Modelling stock price movements

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Modelling stock price movements
By A V Vedpuriswar
July 31, 2009
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Modelling stock price movements

• To measure the market risk of an asset portfolio,
  we have to understand the pattern of movement of
  the underlying.
• We should be able to model the price of the
  underlying.
• The most celebrated modeling has been done for
  stocks.
• But this work can be extended to other asset
  classes too.
• Before we look at the modeling techniques, we
  need to gain a basic understanding of stochastic
  processes.

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Stochastic processes

• When the value of a variable changes over time in
  an uncertain way, we say the variable follows a
  stochastic process.
• In a discrete time stochastic process, the value
  of the variable changes only at certain fixed
  points in time.
• In case of a continuous time stochastic process,
  the changes can take place at any time.
• Stochastic processes may involve discrete or
  continuous variables.
• The continuous variable continuous time
  stochastic process is usually used for
  describing stock price movements.

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Markov Process

• In a Markov process, the past cannot be used to
  predict the future.
• Stock prices are usually assumed to follow a
  Markov process.
• All the past data have been discounted by the
  current stock price.
• Suppose we have a coin tossing game where for
  every head we gain 1 and for every tail, we lose
  1.
• Then the expected value of the gains after i
  tosses will be zero.
• For every toss, the expected value of the gains
  is zero.

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• Suppose we use Si to denote the total amount of
  money we have actually won up to and including
  the ith toss.
• Then the expected value of Si is zero.
• On the other hand, let us say we have already
  had 4 tosses and S4 is the total amount of money
  we have actually won.
• The expected value of the fifth toss is zero.
• Thus the expected value after five tosses is
  nothing but S4.
• That is no change is expected in the variable.
• This leads to the idea of Wiener process.

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Wiener Process

• A stochastic Markov process with mean change 0
  and variance 1 per year is called a Wiener
  process.
• A variable z follows a Wiener process if the
  following conditions hold
• Change ?z during a small period of time ?t is
  given by
• ?z e?t,
• e is a standard normal random variable(mean 0,
  std devn 1)
• The values of ?z for any two different short
  intervals of time, ?t are independent.
• Mean of ?z 0
• Variance of ?z ?t or Standard deviation ? ?t

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Illustration

• To illustrate, say a variable follows a Wiener
  process and has an initial value of 20, at the
  end of one year.
• The expected change in the value of the variable
  is 0.
• The variable will be normally distributed with a
  mean of 20 and a standard deviation of 1.0.
• At the end of 5 years, the mean will remain 20
  but the standard deviation will be ?5.
• This is an extension of the principle used to
  scale volatility as we go further out in time.

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Generalized Wiener Process

• Here the mean does not remain constant.
• Instead, it drifts at a constant rate.
• This is unlike the basic Wiener process which has
  a drift rate of 0 and variance of 1.
• The generalized Wiener process can be written as
• dx a dt bdz
• or dx a dt be??t

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• Suppose the value of a variable is currently 40.
• The drift rate is 10 per year while the variance
  is 900 per year.
• This means the expected change in the variable is
  10 in a year.
• If we consider a period of 1 year the variable
  will be normally distributed, with
• mean of 4010 50
• std deviation of 30.
• If we consider a period of 6 months, the variable
  will be normally distributed with
• mean of 405 45
• std devn of 30?.5 21.21.

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Ito Process

• The generalised Wiener process is useful but
  needs to be modified to make it useful for
  modelling stock prices.
• An Ito process is nothing but a generalized
  Wiener process
• Each of the parameters, a, b, is a function of
  the value of both the underlying variable x and
  time, t.
• Earlier a was a function only of t.
• In an Ito process, the expected drift rate and
  variance rate of an Ito process are both liable
  to change over time.
• ? x a (x, t) ?t b (x, t) e??t
• A process called Itos Lemma will come in handy
  while developing the Black Scholes equation.

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Brownian Motion

• In the coin tossing experiment, the expected
  winnings after any number of tosses is just the
  amount we already hold.
• This is called the Martingale property.
• The quadratic variation of a random walk is
  defined by
• (S1 S0)2 (S2 S1)2 (Si Si-1)2
• For each toss, the outcome is 1 or - 1. So
  each of the terms in the bracket will be (1)2 or
  (-1)2 i.e., exactly equal to 1.
• Since there are i terms within the square
  bracket, the quadratic variation is nothing but i.

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• Let us now advance the discussion by bringing in
  the time element.
• Suppose we have n tosses in the allowed time, t.
  
