seminar 6-1 fincalc

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The fundamentals of The fundamentals of StochasticStochastic Financial Financial

MathematicsMathematics

Seminar 6Seminar 6

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Seminar 6Seminar 6..

Brownian motion Brownian motion

and the assumptions and the assumptions of of

the Black-Scholes the Black-Scholes modelmodel

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The Geometric Brownian Motion

In efficient markets, financial prices should display a random walk pattern.

More precisely, prices are assumed to follow a Markov process, which is a particular stochastic process independent of its history — the entire distribution of the future price relies on the current price only.

The past is irrelevant.

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These processes are built from the following components, described in order of increasing complexity: The Wiener process The generalized Wiener process The Ito process

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A particular example of Ito process is the geometric Brownian motion (GBM), which is described for the variable S as

The process is geometric because the trend and volatility terms are proportional to the current value of S.

This is typically the case for stock prices, for which rates of returns appear to be more stationary than raw dollar returns,

It is also used for currencies

S S t S z

S

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Because represents the capital appreciation only, abstracting from dividend payments, μ represents the expected total rate of return on the asset minus the rate of income payment, or dividend yield in the case of stocks.

S S

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Example 1: A Stock Price Process

Consider a stock that pays no dividends, has an expected return of10%per annum, and volatility of 20% per annum.

If the current price is $100, what is the process for the change in the stock price over the next week?

What if the current price is $10?

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Example 1: Solution (A Stock Price Process)

The process for the stock price is

where is a random draw from a standard normal distribution.

If the interval is one week, or

the mean is

and

S S t t

1 52 0.01923t

0.10 1 52 0.001923t

0.20 1 52 0.027735t

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Example 1: Solution (A Stock Price Process)

The process is

With an initial stock price at $100, this gives

With an initial stock price at $10, this gives

The trend and volatility are scaled down by a factor

of ten

0.1923 2.7735S

0.01923 0.27735S

$100 0.001923 0.027735S

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This model is particularly important because it is the

underlying process forthe Black-Scholes formula

The key feature of this distribution is the fact that the volatility is proportional to S.

This ensures that the stock price will stay positive.

Indeed, as the stock price falls, its variance decreases, which makes it unlikely to experience a large down move that would push the price into negative values

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As the limit of this model is a normal distribution for dS/S = d

ln(S), S follows a lognormal

distribution. This process implies that, over an interval

the logarithm of the ending price is distributed as

where is a standardized normal variable.

T t

2

ln ln2TS S

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Example: A Stock Price Process (Continued)

Assume the price in one week is given by S = $100exp(R), where R has annual expected value of 10% and volatility of 20%.

Construct a 95% confidence interval for S

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Example: A Stock Price Process (Continued)

The standard normal deviates that corresponds to a 95% confidence interval are αmin = −1.96 and αmax

= 1.96.

In other words, we have 2.5% in each tail. The 95% confidence band for R is then

and

This gives Smin = $100exp(−0.0524) = $94.89,

and Smax = $100exp(0.0563) = $105.79.

min 1.96

0.001923 1.96 0.027735 0.0524

R t t

max 1.96

0.001923 1.96 0.027735 0.0563

R t t

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The choice lognormal & normal assumption

Whether a lognormal distribution is much better than the normal distribution depends on the horizon considered. If the horizon is one day only, the choice of the lognormal versus normal assumption does not really matter.

It is highly unlikely that the stock price would drop below zero in one day, given typical volatilities.

On the other hand, if the horizon is measured in years, the two assumptions do lead to different results.

The lognormal distribution is more realistic, as it prevents prices from turning negative.

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In simulations:

this process is approximated by small steps with a normal distribution with mean and variance given by

To simulate the future price path for S, we start from the current price St and generate a sequence of independent standard normal variables , for i = 1, 2, . . . , n.

2,S

N t tS

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In simulations: The next price is built as

The following price is taken as

and so on until we reach the target horizon, at which point the price should have

a distribution close to the lognormal.

