第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model

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1 1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American Option Prices

2 Black-Scholes 期 权 定 价 理 论 The two papers by B-S (The pricing of options and corporate liabilities, JPE, 1973) and Merton (The theory of rational option pricing, BEMS, 1973) provide the first analytical framework for option price and option hedging. Option pricing techniques are often considered among the most mathematically complex of all applied areas of finance. Scholes and Merton were awarded Nobel prize for economics in 1997.

3 3 Black-Scholes 随 机 漫 步 假 设 The basic assumption under B-S model is that stock prices follow random walk. A Markov stochastic process ( 马 尔 科 夫 过 程 )is a description of random walk. B-S 模 型 的 基 本 假 设 是 股 票 价 格 服 从 随 机 游 走 马 尔 科 夫 过 程 是 对 随 机 游 走 的 一 种 描 述 Consider a stock whose price is S In a short ds time of length dt, the change in the stock price S is normal with mean mdt and standard deviation s dt m is the expected return and s is volatility per annum

4 4 Random Walk 随 机 漫 步 A random walk is defined as the one in which future steps or directions cannot be predicted on the basis of past actions Burton Malkiel A Random Walk Down Wall Streets ( 漫 步 华 尔 街 ) 随 机 游 走 表 示 未 来 的 路 径 和 方 向 不 能 通 过 过 去 的 行 动 进 行 预 测 In financial markets, short run changes in stock prices cannot be predicted.

5 5 Cont. 几 何 布 朗 运 动 Geometric Brownian Motion (continuous time) ds msdt s Sdz or ds S mdt sdz ds is the change in stock price over a small interval dt; dz dt and is a random drawing from a standardized normal distribution. The above process is also called a diffusion process.

6 6 Cont. Geometric Brownian Motion (continuous time) ds msdt s Sdz or ds is the change in stock price over a small interval dt; dt and is a random drawing from a standardized normal distribution. dz ds S mdt sdz A Wiener process, with mean of 0, and st dev. of dt The above process is also called diffusion process ( 扩 散 过 程 ).

7 7 Example Consider a stock that pays no dividends, has a volatility of 30% per annum, and provides an expected return of 15% per annum with continuous compounding. The initial price is $100. The distribution of the price in one week is ds S 0.15dt 0.30 dt ds=100( ) or ds=

8 8 The Lognormal Property( 对 数 正 态 ) Random walk implies that S T is lognormally distributed, and lns T is normally distributed. ln S T is normally distributed with mean: lns ( m - / )T 0 s and standard deviation: s T

9 9 The Lognormal Distribution

10 10 The Lognormal Property ln or S T ] m - s s ln S 0 ( ) T, T ln S S T 0 ] m - s s ( ) T, T where m,s] is a normal distribution with mean m and standard deviation s

11 11 An Example A stock with a price of $40, an expected return of 16% per annum and a volatility of 0% per annum. What is the possible price over next 6 months. The distribution of stock price S T in 6 months is 0. lnst ~ [ln 40 ( )0.5,0. 0.5] or ln S ~ (3.795,0.14) T With 95% confidence, lns T is *0.141 < lns T < *0.141 For the stock price e < S T < e or 3.55 < S T < What is the mean stock price?

12 1 Another Example A stock with an expected return of 17% per annum and a volatility of 0% per annum. What is the mean return and standard deviation over one year time? What is the return distribution with 95% confidence level? The mean return = / = Standard deviation is # With 95% confidence, the return is -4.% < u < 54.%.

13 13 波 动 性 Volatility The volatility is the standard deviation of the continuously compounded rate of return in 1 year 波 动 性 是 指 1 年 期 连 续 复 利 收 益 的 标 准 差 The standard deviation of the return in time dt is s dt If a stock price is $50 and its volatility is 30% per year what is the standard deviation of the price change in one week?

14 Estimating Volatility from Historical Data( 从 历 史 数 据 中 估 计 波 动 性 ) Take observations S 0, S 1,..., S n at the interval of t years. Define the continuously compounded return as: S i u i ln S i Calculate the standard deviation, s, of the u i s s 1 n ( u i - u) n -1 i1.

15 15 Volatility Estimation( 波 动 性 的 估 计 ) 4. The historical volatility estimate is: sˆ s t Given the data, the volatility can be calculated easily using Excel. Time varying volatility can be estimated using statistical softwares.

