A Closed-Form Approximation for Valuing European Basket Warrants under Credit Risk and Interest Rate Risk

[[abstract]]摘要 近幾年由於全球金融市場之自由化，產生了許多經濟上不確定之因素。為了滿足投資人更多避險之需求，因而發展出許多衍生性金融商品。在衍生性金融商品中，組合型權證(Basket warrant)是屬於高槓桿操作，選擇權形式之金融工具。在衍生性商品評價中，組合型權證一直是缺乏信用風險考量的評價公式(Genlte 1993; Huynh 1994; Milevsky and Posner 1998)。在考慮信用風險之類別中(Klein and Inglis 1999; Klein 1996)，現有評價公式均屬單一型權證(Single-tock warrant)，無法直接用於有信用風險之組合型權證評價。 本文探討連續時間模型下，同時考慮信用風險、利率風險，以及不同風險性資產間之關連性；利用平賭過程法(Martingale method) 推導脆弱歐式組合型認購權證(Vulnerable european basket warrants)的買權及賣權之近似評價公式，來填補信用風險下組合型權證評價不足之處，並證明此評價公式為相關權證評價公式之一般解(General solution)，不但可以省去重新推導評價公式之困擾，而且在評價公式之計算上更加的方便。最後，以數值範例(Numerical examples)，利用Cox et al (1985a) CIR利率模型，假設在不同股價、權重、波動度及資產相關性條件下，說明對脆弱歐式組合型權證價格之影響。 此評價公式的驗證及結果如下:當組合型權證中股價波動度越大或波動度越大的股價佔組合權重越高時，脆弱歐式組合型權證買權價格越大。當公司負債比率越高時，脆弱歐式組合型權證買權價格遞減。當公司資產價值與股價呈現高度負相關時，脆弱與非脆弱歐式組合型買權價格差異大，並且股價波動度越大時其價差越大。公司資產價值與利率之相關性，由於權證有效期間很短，對脆弱歐式組合型買權價值之影響有限。當權證標的個股為負相關時，可減少權證之組合風險，使得脆弱歐式組合型買權之權利金較低。此評價公式不僅將組合型權證與不同風險因素之影響加以整合，而且優於Klein and Inglis(1999)脆弱歐式單一型權證在評價之詮釋，在應用上更為廣泛且具有彈性。[[abstract]]ABSTRACT Recently a free global financial market has resulted in many economic uncertainties To meet the need of hedging among more investors financial derivatives have therefore been developed Among those derivatives basket warrants are of high leverage trading and options In the evaluations of derivatives basket warrants have long been lacking a formula to properly evaluate credit risk (Genlte 1993; Huynh 1994; Milevsky and Posner 1998) In the consideration of credit risks (Klein and Inglis 1999; Klein 1996) current evaluation formulae are aiming at single-stock warrants and cannot be directly applied in the evaluation of basket warrants with high credit risks Under a consecutive time model this dissertation considers credit interest rate risks and the relationships between properties with different risks; it also adopts martingale method to deduct an evaluation formula of call and put options of the vulnerable European basket warrants to make up for the deficiencies of basket warrants under different credit risks Through examination the deducted formula proposed in this dissertation proves to be a general solution to related evaluation formulae of warrants; it saves the trouble of re-deducting another new evaluation formula in the future and is more efficient in the process of calculation Finally this dissertation gives numerical examples and uses the interest rate model of Cox et al (1985a) to illustrate the influences on the price range of vulnerable European basket warrants under different stock prices weights volatility of stock prices and asset-related conditions The result of this evaluation formula goes as follows: the higher the price volatility of basket warrants or the percentage of stocks with high volatility the higher the call options of basket warrants When the debt ratio of the issuing firm is higher the call option prices of the vulnerable European basket warrants are lower While the value of the firm is in negative correlation with the stock price the prices of call options of the vulnerable and non-vulnerable European basket warrants vary greatly; the higher the volatility of stock prices the greater the prices vary Due to the short-term validity of warrants the value of a firm and the interest rate has limited influence on that of call options of the vulnerable European basket warrants When the stocks targeted at by the warrants show their negative correlations they can reduce the risks of different combinations of warrants lowering the premium of the call options of the vulnerable European basket warrants This evaluation formula not only combines basket warrants with different risk factors but is more elastic and superior to the interpretation of the vulnerable European basket warrants offered by Klein and Inglis (1999) and can be more widely applie