1 1t t tS S S t t

2 1 1 2t t tS S S t t 2tS

1tS

t n TS S

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Simulating a Price Path

http://www.mathworks.com/matlabcentral/fx_files/28056/5/content/html/ModelNGPrice_01.png

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http://www.mathworks.com/matlabcentral/fx_files/28056/5/content/html/ModelNGPrice_02.png

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Shortcomings of the model

While very useful to model stock prices, this model has shortcomings.

Price increments are assumed to have a normal distribution.

In practice, we observe that price changes have fatter tails than the normal distribution.

Returns may also experience changing variances.

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Shortcomings of the model

In addition, as the time interval shrinks, the volatility shrinks as well.

This implies that large discontinuities cannot occur over short intervals.

In reality, some assets experience discrete jumps, such as commodities. The stochastic process, therefore, may have to be changed to accommodate these observations

t

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1. A fundamental assumption of the random walk hypothesis of market

returns is that returns from one time period to the next are statistically

independent. This assumption implies

a. Returns from one time period to the next can never be equal.

b. Returns from one time period to the next are uncorrelated.

c. Knowledge of the returns from one time period does not help in predicting returns from the next time period.

d. Both b) and c) are true.

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2. Consider a stock with daily returns that follow a random

walk. The annualized volatility is

34%. Estimate the weekly volatility of this

stock assuming that the year has 52 weeks. a. 6.80% b. 5.83% c. 4.85% d. 4.71%

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3. Assume an asset price variance increases linearly with time. Suppose

the expectedasset price volatility for the next two

months is 15% (annualized), and for the one month that follows, the expected

volatility is 35% (annualized). What is the average expected volatility

over the next three months? a. 22% b. 24% c. 25% d. 35%

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44. In the geometric Brown . In the geometric Brown motion process for a variable motion process for a variable

SS,, I. I. S is normally distributed.is normally distributed. II. II. d ln(S ) is normally distributed. is normally distributed. III. III. dS/S is normally distributed. is normally distributed. IV. IV. S is lognormally distributed.is lognormally distributed.

a. a. I onlyI only b. b. II, III, and IVII, III, and IV c. c. IV onlyIV only d. d. III and IV III and IV

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55. Consider that a stock price . Consider that a stock price S S that that follows a geometric Brownian motion follows a geometric Brownian motion dS =

aSdt + bSdz, , with with b strictly positivestrictly positive

Which of the following statements is false?

a. If the drift a is positive, the price one year from now will be above today’s price.

b. The instantaneous rate of return on the stock follows a normal distribution.

c. The stock price S follows a lognormal distribution.

d. This model does not impose mean reversion

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66. If . If follows a geometric Brownian follows a geometric Brownian motion and motion and follows a geometric follows a geometric

Brownian motionBrownian motion

which of the following is which of the following is truetrue?? a. a. ln(ln(SS1 + 1 + SS2) is normally distributed.2) is normally distributed. b. b. SS1 × 1 × SS2 is lognormally distributed.2 is lognormally distributed. c. c. SS1 × 1 × SS2 is normally distributed.2 is normally distributed. d. d. SS1 + 1 + SS2 is normally distributed2 is normally distributed

2S1S

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77. Which of the following statements . Which of the following statements best characterizes the relationship best characterizes the relationship

betweenbetweenthe normal and lognormal the normal and lognormal

distributions?distributions? a. a. The lognormal distribution is the logarithm of the The lognormal distribution is the logarithm of the

normal distribution.normal distribution. b. b. If the natural log of the random variable If the natural log of the random variable X X is is

lognormally distributed, then lognormally distributed, then X X is normally is normally distributed.distributed.

c. c. If If X X is lognormally distributed, then the natural is lognormally distributed, then the natural log of log of x x is normally is normally distributed.distributed.

d. d. The two distributions have nothing to do with The two distributions have nothing to do with one anotherone another

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88. . Consider a stock with an initial price of Consider a stock with an initial price of $100. $100.