16 16 Black-Scholes 定 价 理 论 的 基 本 原 理 The option price and the stock price depend on the same underlying source of uncertainty( 期 权 和 股 票 价 格 依 赖 于 同 一 种 不 确 定 性 ) We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty 可 以 构 造 一 个 股 票 和 期 权 的 组 合 来 消 除 这 种 不 确 定 性 The portfolio is instantaneously riskless and must earn the riskfree rate 因 此 这 个 组 合 是 无 风 险 的 且 其 收 益 应 等 于 无 风 险 利 率 The B-S approaches the problem by considering a short period of time. Thus, B-S formula corresponds to a situation where stock trade continuously during the day. 要 求 股 票 连 续 交 易

17 17 An Analogy The B-S model is related to a binomial model ( 二 叉 树 模 型 )with a large number of intervals. How continuous time process is linked to a binomial tree. Consider the following analogy: B-S 模 型 与 一 个 划 分 了 大 量 阶 段 的 二 叉 树 模 型 本 质 上 是 类 似 的 Consider an old fashion film An action is achieved by moving through the frames quickly The time for a frame is the length of a period in the binomial model What B-S modeled is a world where the frames go by so quickly that you see a movie rather than individual frames.

18 18

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22 The B-S Arguments Consider an arbitrage portfolio consisting of 1 option and units of the underlying asset. For simplicity, consider a European call, price c on a share, price S. Arbitrage portfolio value is V = c + S. is chosen so that V is independent of the asset price S. The arbitrage portfolio is the riskless and must earn risk free rate r. Consider a short interval, solve the diffusion equation to get the Black-Scholes prices. d( c S) dt r

23 连 续 时 间 随 机 过 程 Continuous Time Stochastic Process A variable z follows a Wiener process ( 维 纳 过 程 )if it has the following properties: The change of z during a short period of time dt 3 The values of dz for any two different short intervals of time dt are independent. Generalized Wiener process dx = adt+bdz dz dt where a and b are constant, adt denotes that x has an expected drift rate of a per unit of time, and dz is a basic Wiener The Lecture process. 8: B-S-M Model

24 4 Ito Process( 伊 藤 过 程 ) It is a generalized Wiener process ( 维 纳 过 程 ) where the parameters a and b are functions of the value of the underlying variable, x, and t. dx = a(x,t)dt +b(x,t) dz Ito s Lemma given that ds =usdt + ssdz If a variable G (a function of S and t), then dg G G G 1 ms s S ) dt S t S G ssdz S

25 5 The Black-Scholes Formulas The price of a call is f=f(s,t). From Ito s lemma( 伊 藤 定 理 ), we have df f f f 1 ms s S ) dt S t S f ssdz S Form a portfolio to eliminate the Wiener process in the equation as -1: derivative f S shares

26 6 Cont. The value of the portfolio is - f f S The change of value over dt is S We have -df Note: this equation does not contain dz, the portfolio must be riskless( 无 风 险 ). - f S f t ds - 1 S f s S t

27 7 Cont. Because the portfolio is risk free, we have =rt. Substituting previous equations gives The last equation is the Black-Scholes-Merton differential equation. These authors solved the equation under boundary conditions. rf S f S t f rs t f that so t S S f f r t S S f t f - 1, 1 s s

28 8 The Black-Scholes Formulas(B-S 公 式 ) -rt c S N( d ) - X e N( d ) 0 1 -rt p X e N( -d ) - S N( -d ) where d 0 1 ln( S X r 0 / ) ( s / ) T d 1 s T ln( S X r - 0 / ) ( s / ) T d - s 1 s T T

29 9 The N(x) Function N(x) is the cumulative probability function for a standard normal variable. It is the probability that a variable with a mean of zero and a standard deviation of 1 is less than x See tables at the end of the book

30 30 B-S 模 型 假 设 Assumptions Used The stock pays no dividends during the option's life 不 支 付 股 利 European exercise terms are used 欧 式 期 权 Markets are efficient 市 场 有 效 No commissions are charged 无 交 易 费 用 Interest rates remain constant and known 利 率 已 知 且 不 变 Stock prices are lognormally distributed 股 票 价 格 服 从 对 数 正 态 分 布

31 31 An Example X, Inc s price is $75.5 at the end of Fed 000. July call options, with 18 weeks to maturity, traded with strikes of $750 and $800. No dividend was expected. The risk free rate is 7.41%. Volatility is 37.5%. The price of the 750 call is $ S 75.5 d N(d1) X 750 d 0.06 N(d) R 7.41% s 37.5% c T What is c for X=800?

