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陈平教授复杂演化经济学
Complex Evolutionary Economics：
A General Theory on Micro, Meso , Macro , and Clio Economics
of An Observable World
第十一周（5/2-3/09)
金融工具:期权定价
国家发展研究院
双学位2009春季课程
周六与周日，11-12节（19：10 – 21：00）
地点：理教211
重新认识基于几何布朗运动的
期权定价模型
- 均衡理论的顶峰-期权定价模型：
- 布朗运动理论在经济学与物理学中应用的差别：Bachelier(1900)》唯像理论=无结构, Einstein (1904)》联系分子运动论与粘滞流体中的扩散过程=估计出分子数》否定奥斯瓦尔德的唯能论》发展热运动的统计物理
- Samuelson(1965), Merton (1973), Black & Scholes (1973)》期权定价模型：从多参数到单参数(volatility)》数学美化，物理简化？
- 均衡理论的罕见创新》期权市场的同步发展
- 金融危机及其对均衡理论的挑战：代表者模型+二阶矩假设的局限（陈平，李华俊，曾伟，唐毅南）
Two Equivalent Ways to Derive the Black-Scholes-Merton Equation
Presented by
LIU, Ruixuan
刘睿轩
Dep. Finance, School of Econ.
May2, 2009
Major Focus
- Two Ways to Derive Option Pricing Equation:Black-Scholes and Merton’s Methods
- Common Regularity Conditions:
- Geometric Brownian Motion of the Stock Price
- Existence of Riskless Return, r
- Adapted ( Dynamic) Trading Strategy – So-called Delta Hedge
- Different Methods ( at first glance):
- Black-Scholes: To solve a specific stochastic differential equation
- Merton: Take the expectation under risk-neutral measure using martingale’s property
Major Focus
- Common Intuition behind Abstract Derivation:
- Movement of stock price is much more volatile than the riskless rate of return.
- We need duplicate the pay of the call option ( using stock and riskless return) almost surely along the time span.
- Only the noise term contain information: Delta Hedge Strategy is played through the noise term.
Black-Scholes’ Stochastic Differential Equation
- European Call Option Pays at time T.
- Let the option’s value be c(t,S(t)), obviously it depends on the random S(t) and it is random itself.
- If we can construct a portfolio, X(t), which pays exactly as the European call option along the time span, then their present value should be equal according to No-Arbitrage Theory
- How can we be sure the payment is exactly the same almost at every point?
- Even slight changes( that’s where stochastic calculus is applied) in those two portfolios should be equal
Black-Scholes’ Stochastic Differential Equation
- Basic Setting for BS Derivation
- Geometric Brownian Motion of S(t) ( Put some structure on the randomness of S(t))
Its differential (in the Ito sense) form is more intuitive
- We construct a two-factor portfolio X(t): ⊿(t)S(t) in stock market and the rest X(t)-⊿(t)S(t) in the money market which pays r.
- The evolution of the portfolio value is coming from the above two factors
Black-Scholes’ Stochastic Differential Equation
- Now we can equate those two slight changes( after properly discounted, since we consider the present value)
- Let’s separate the trend and noise terms on both sides ( in rather scaring form)
Black-Scholes’ Stochastic Differential Equation
- In order to let the above equation to hold everywhere, we just equate the relevant terms on dt and dW(t)
- dt and dW(t) are of different magnitudes, dW(t) is much more volatile. Rigorously, dW(t) contains quadratic variation whereas dt doesn’t.
- First, equate the W(t) term, that is so-called Delta-Hedge
- Then, after equating the dt term, we have already got BS Equation
- Note we have substitute x for S(t), because the above one is just simple differential equation, no probability is involved.
- BS solve that equation under some boundary conditions
Merton’s Risk-Neutral Measure
- Motivation: as we know the discounted payoff of the call option is
so why cannot we take the conditional expectation at time t and let it be the call’s value, c(t,S(t))?
- Generally, this is not the case! The property of martingale is needed and the c(t,S(t)) driven by W(t) is not martingale.
- Recall we need X(t) = c(t,S(t)) almost surely, that is strong convergence, even if we change the probability of particular event a bit, that doesn’t matter.