• We define the game in such a way that each time
  we toss the coin, we may gain or lose an amount
  of ?(t/n) .
• Now each term in the small bracket is ?(t/)n2
  or t/n
• Since there are n tosses, the quadratic variation
  is (t/n) (n) t.
• Thus the expected value of the pay off is zero
  and that of the variance is t.

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Brownian Motion

• The limiting process as time steps go to zero is
  called Brownian motion.
• Wilmott has summarized the properties of Brownian
  motion
• The increment scales with the square root of the
  time step.
• The paths are continuous.
• The distribution follows the Markov property.
• The quadratic variation is t.
• Over finite time increments, ti-1 to ti, x(ti)
  x (ti-1) is normally distributed with mean zero
  and variance (ti ti-1).

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Geometric Brownian Motion

• It is tempting to write ds µdt ?dz, where ds
  is the change in stock price.
• But this is not logical.
• The most widely used model of stock price
  behaviour is given by the equation
• µdt ?dz
• ? is the volatility of the stock price
• µ is the expected return.
• This model is called Geometric Brownian motion.
• The first term on the right is the expected
  return and the second is the stochastic
  component.

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Riskless protfolio

• In general, many variables can be broken down
  into a predictable deterministic component and a
  risky stochastic or random component.
• When we construct a risk free portfolio, our aim
  will be to eliminate the stochastic component.
• The component which moves linearly with time has
  no risk.

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Illustration

• Suppose a stock has a volatility of 20 per annum
  and provides an expected return of 15 per annum
  with continuous compounding.
• The process for the stock price can be written
  as
• .15dt .20dz
• or .15 ?t .20 ?z
• or .15 ?t .20 e ??t
• If the time interval 1 week .0192 years and
  the initial stock price is 50.
• 50 (.15 x .0192 .20 e ?.0192)
• .144 1.3856 e

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Understanding Geometric Brownian Motion

• To get a good intuitive understanding of
  Geometric Brownian motion, we draw on the work of
  Neil A Chriss.
• Consider a heavy particle suspended in a medium
  of light particles.
• These particles move around and crash into the
  heavy article.
• Each collision slightly displaces the heavy
  particle.
• The direction and magnitude of this displacement
  is random.
• It is independent of other collisions.
• Each collision is an independent, identically
  distributed random event.

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• The stock price is equivalent to the heavy
  article.
• Trades are equivalent to the light particles.
• We can expect stock prices will change in
  proportion to their size.
• As the returns we expect do not change with the
  stock prices.

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• Thus we would expect 20 return on Reliance
  shares whether they are trading at Rs. 50 or Rs.
  500.
• So the expected price change will depend on the
  current price of the stock.
• So we write
• ?s S (µdt ?dz).
• Because we scale by S, it is called Geometric
  Brownian Motion.

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Short and long run

• In the short run, the return of the stock price
  is normally distributed.
• The mean of the distribution is µ?t.
• The std devn is ???t.
• µ is the instantaneous expected return.
• ? is the instantaneous standard deviation.

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The long run

• In the long term, things are different.
• Let S be the stock price at time, t.
• Let µ be the instantaneous mean.
• Let ? be the instantaneous standard deviation.
• The return on S between now (time t) and future
  time, T is normally distributed with a mean of
  (µ-?2/2) (T-t) and std devn of ??T-t.
• Why do we write (µ-?2/2) and not µ?
• What is the intuitive explanation?

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Geometric Brownian Motion

• We need to first understand that volatility tends
  to depress the returns below what the short term
  returns suggest.
• Expected returns reduce because volatility jumps
  do not cancel themselves.
• A 5 jump multiplies the current stock price by
  1.05 A 5 fall multiplies the amount by .95.
• If a 5 jump is followed by a 5 fall or vice
  versa, the stock price will reach 0.9975, not 1!
• In general, if a positive return x is followed
  by a negative return x, the price will reach
  (1x) (1-x) 1- x2
• How do we estimate the value of x?
• Consider a random variable x. We can calculate
  the variance of x as follows

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• ?2 E x2 Ex2 E x2 (assuming Ex
  0, ie., ups and downs in x cancel out)
• Thus the expected value of x2 is the variance.
• But the amount by which the returns are depressed
  when a positive movement of x is followed by an
  equal negative movement is x2.
• For two moves, the depression is x2.
• So we could say that the average depression per
  move is x2/2.
• But the expected value of x2 is ?2 .
• So we can write ?2 /2 as the expected value of
  the amount by which the returns fall from the
  mean.
• That is why we write (µ-?2/2) and not µ.

