L11 期权定价：B-S-M模型

Outlines

Wiener Processes and Itô's Lemma

The Black-Scholes-Merton Model

Option Sensitivities: Greeks

Wiener Processes and Itô's Lemma

Model the Dynamics of Stock Prices / Returns

ABC of Stochastic Processes

• Stochastic Processes

  • Describes the way in which a variable such as a stock price, exchange rate or interest rate changes through time
  • Incorporates uncertainties

• Example of Stochastic Processes

  • Each day a stock price: increases by $1 with probability 30, stays the same with probability 50%, reduces by $1 with probability 20%
  • Each day a stock price change is drawn from a normal distribution with mean $0.2 and standard deviation $1

• Markov Processes

  • In a Markov process future movements in a variable depend only on where we are, not the history of how we got to where we are
  • Is the process followed by the temperature at a certain place Markov?
  • We assume that stock prices follow Markov processes

• Weak-Form Market Efficiency

  • This asserts that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices. In other words technical analysis does not work.
  • A Markov process for stock prices is consistent with weak-form market efficiency

Variances & Standard Deviations

• In Markov processes changes in successive periods of time are independent

• This means that variances are additive

• Standard deviations are not additive

• In our example it is correct to say that the variance is 100 per year.

• It is strictly speaking not correct to say that the standard deviation is 10 per year.

A Wiener Process

• Define as a normal distribution with mean and variance

• A variable follows a Wiener process if

  • The change in in a small interval of time is
  • where
  • The values of for any different (non-overlapping) periods of time are independent

• Properties of a Wiener Process

  • Mean of is
  • Variance of is
  • Standard deviation of is

Norbert Wiener (诺伯特维纳)

Brownian Motion

Louis Bachelier (路易斯巴舍利耶)

Generalized Wiener Processes

• A Wiener process has a drift rate (i.e. average change per unit time) of and a variance rate of

• In a generalized Wiener process the drift rate and the variance rate can be set equal to any chosen constants

  • Mean change in per unit time is
  • Variance of change in per unit time is

• Taking Limits

  • What does an expression involving and mean?
  • It should be interpreted as meaning that the corresponding expression involving and is true in the limit as tends to zero
  • In this respect, stochastic calculus is analogous to ordinary calculus

Generalized Wiener Process

A Model for Stock Prices

• Itô Process
  • In an Itô process the drift rate and the variance rate are functions of time

• The discrete time equivalent

is true in the limit as tends to zero

• A generalized Wiener process is not appropriate for stocks

  • For a stock price we can conjecture that its expected percentage change in a short period of time remains constant (not its expected actual change)
  • We can also conjecture that our uncertainty as to the size of future stock price movements is proportional to the level of the stock price

• Geometric Brownian Motion

Itô's lemma

Kiyosi Itô (伊藤清)

Itô's lemma

• If we know the stochastic process followed by , Itô's lemma tells us the stochastic process followed by some function . When then

• Since a derivative is a function of the price of the underlying asset and time, Itô's lemma plays an important part in the analysis of derivatives

• Applications of Itô's Lemma to A Stock Price Process

  • Suppose the stock price process is $$dS=\mu Sdt+\sigma Sdz$$
  • For a function of and we have

The Black-Scholes-Merton Model

The Stock Price Assumption

• Consider a stock whose price is
• In a short period of time of length , the return on the stock is normally distributed:

• So the price is lognormal distributed

or

Continuously Compounded Return

• If is the realized continuously compounded return we have

• The Expected Return

  • The expected value of the stock price is
  • The expected return on the stock is not
  • The reason is that, and are not the same

• vs.

  • is the expected return in a very short time, , expressed with a compounding frequency of
  • is the expected return in a long period of time expressed with continuous compounding (or, to a good approximation, with a compounding frequency of )

Volatility

• The Volatility
  • The volatility is the standard deviation of the continuously compounded rate of return in year
  • The standard deviation of the return in a short time period time is approximately
  • If a stock price is $50 and its volatility is 25% per year what is the standard deviation of the price change in one day?

• Estimating Volatility
  • Take observations at intervals of years (e.g. for weekly data )
  • Calculate the continuously compounded return in each interval as:
  • Calculate the standard deviation, , of the 's
  • The historical volatility estimate is:

• Nature of Volatility
  • Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed
  • For this reason time is usually measured in "trading days" not calendar days when options are valued
  • It is assumed that there are trading days in one year for most assets

The Black-Scholes-Merton Model

Black-Scholes-Merton: The Big Idea

• The option price and the stock price depend on the same underlying source of uncertainty

• We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty

• The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate

• This leads to the Black-Scholes-Merton differential equation

The Derivation of the Black-Scholes-Merton Differential Equation

• Stock price & derivative price

• Set up a portfolio

  • derivative:
  • shares:

• The value of the portfolio

Since the portfolio is risk-free, we have

So, the Black-Scholes-Merton Differential Equation is

• Any security whose price is dependent on the stock price satisfies the differential equation
• The particular security being valued is determined by the boundary conditions of the differential equation
• In a forward contract the boundary condition is when
• The solution to the equation is

Black-Scholes-Merton Pricing for Options

where

• Properties of Black-Scholes Formula
  • As becomes very large tends to and p tends to zero
  • As becomes very small tends to zero and tends to
  • What happens as becomes very large?
  • What happens as becomes very large?

Understanding Black-Scholes

The formula

• : Discount rate
• : Probability of exercise
• : Expected percentage increase in stock price if option is exercised
• : Strike price paid if option is exercised

Risk-Neutral Valuation

Risk-Neutral Valuation

• Risk-Neutral Valuation

  • The variable does not appear in the Black-Scholes-Merton differential equation
  • The equation is independent of all variables affected by risk preference
  • The solution to the differential equation is therefore the same in a risk-free world as it is in the real world
  • This leads to the principle of risk-neutral valuation

• Applying Risk-Neutral Valuation

  • [1] Assume that the expected return from the stock price is the risk-free rate
  • [2] Calculate the expected payoff from the option
  • [3] Discount at the risk-free rate

Option Sensitivities: Greeks

Greeks

The Taylor Expansion

• Delta:
• Gamma:
• Vega:
• Rho:
• Theta:

认股权证与可转换债券

认股权证与可转换债券

• 认股权证(Warrants) 是公司以自己的股票作为标的发行的长期看涨期权

  • 认股权证通常在发行债券的时候被创造出来
  • 认股权证通常与债券分开交易
  • 当认股权证被执行时，发行人对持有人结清
  • 当认股权证被执行时，公司获得现金同时发行新的股票

• 可转换债券(Convertible bonds) 是公司发行的，在特定时间可以按照事先确定的转换比率转换为股票的债券

  • 可转换债券等价于普通债券加认股权证
  • 可转换债券使公司可以以较低的票息率发行债务

定价

认股权证和可转债可以用标准的期权定价模型定价，我们只需将摊薄效应(dilution effect)考虑在内。假设公司现有股股票和份认股权证，每份权证允许持有人以协议价格购买股股票。在期初包括股票和认股权证的公司价值为：

在新股摊薄之后：

简化后，我们得到：

这等于份股票期权的价值：

课后阅读与练习

课后阅读与练习

• 课后阅读：教材第十一章第三节、十二章第一节、第六章、第七章相关内容

• 练习

  • 教材pp186：7，10
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