- Merton’s method involves change of probability measure so that we can have c(t,S(t)) as a martingale under the alternative probability measure, thus we can just take the conditional expectation of c(t, S(t)).
Merton’s Risk-Neutral Measure
- What is change of probability measure?
We toss a coin, the outcome is head or tail. Normally the probability is equal for both outcomes, when we e.g. assign P(H)=0.6, P(T)=0.4, we change the measure but the number of outcome is the same.
- Mathematically, we can state: a positive random Z, satisfying EZ=1, can act as the switch
- For a stochastic process, we need a sequence of Z(t) so a martingale under alternative probability measure can be obtained.
Merton’s Risk-Neutral Measure
- For the previously defined portfolio X(t), Merton finds the appropriate change of measure and martingale sequence, according to Girsanov Theorem.
- Now we can just take the conditional expectation and get the result
- Notice should be taken since now we need to derive the above expectation using and
Equivalence of Two Methods
- Common Intuition and Abstract Derivation:
- Movement of stock price is much more volatile than the riskless rate of return => dW(t) is more important than dt, for M-Method, the dt is simply moved by change of measure.
- We need duplicate the pay of the call option ( using stock and riskless return) almost surely along the time span => Almost surely convergence guarantee the change of measure in M-Method
- Only the noise term contain information: Delta Hedge Strategy is played through the noise term => Although not explicitly specified, M-Method contains the dynamic ⊿(t), which can be solved afterwards.
Major Focus
- Two Ways to Derive Option Pricing Equation:Black-Scholes and Merton’s Methods
- Common Regularity Conditions:
- Geometric Brownian Motion of the Stock Price
- Existence of Riskless Return, r
- Adapted ( Dynamic) Trading Strategy – So-called Delta Hedge
- Different Methods ( at first glance):
- Black-Scholes: To solve a specific stochastic differential equation
- Merton: Take the expectation under risk-neutral measure using martingale’s property
集体模型下的金融市场行为和定价
唐毅南
09/05/02
研究动机:群体模型VS代表者模型
- 新古典金融学→稳态线性随机过程→ 完全套利→无参数的定价模型(Black-Scholes等)→但是和现实不符(金融危机必须外生) →不可证伪(Fama 1991)
- 行为金融学→引入人的行为和心理→ 描写内生的泡沫和危机→沿用线性随机过程→模型依赖参数
- 凯恩斯主义→用宏观变量描写经济→投机家阴谋+从众行为=宏观失衡(Minsky) →内生的危机 →新凯恩斯主义寻找微观基础(回到个人理性难以和新古典争论)
- 奥地利学派→企业家是自身稳定（哈耶克）→危机外生(政府）-取消央行
- 演化学派→创造性毁灭过程（熊彼特-社会主义）
- 金融物理→非高斯分布的金融市场→复杂性→so what?
我们的发现
- 能支持群体模型的经验证据是系统的空间非线性
- 描写群体行为的随机过程是生灭过程
- 金融危机的群体动力学判据是趋势瓦解和高阶矩发散
- Black-Scholes模型的极值之谜要用群体行为的高阶矩风险升水解决
支持群体模型
