550.444 Modeling and Analysis Securities and Markets

1
550.444Modeling and AnalysisSecurities and
Markets

• Week of November 17, 2008
• Modeling the Stochastic Process for Derivative
  Analysis

2
Where we are

• Last Week Modeling the Stochastic Process for
  Derivative Analysis (Chapter 12, OFOD)
• This Week Modeling the Stochastic Process for
  Derivative Analysis (Chapter 12-13, OFOD)
• Next Week Black-Scholes-Merton Model for Options
  (Chapter 13, OFOD)
• Final Exam Thursday, Dec 18th 9-12pm

3
Assignment

• For This Week (Nov 17)
• Read Hull Chapter 12 13
• Problems
• Chapter 12 3,5,9,1112
• Chapter 13 3,11,2328
• Due Nov 21st

4
Plan for Today

• Modeling the Stochastic Process for Derivative
  Analysis
• Ito Process and Itos Lemma
• Stock Price Process
• Lognormal Property of Stock Price Process
• Expected Return
• Whats Ahead Black-Scholes-Merton Model and
  Option Pricing (13 -14)

5
Itôs Lemma

• If we know the stochastic process followed by x,
  Itôs lemma tells us about the stochastic process
  followed by some function G (x, t )
• Since a derivative security is a function of the
  price of the underlying and time, Itôs lemma
  plays an important part in the analysis of
  derivative securities

6
Application of Itos Lemmato a Stock Price
Process

• When the stock price process is
• Then for a function G of S and t Itos Lemma
  gives that

7
Applications of Itos Lemma

• For the forward contract of term T on a
  non-dividend paying stock
• Or more generally
• We can use Itos Lemma to determine the process
  for F where the process for S is
• and we find
• So
• Like S, the forward price process F follows an
  Ito Process, but has growth rate µ r rather
  than µ
• The growth rate in F is the excess return of S
  over the risk-free rate

8
Applications of Itos Lemma

• Suppose G ln S where S is the stock price
  process,
• then
• an generalized Wiener process
• Thus, the stock price process is a lognormal
  process
• If a variable has a process whose natural
  logarithm is normally distributed, the process is
  said to be a lognormal process
• Clearly this is so for our stock process (as we
  will see next)

9
The Stock Price Assumption

• Consider a stock whose price is S and (as we have
  seen) follows the Ito Process,
• dS µS dt sS dz
• where m is expected return and s is volatility
  (per year)
• In a short period of time of length Dt, the
  return on the stock can be characterized as
• with e f (0,1), the generalized normal
  distribution
• So it follows that ?S/S is normally distributed
  as well

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Lognormal Property of Stock Prices

• From Itos Lemma, letting G ln S, gives
• So the change in ln S between 0 and some future T
  is normally distributed with mean (µ - s2/2)T
  and variance s2T
• Which we can express as
• or
• Since ln ST is normal, ST is lognormally
  distributed

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Lognormal Property of Stock Prices

• A lognormal variable can take any value from 0 to
  8 and has distribution
• From our expression for ln ST we can show that
• with a straightforward exercise in integrating
  the pdf of a normal distribution

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Lognormal Property of Stock Prices

• If we define the continuously compounded rate of
  return per annum realized between 0 and T as x ,
  then
• or
• and as a consequence of the lognormal property
  of price
• So the realized cc rate of return per annum is
  normally distributed
• As T increases, the std dev of x declines
• We are more certain about the average rate of
  return, the longer the period, T

13
The Expected Returns m and m-s2/2

• Suppose we have daily data for a period of
  several months
• S0, S1 , S2 ,
• m is the average of the returns in each day
  E(DS/S)
• m-s2/2 is the expected return over the whole
  period covered by the data measured with
  continuous compounding (or daily compounding,
  which is almost the same)

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The Expected Return

• From the Ito Process stock-price equation, m?t ,
  is the expected percentage change in a small
  interval, ?t
• This is not the expected continuously compounded
  return on the stock actually realized over a
  period of time of length T, defined
• where Ex m s2/2 as was shown
• The expected return on the stock is m s2/2 not
  m
• We derived that the expected value of the stock
  price, a consequence of the lognormal price
  process, as S0emT
• Taking logs of gives
• If then we could write
• But Jensens Inequality tells us that in fact
• so or
• From the 2nd previous slide

