歡迎，但不適合於模擬有長期影響的流動性不足現象，使用Fractional Hawkes process的均值回歸跳躍模型來處理對數Amihud指標，使模型能夠準確地模擬出金融市場中真實出現的流動性短缺情況，同時在模型中引入長期依賴性和易處理性。這是現有 Hawkes 過程無法實現的。因此，我們可以使用這個模型來進行流動性不足的風險管理，並且可以引入和定價基於Amihud 指標的流動性衍生品。總結以上所述，本文主要將探討跳躍群聚的擴散模型下納入Fractional Hawkes process，建立出一個脆弱歐式選擇權定價公式。

Vulnerable options are a type of derivative financial instrument traded in the over-the-counter market and are characterized by the presence of default risk. Traditional literature commonly assumes that the underlying assets of options follow geometric Brownian motion. However, given the significant volatility in financial markets, many scholars have substituted geometric Brownian motion with jump diffusion or clustered jump models to represent the fluctuations in underlying asset values. In financial markets, liquidity is a critical factor in default risk because it directly affects the ability to buy and sell assets and transaction costs. A market with good liquidity means that assets can be bought and sold quickly and easily, with relatively low transaction costs. This is very important for investors and traders as it enables easier market entry and exit, portfolio adjustment, or swift execution of trading strategies. However, a lack of liquidity can lead to drastic price fluctuations in the market, as even small transaction volumes may have a significant impact on asset prices. Additionally, low liquidity can make it difficult for investors to execute trades promptly, potentially exposing them to greater risks. Therefore, liquidity is one of the key factors ensuring efficient market operation and attracting investors in financial markets. Counterparty assets are rarely traded directly in actual markets, meaning that the liquidity of counterparty assets can be quite limited. Although Amihud's illiquidity measure is very popular in empirical literature, it is not suitable for simulating phenomena of long-term illiquidity. By employing a mean-reverting jump model using the Fractional Hawkes process to handle the logarithmic Amihud index, the model can accurately simulate the liquidity shortages observed in financial markets, while introducing long-term dependencies and tractability into the model. This is something that the existing Hawkes processes cannot achieve. Thus, we can use this model for liquidity risk management and to introduce and price liquidity derivatives based on the Amihud index. In summary, this paper primarily discusses the integration of the Fractional Hawkes process into a clustered jump diffusion model to establish a pricing formula for vulnerable European options.

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