| 课程号 |
06239122 |
学分 |
3 |
| 英文名称 |
Stochastic Calculus and Its Applications in Quantitative Finance |
| 先修课程 |
微积分，线性代数，概率论，随机过程 |
| 中文简介 |
本课程旨在教授学生研习及分析连续时间随机模型所须具备的知识与技术。此类模型主要源自于随机微分方程及带跳的扩散过程。其在量化金融的主要应用包含风险中立定价理论，期权定价，奇异期权定价，以及计价标准转换等等。学生修完成本课程应具备随机微积分的基础知识与操作技术，及其在量化金融上的相关应用。 |
| 英文简介 |
The course aims to provide the students with knowledge and skills for the study and analysis of continuous-time stochastic models that are originated from stochastic differential equations and jump diffusions. Applications of the stochastic calculus methods focus mainly in the field of quantitative finance including risk neutral pricing theory, options pricing, pricing of exotic options, and the technique of change of numeraire, etc. Upon completion, students are expected to understand and to be capable of manipulating stochastic calculus techniques and their applications in quantitative finance. |
| 开课院系 |
国家发展研究院 |
| 通选课领域 |
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| 是否属于艺术与美育 |
否 |
| 平台课性质 |
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| 平台课类型 |
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| 授课语言 |
英文 |
| 教材 |
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| 参考书 |
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| 教学大纲 |
1. 了解并熟悉随机微积分的理论及其演算方式。
2. 了解并熟悉金融衍生性商品以及随机微积分在关于此类商品定价模型上的应用。
上课时间：每周一、周三、周五下午7-9节。其中， 7月8日、10日停课，7月13日、14日7-9节补课。
Discrete time martingale
1. Conditional probability and conditional expectation
2. Uniform integrability
3. Stochastic processes in general
4. Martingale, submartingale, supermartingale
5. Doob`s inequalities
6. Predictable processes
7. Doob decomposition theorem
8. Stopping times
9. Martingale transformation and discrete stochastic integral
10. Optional stopping theorem
？ Brownian motion
1. Definition, construction and properties of Brownian motions
2. Reflection principle
3. Brownian motion with drift
4. Cameron-Martin theorem
？ Stochastic calculus
1. Construction of Ito’s integral
2. Martingale property
3. Ito`s isometry
4. Quadratic variation
5. Ito`s formula
6. Girsanov theorem
7. Stratonovich integral
？ Markov processes
1. Markov property and strong Markov property
2. Transition matrix, transition density
3. Chapman-Kolmogorov equation
4. The generator and the infinitesimal generator
5. Markov martingales
6. Dynkin formula
？ Stochastic differential equations
1. Existence and uniqueness
2. Diffusion processes
3. Relationship to partial differential equations
4. Kolmogorov forward and backward equations
5. Feynman-Kac formula
？ Risk neutral pricing
1. Arbitrage pricing theory and risk neutral probability
2. Fundamental theorem of asset pricing
3. Black-Scholes model and Black-Scholes formula
4. Exotic option pricing
5. Change of numeraire
每周一、周三、周五下午7-9节，
？ Stochastic calculus with jump processes
1. Poisson and compound Poisson process
2. Ito`s formula for processes with jump
3. Change of measure for jump processes
4. Option pricing in processes with jump
？ Numerical stochastic differential equation
1. Ito-Taylor expansion
2. Strong and weak convergence
3. Variance reduction techniques
课堂讲授 100%
作业 30%, 期中考 30%, 期末考 40%
期中考及期末考皆闭卷
初定7月19日期中考试，8月2日期末考试。
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| 教学评估 |
Tai-Ho Wang：
学年度学期：17-18-2，课程班：随机微积分及其在量化金融的应用1，课程推荐得分：4.76，教师推荐得分：4.76，课程得分分数段：95-100；
学年度学期：18-19-3，课程班：随机微积分及其在量化金融的应用1，课程推荐得分：0.0，教师推荐得分：9.22，课程得分分数段：95-100；
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