32 3 Cont. What is the call price for X=800? d1= [ln(75.5/800)+( ^)80.346]/[0.375*sqrt(0.346)] = S=75.5 d N(d1) X=800 d N(d) R=7.41% s=37.5% c T=0.346

33 33 Graph of B-S Model

34 34 Black-Scholes Formula 属 性 As S 0 becomes very large c tends to S Xe -rt and p tends to zero As S 0 becomes very small c tends to zero and p tends to Xe -rt S

35 Call Price and Maturity in the B-S Model( 看 涨 期 权 价 格 与 期 限 的 关 系 ) The following 3 graphs show the impact of deminishing time remaining on a call with: S = $48; X = $50; r = 6%; sigma = 40% 35

36 36 Cont.

37 B-S Call Price and Volatility( 看 涨 期 权 价 格 与 波 动 性 的 关 系 ) S = $48; X = $50; r = 6% 37

38 38 Cont.

39 39 Option Price Calculator A lot of online B-S option calculators use an adjusted Black- Scholes model to value European options. The model is adjusted to take into account dividends paid on the underlying security. The calculators can in fact be used for: stock options index options currency options options on futures. Availability: for example

40 40 风 险 中 性 定 价 原 则 Risk-Neutral Valuation The variable m does not appear in the Black- Scholes equation The equation is independent of all variables affected by risk preference This is consistent with the risk-neutral valuation principle, which is. Noted: risk-neutral valuation does not state that investors are risk neutral. ( 风 险 中 性 定 价 并 不 代 表 投 资 者 是 风 险 中 性 的 )

41 41 隐 性 波 动 性 Implied Volatility The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price 隐 性 波 动 性 是 指 Black-Scholes 模 型 所 计 算 出 来 的 价 格 等 于 市 场 价 格 时, 公 式 中 所 暗 含 的 股 票 波 动 性 The volatility is the market s expectation of the volatility prevailing over the period of time assuming the B-S model is a correct model. The is a one-to-one correspondence( 一 一 对 应 ) between option price and implied volatility Traders and brokers often quote implied volatility rather than dollar price

42 4 Implied Volatility( 隐 性 波 动 性 ) There are many option prices available from which we obtain an implied volatility. Different options on the same stock can have different implied volatilities. A simple way is to compute equal weighted average of these volatilities (eliminate deep outof-money and deep in-the-money options) Alternatively, one can find the implied volatility that minimizes the absolute deviations of option prices from the B-S price.

43 43 波 动 性 属 性 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 Volatility per annum = Volatility per day * Sqrt(Number of trading days per annum) Market practice takes 1 trading days per month and 5 days per year.

44 44 现 金 红 利 与 B-S 模 型 European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into the Black-Scholes formula Only dividends with ex-dividend dates during life of option should be included The dividend should be the expected reduction in the stock price

45 45 Example A European call with ex-dividend dates in two months and five months. each dividend is expected to be $0.50. S=$40; X=$40; sigma = 30% per year; r=9% per year; T=6 m. What is option price? The PV of dividends = 0.5e -0.09*/ e -0.09*5/1 = The stock price used is = Based on the B-S model, d1 = 0.017; d = This corresponds to N(d1) = 0.580; N(d) = The call option price = * e -0.09*0.5 * = 3.67.

46 46 B-S 模 型 与 美 式 看 涨 期 权 定 价 An American call on a non-dividend-paying stock should never be exercised early An American call on a dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date

47 Black s Approach to Dealing with Dividends in American Call Options( 用 B S 公 式 计 算 带 红 利 的 美 式 看 涨 期 权 价 格 ) 47 Set the American option price equal to the maximum of two European option prices: 1. The 1st European price is for an option maturing at the same time as the American option. The nd European price is for an option maturing prior to the latest ex-dividend date The American option price is set equal to the higher of these two European option prices.( 美 式 期 权 价 格 被 设 定 等 于 两 个 欧 式 期 权 中 较 高 的 那 个 )

48 48 Example The same example as before but the option is an American option. What is the option price? The PV of the first dividend = The price of the option that expires just before the final ex-dividend date is: S= ; X=40; sigma=0.30; T= c = 3.5. If the option expires in 6 months, c=3.67. Black s approximation gives the value of American call option $3.67.

49 49 B-S 模 型 与 美 式 看 跌 期 权 定 价 Dividends make an American put less likely to be exercised early. If dividends exceed a certain level, it is never optimal to exercise the option early. This is, D i -r 1 X[ 1- e ( t i -ti ]. In other cases, early exercise is optimal in which case numerical procedure must be used to value the options.

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