Its price one year from now is given by Its price one year from now is given by S S = 100 × exp(= 100 × exp(r r )),,

The rate of return The rate of return r r is normally distributed is normally distributed with a mean of 0.1 and a standard deviation with a mean of 0.1 and a standard deviation of 0.2. of 0.2.

With 95% confidence, after rounding, With 95% confidence, after rounding, S S will will be betweenbe between a. a. $67.57 and $147.99$67.57 and $147.99 b. b. $70.80 and $149.20$70.80 and $149.20 c. c. $74.68 and $163.56$74.68 and $163.56 d. d. $102.18 and $119.53$102.18 and $119.53

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99. Which of the following . Which of the following statements are statements are truetrue??

I. The sum of two random normal variables is also I. The sum of two random normal variables is also a random normal variable.a random normal variable.

II. The product of two random normal variables is II. The product of two random normal variables is also a random normal variable.also a random normal variable.

III. The sum of two random lognormal variables is III. The sum of two random lognormal variables is also a random lognormal variable.also a random lognormal variable.

IV. The product of two random lognormal IV. The product of two random lognormal variables is also a random lognormal variable.variables is also a random lognormal variable. a. a. I and II onlyI and II only b. b. II and III onlyII and III only c. c. III and IV onlyIII and IV only d. d. I and IV only I and IV only

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1010. For a lognormal variable . For a lognormal variable XX, we know , we know that ln(that ln(XX) has a normal distribution) has a normal distributionwith a mean of zero and a standard with a mean of zero and a standard

deviation of 0.5deviation of 0.5

What are the expected value and the What are the expected value and the variance of variance of XX?? a. a. 1.025 and 0.1871.025 and 0.187 b. b. 1.126 and 0.2171.126 and 0.217 c. c. 1.133 and 0.3651.133 and 0.365 d. d. 1.203 and 0.3991.203 and 0.399

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Question 1Question 1

What would it mean to assert that What would it mean to assert that the temperature at a certain place the temperature at a certain place follows a Markov process? Do you follows a Markov process? Do you think that temperatures do, in fact, think that temperatures do, in fact, follow a Markov process? follow a Markov process?

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Question 1. SolutionQuestion 1. Solution Imagine that you have to forecast the future Imagine that you have to forecast the future

temperature fromtemperature from a) the current temperature, a) the current temperature, b) the history of the temperature in the last week, and b) the history of the temperature in the last week, and c) a knowledge of seasonal averages and seasonal trends. c) a knowledge of seasonal averages and seasonal trends.

If temperature followed a Markov process, the If temperature followed a Markov process, the history of the temperature in the last week would history of the temperature in the last week would be irrelevant. be irrelevant.

To answer the second part of the question you To answer the second part of the question you might like to consider the following scenario for might like to consider the following scenario for the first week in May: the first week in May: (i) Monday to Thursday are warm days; today, (i) Monday to Thursday are warm days; today,

Friday, is a very cold day. Friday, is a very cold day. (ii) Monday to Friday are all very cold days. (ii) Monday to Friday are all very cold days.

What is your forecast for the weekend? If you are What is your forecast for the weekend? If you are more pessimistic in the case of the second more pessimistic in the case of the second scenario, temperatures do not follow a Markov scenario, temperatures do not follow a Markov process. process.

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Can a trading rule based on the past Can a trading rule based on the past history of a stock's price ever history of a stock's price ever produce returns that are consistently produce returns that are consistently above average? Discussabove average? Discuss

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Question 2. SolutionQuestion 2. Solution

The first point to make is that any trading The first point to make is that any trading strategy can, just because of good luck, strategy can, just because of good luck, produce above average returns. produce above average returns.

The key question is The key question is whether a trading whether a trading strategy consistently outperforms the strategy consistently outperforms the market when adjustments are made for riskmarket when adjustments are made for risk. .

It is certainly possible that a trading strategy It is certainly possible that a trading strategy could do this. could do this.