- 必须承认金融系统内部存在不稳定性，金融危机是内生的(陈平 1988）
- 必须能够观察到人的行为和心理的宏观效应
- 以此出发建立基于可观测量的新随机过程，应能容纳通常状况和金融危机
- 建立基于可观测量的定价模型，其推论应与经验相一致
- 相应的微观机制应该得到说明
为什么要用生灭过程？
- 观察到的股价指数在金融危机的冲击下相对偏差仍然随时间稳定
- 理论分析（陈平2002，2005，李华俊2002）：
- 代表者模型（几何布朗运动，随机游走）都不稳定
- 群体模型（生灭过程）随时间稳定→可以取代几何布朗运动作线性区域内的期权定价的基础
标准-普尔500股价指数相对偏差的稳定性
(RD = sHPc/mHPg)
动态随机过程相对偏差的稳定性（陈平2005）
- 布朗运动=扩散过程的代表者模型》随时间爆炸
- 生灭过程=群体模型》长时间稳定，短时间则视情况而定
- 随机游走=代表者模型》随时间衰减
金融危机：趋势消失和高阶矩发散
道·琼斯工业指数，大萧条期间
variance
Skewness
kurtosis
5thorder cumulant
…
0.0196
0.4537
2.8378
2.5448
…
群体模型的期权定价模型
- 欧式看涨期权的定价公式
和Black-Scholes模型对比
- BS模型只含有二阶矩升水（隐含波动率），并且时间上的频率成分为0，可称为 模型
- 直接测度高阶矩的模型含有m阶升水，n个频率成分，可称为 阶模型
- 线性过程逼近的模型含有所有阶矩的升水，n个频率，可称为 阶模型
- 有效市场理论是 模型，相当于
和Black-Scholes模型对比
可以解释 和市场的偏离
结论
- 生灭过程考虑了群体行为，是真实市场更好的描述
- 套利的可能性取决于市场过程的线性程度，因此BS模型采取的线性随机过程不能更好的描述市场
- 可以构造定价模型 ，金融危机和BS模型是它的两个极端状况
国际金融的监管和改革
- 公允价值的问题：资产价格波动率》债务的真实价值波动》放大不稳定性
- 开卷考试补充问题：
- 计算自相关函数的 To
- 计算特征时间尺度 Tc = 4*To
- Tc 可用来作为资产估价时选择时间窗户的依据
HPc 的自相关函数与特征时间尺度 Tc=4*To 》资产估值时间窗户的参照
物理学与经济学的联系与异同
- 平衡态物理学的条件：孤立或封闭体系》能量守恒》动力学方程满足变分原理》变为优化问题》哈密顿经济学体系》新古典
- 三类物理问题：
- 波动方程：晶格点阵+时间的两阶导数》正弦或余弦函数
- 扩散（热传导）方程：连续性方程+热传导的一阶空间梯度》指数函数
- 化学反应方程》人口动力学方程（生态方程）》开放系统》非线性+非均衡+群体》复杂系统
- 问题：宏观、金融波动该用哪类方程？
经济学中的封闭系统=没有创新空间
- 需求曲线斜率向下的条件：
- 弗里德曼-张五常条件：真实收入不变=宏观没有非自愿失业
- 陈平条件：没有新产品=产品种类可区分=产品周期为无穷大（没有新陈代谢）
- 类似问题：产权界定：选择=选优？
- 自私是天生的？有无自私基因？利他行为的起源？
- 演化博弈论能回答生物学与经济学的基本问题，如劳动分工，家庭，市场，组织，国家的起源吗？
- 1维人（效用排序）与多维人》理性、感性、与社会性的矛盾与取舍》优化还是演化？
从变中寻找不变：标准-普尔指数为例
HP趋势》h-series
HP 波动》p-series
相对偏差》比波动率(volatility)稳定
波动率(volatility)
资产定价线性理论的问题
- 只有（用概率论）可预测的风险(risk)
- 没有（用概率论）不可预测的不确定性(uncertainty)
- 观察资产价格波动的短期行为（单位时间的变化率）》忽略中长期的趋势与波动
- 均值、方差不足以刻画系统行为》高阶矩的影响（在市场动荡时）不可忽略
- 无风险利率不存在》国债利率有通胀风险+汇率贬值风险+政府破产（拒付）风险
Class Information
- See Ping Chen’s CCER website
- For registered students, send email to: pchen@ccer.edu.cn （rarely has time to answer, better ask in class）
- Video on Great Depression:
- http://v.youku.com/v_show/id_XNjk1MDYxMzI=.html
- TA: 许富强，6275-4117
- Email：hitxfq@163.com
References
- Samuelson, P. A. "Rational Theory of Warrant Pricing," Industrial Management Review, 6(2), 13-31 (1965).
- Black, F. and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, 81, 637-654 (1973).
- Merton, Robert C. “Theory of Rational Option Pricing,” Bell Journal of Eco. and management Science, 4(1), 141-183 (1973).
- Merton, Robert C. “Applications of Option-Pricing Theory: 25 Years Later,” Amer. Eco. Rev. 88(3), 323-349 (1998).
- 陈平,文明分岔、经济混沌、和演化经济动力学，北京大学出版社，北京2004年出版。
- Chen, P. “Equilibrium Illusion, Economic Complexity, and Evolutionary Foundation of Economic Analysis,” Evolutionary and Institutional Economics Review, 5(1), 81-127 (2008).
