Fu za jin rong jing ji xi tong zhong de sui ji guo cheng

This thesis contains two different topics under the common umbrella of analysing and solving quantitative finance problems subject to the Cox-Ingersoll-Ross (CIR) process with physics approach. In the first part of the thesis, we focus on option pricing under the Heston stochastic volatility with time-dependent parameters. The Heston model assumes that the price of the underlying is determined by the Black-Scholes type stochastic process while the variance is a time-dependent CIR process. Starting with the special case, where the volatility of the volatility is zero and applying the Wei-Norman theorem, an analytical formula for pricing European vanilla options, with an arbitrary time-dependent long-term variance, is derived in terms of the Black-Scholes formula. Subsequently, we determine the exact error correction function for non-zero volatility of the variance and approximate the error correction function by Lie-Trotter splitting and Taylor expansion, which can be computed numerically and gives accurate results. We also improve the accuracy by reiterating the calculation of the error correction function. In addition, we obtain the approximation for option pricing subjects to one or two barriers by replacing the Black-Scholes solution of vanilla options with barrier options' solution and kernels.In the second part of the thesis, we focus on the fundamental dynamic of the exchange rate. We present an exchange rate model which the spot of exchange rate is confined in either a wide moving band or target zone and a currency crash occurs when the spot of exchange rate breaches the defined boundary. We derived a solution from the standard log exchange rate equation for the model with a smooth-pasting condition at the defined boundary. The probability of leakage for the exchange rate to cross the boundary increases when the mean-reverting force of the exchange rate is weakened, which implies a rise in currency crash risk. Moreover, we have succeeded in performing explicit calibration of the proposed fundamental dynamics, thus confirming the validity of our model. Unlike the Krugman model, our model is capable of explaining the empirical observations that the exchange rate dynamics is sticky near the edges of the band, and that both positive and negative interest rate differentials can appear in the entire target zone. In addition, the model is also able to determine the crash risk of the G10 currencies during the Global Financial Crisis in 2008 and gives evidence that the transitions of the exchange rate between the two regimes cause changes in the level of the exchange rate volatility but not the dynamical structure本論文以物理方法去分析和解決兩個有關Cox-Ingersoll-Ross（CIR）模型的計量金融問題。在本文的第一部分中，我們重點研究帶有時變參數的Heston模型中的期權定價，該相關資產的價格遵循Black-Scholes過程，但該隨機波動過程則遵循CIR過程。從以方差的波動率趨於零為正的極限開始，並採用Wei-Norman定理，得出歐式期權定價的解析公式，其中長期方差是隨時間變化的任意函數，而公式是以Black-Scholes模型中的公式表達。隨即，我們找出方差的波動率在非零時的確切誤差校正函數，並通過算符分割方法(Lie-Trottersplitting)和泰勒展開(Taylor expansion) 對誤差校正函數進行估算，該誤差校正函數的估算可以進行數值計算，並能在參數與時間相關的情況下給出準確的結果。我們還通過重複計算錯誤校正函數來提高準確性。除此之外，我們將整個公式應用於有一個或兩個界限的界限期權定價上，方法是將本來的歐式期權定價的解析公式中的Black-Scholes模型中的公式替換為界限期權的公式和核(kernel)。論文的第二部分集中在匯率的基本動態。我們展現了一種匯率模型，該模型將匯率的現貨(spot) 限制在較寬的移動區間或目標區間(target-zone)內，並且當匯率突破區間的下限時會發生貨幣衝擊。我們從標準對數匯率方程式中得出了模型的解(solution)，該模型的下邊界具有平滑粘貼(smooth-pasting) 的邊值條件(boundary condition) 。當匯率的均值回復力減弱時， 匯率越過邊界的可能性增加，這意味著貨幣衝擊的風險增加。此外，我們成功對匯率的基本動態進行了顯式校準，從而證實了我們模型的有效性。與克魯格曼模型(Krugman model)不同，我們的模型能夠解釋經驗性的觀察，即匯率的動態在該區間邊緣附近具有粘性，以及正利率差和負利率差都可能出現在整個目標區間中。此外，這模型亦能夠測定2008年全球金融危機期間G10貨幣衝擊的風險，並提供證據表明兩種區間之間的匯率轉換只會導致匯率波動程度的變化，但不會導致匯率的基本動態改變。本動態改變。Liu, Chi Hei Christopher = 複雜金融經濟系統中的隨機過程 / 劉梓熙.Ph.D. Chinese University of Hong Kong 2021.Includes bibliographical references (leaves 120-130).Abstracts also in Chinese.Title from PDF title page (viewed on November 03, 2022).Liu, Chi Hei Christopher = Fu za jin rong jing ji xi tong zhong de sui ji guo cheng / Liu Zixi