However, when enough investors know However, when enough investors know about the strategy and trade on the basis of about the strategy and trade on the basis of the strategy, the profit will disappear. the strategy, the profit will disappear.

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Question 2. SolutionQuestion 2. Solution As an illustration of this, consider a As an illustration of this, consider a

phenomenon known as phenomenon known as the small firm the small firm effecteffect..

Portfolios of stocks in small firms appear to have Portfolios of stocks in small firms appear to have outperformed portfolios of stocks in large firms outperformed portfolios of stocks in large firms when appropriate adjustments are made for riskwhen appropriate adjustments are made for risk..

Papers were published about this in the early Papers were published about this in the early 1980s and mutual funds were set up to take 1980s and mutual funds were set up to take advantage of the phenomenonadvantage of the phenomenon..

There is some evidence that this has resulted in There is some evidence that this has resulted in the phenomenon disappearingthe phenomenon disappearing..

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A company's cash position (in millions of A company's cash position (in millions of dollars) follows a generalized Wiener dollars) follows a generalized Wiener process withprocess with a drift rate of 0.5 per quarter a drift rate of 0.5 per quarter a variance rate of 4.0 per quarter. a variance rate of 4.0 per quarter.

How high does the company's initial cash How high does the company's initial cash position have to be for the company to position have to be for the company to have a less than 5% chance of a negative have a less than 5% chance of a negative cash position by the end of one year? cash position by the end of one year?

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Suppose that the company's initial cash Suppose that the company's initial cash position is x. position is x.

The probability distribution of the cash The probability distribution of the cash position at the end of one year is position at the end of one year is

where is a normal probability where is a normal probability distribution with mean and standard distribution with mean and standard deviation . deviation .

4 0.5, 4 4 2.0, 4x x

,

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Question 3. SolutionQuestion 3. Solution The probability of a negative cash position The probability of a negative cash position

at the end of one year isat the end of one year is

where is the cumulative probability where is the cumulative probability that a standardized normal variable (with that a standardized normal variable (with mean zero and standard deviation 1.0) is mean zero and standard deviation 1.0) is less than x. less than x.

N x

2.0

4

xN

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From normal distribution tablesFrom normal distribution tables

when: when:

i.e., when x = 4.5796. i.e., when x = 4.5796. The initial cash position must therefore be $4.56 The initial cash position must therefore be $4.56

million. million.

2.01.6449

4

x

2.00.05

4

xN

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Consider a variable S that follows the process Consider a variable S that follows the process

For the first three years, and ;For the first three years, and ;

for the next three years, and .for the next three years, and .

If the initial value of the variable is 5, what is If the initial value of the variable is 5, what is the probability distribution of the value of the the probability distribution of the value of the variable at the end of year 6? variable at the end of year 6?

dS dt dz

2 3 3 4

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The change in S during the first three The change in S during the first three years has the probability distributionyears has the probability distribution

The change during the next three years The change during the next three years has the probability distribution has the probability distribution

2 3, 3 3 6, 5.20

3 3, 4 3 9, 6.93

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Question 4. SolutionQuestion 4. Solution The change during the six years is the sum The change during the six years is the sum

of:of: a variable with probability distribution a variable with probability distribution and a variable with probability distribution and a variable with probability distribution

The probability distribution of the change The probability distribution of the change is therefore is therefore

Since the initial value of the variable is 5, Since the initial value of the variable is 5, the probability distribution of the value of the probability distribution of the value of the variable at the end of year six is the variable at the end of year six is

6, 5.20

9, 6.93

20, 8.66

2 26 9, 5.20 6.93 15, 8.66

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Stock A and stock Stock A and stock ВВ both follow both follow geometric Brownian motion. Changes geometric Brownian motion. Changes in any short interval of time are in any short interval of time are uncorrelated with each other. uncorrelated with each other.

Does the value of a portfolio Does the value of a portfolio consisting of one of stock A and one consisting of one of stock A and one of stock of stock ВВ follow geometric Brownian follow geometric Brownian motion? Explain your answer. motion? Explain your answer.

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