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The Expected Return

• Consider the two equations
• Show as in Jensons Inequality
• So

16
The Volatility

• The volatility is the standard deviation of the
  continuously compounded rate of return in 1 year
• The standard deviation of the return in time Dt
  is
• If a stock price is 50 and its volatility is 25
  per year what the standard deviation of the price
  change in one day is
• From
• We can see the price change is .25 (50)
  sqrt(1/252) .79

17
Estimating Volatility from Historical Data

• Take observations S0, S1, . . . , Sn at
  intervals of t years (the interval is best if
  daily, t .00397 or .00274 )
• Calculate the continuously compounded return in
  each interval as
• Calculate the standard deviation, s , of the uis
• The historical volatility estimate is
• with error

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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
• 252 days per year vs 365 or 360
• 1 day is (1/252) year or .00397 (or .00274 or
  .00278) year

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The Concepts Underlying Black-Scholes

• 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 differential
  equation

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Derivation of the Black-Scholes Differential
Equation

• We start with the Stock Price Process and Itos
  Lemma for a function of the price process
• Then form the candidate riskless portfolio

21
Derivation of the Black-Scholes Differential
Equation

• The value ? of the portfolio is given by
• The change in its value in time ?t is given by
• Substituting the stock price equation for ?S ,
  and using the Ito result for ?f , gives

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Derivation of the Black-Scholes Differential
Equation

• As equation for ?? does not involve ?z , the
  portfolio is riskless during time ?t , and must
  return the risk-free rate
• Substituting for ?? and ? gives
• Which results in the Black-Scholes-Merton
  differential equation

23
Black-Scholes-Merton Differential Equation

• It has many solutions corresponding to all the
  different derivatives that can be defined with S
  , the stock price, as the underlying variable
• The particular derivative that is obtained when
  the equation is solved depends on the boundary
  conditions that are used.
• For a European Call, the key boundary condition
  is
• f max (S K, 0) when t T
• For a European Put, the key boundary condition is
• f max (K S, 0) when t T

24
Black-Scholes-Merton Differential Equation

• For a Forward Contract on a non-dividend paying
  stock, the key boundary condition is
• S K when t T
• In general, we know that the value of a forward
  contract, f , at any time is given in terms
  of the stock price S at time t by S K er
  (T t )
• Substituting into the BSM differential equation
• Showing that the differential equation is indeed
  satisfied

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Black-Scholes-Merton Differential Equation

• Any function f(S , t) that is a solution of the
  BSM differential equation is the theoretical
  price of a derivative that could be traded.
• If such a derivative existed, it would not create
  any arbitrage opportunities
• Conversely, if a function f(S , t) does not
  satisfy the BSM differential equation, it cannot
  be the price of a derivative without creating
  arbitrage opportunities for traders
• If f(S , t) eS , it does not satisfy BSM so it
  is not a candidate for being the price of a
  derivative dependent on the stock price
• If , it does satisfy BSM, and so
  could be the price of a tradable derivative
• One that pays off 1/ST at tT

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Risk-Neutral Valuation

• The concept of risk-neutral valuation was
  introduced in the context of the binomial model
• Single most important result for derivative
  analysis
• The Black-Scholes equation is independent of all
  variables affected by risk preference
• Only variables are S, t, s, and r no expected
  return, µ
• 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
• Assume the expected return of the underlying
  asset is r , i.e. µ r
• Calculate the expected payoff from the derivative
• Discount the expected payoff at the risk-free
  rate, r

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Risk-Neutral Valuation

• The principle of risk-neutral valuation is the
  same as we saw for the binomial model
• An artificial device for obtaining solutions to
  the Black-Scholes equation
• Such solutions are still valid in worlds where
  investors are risk-averse
• When we move from a risk-neutral world to a
  risk-averse world two things happen
• The expected growth rate in the stock price
  changes
• The discount rate for payoffs from the derivative
  changes
• These two changes always offset each other exactly

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Valuing a Forward Contract with Risk-Neutral
Valuation

• Consider a long forward contract on a
  non-dividend paying stock, S, that matures at T
  with delivery price K
• The payoff at maturity is ST K
• Denoting the value of the forward contract at
  time zero by f , means that , discounting
  the the expected payoff in a risk neutral world
  
• Expected return on the underlying asset is r so
  
• Present value of expected payoff is

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The End for Today

• Questions?