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Geometric Brownian Motion

• Can we make some prediction about the kind of
  distribution followed by the stock price under
  the assumption of a Geometric Brownian Motion?
  Let us begin with the assumption that the stock
  returns are normally distributed.
• Annualised return from t0 to T
• ST future price St0 current price, T-t0 is
  expressed in years.
• Annualised return
• Let random variable X
• Let us define a new random variable
• X
• The second term of the expression is a constant.
  So the basic characteristics of the distribution
  are not affected. Only the mean changes.
• Also X
• or (T-t0) X ln St0 ln ST

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Geometric Brownian Motion

• The mean return on S from time t to time T is
  (T-t) (r-?2/2), while the std devn is ??T-t
• The return on S from time t to T ln ST /St
• The random variable is normally distributed
  with mean 0 and std devn 1.
• Suppose a call option on the stock with strike
  price, K is in the money at expiration.
• We want to estimate the probability of the stock
  price exceeding the strike price.
• ? ST K
• ? ST/St K/St
• ln (ST/St) ln (K/St)
  

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Geometric Brownian Motion

• ? (Taking the negative of both sides
  and noting that
• The probability of the stock price exceeding the
  strike price can be written as
• P (ST K)
• But r(T-t) ln er(T-t) (? x elnx)
• Or P (ST K)

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Geometric Brownian Motion

• This expression reminds us of the Black Scholes
  formula!
• Indeed, GBM is central to Black Scholes pricing.
• GBM assumes stock returns are normally
  distributed.
• But empirical data reveals that large movements
  in stock price are more likely than a normally
  distributed stock price model suggests.
• The likelihood of returns near the mean and of
  large returns is greater than that predicted by
  GBM while other returns tend to be less likely.
• There is also evidence that monthly and quarterly
  volatilities are higher than annual volatility.
• Daily volatilities are lower than annual
  volatilities.
• So stock returns do not scale as they are
  supposed to.

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Itos lemma

• Let us move closer to the Black Scholes formula.
  
• Black and Scholes formulated a partial
  differential equation which they later solved,
  with the help of Merton by setting up boundary
  conditions.
• To understand the basis for their differential
  equation, we need to appreciate Itos lemma.
• Consider G, a function of x.
• The change in G for a small change is x can be
  written as
• ?G
• We can understand this intuitively by stating
  that the change in G is nothing but the rate of
  change with respect to x multiplied by the change
  in x.
• If we want a more precise estimate, we can use
  the Taylor series
• ?G

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Itos lemma

• Now suppose G is a function of two variables, x
  and t.
• We will have to work with partial derivatives.
• This means we must differentiate with respect to
  one variable at a time, keeping the other
  variable constant. We could write
• ?G
• Again, if we want to get a more accurate
  estimate, we could use the Taylor series
• ?G
• Suppose we have a variable x that follows the Ito
  process.
• dx a (x,t) dt b(x,t) dz
• or ?x a(x,t) ?t b(x,t)???t
• or ?x a ?t b ???t
• ? follows a standard normal distribution, with
  mean 0 and standard deviation
  1.

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Itos lemma

• We can write (?x)2 b2?2 ?t other terms where
  the power of ?t is higher.
• If we ignore these terms assuming they are too
  small, we can write
• ?x2 b2?2 ?t
• All the other terms have ?t with power 2 or
  more. They can be ignored. But ?x2 itself is big
  enough and cannot be ignored.
• Let us now go back to G and write
• ?G
• But (?x)2 b2?2 ?t as we just saw a little
  earlier.
• It can be shown (beyond the scope of this
  coverage) that the expected value of ?2 ?t is ?t,
  as ?t becomes very small.
• Thus (?x)2 b2?t

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Itos lemma

• But dx a(x,t) dt b(x,t) dz
• So we can rewrite
• dG
• This is called Itos lemma.
• It is very much a type of generalised Weiner
  process.

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The Black Scholes differential equation

• The Itos lemma is very useful when it comes to
  framing the Black Scholes differential equation.
  
• Let us assume that the stock price follows
  Geometric Brownian motion, i.e.,
• Or ?S µS?t ?S?z
• Let f be the price of a call option written on
  the stock whose price is modeled as S.
• f is a function of S and t.
• or ?S a (S,t) dt b (S,t) dS.
• Applying Itos lemma, we can relate the change in
  f to the change in S .
• Comparing with the general expression for Itos
  Lemma, we get
• G f , a µS and b ?S, x S,
• or

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The Black Scholes differential equation

• Our aim is to create a risk free portfolio whose
  value does not depend on the S, the stochastic
  variable. Suppose we create a portfolio with a
  long position of shares and a short
  position of one call option.
• The value of the portfolio will be
• ? -f S
• (Value means the net positive investment made.
  So a purchase gets a plus sign and a short sale
  gets a negative sign.)
• We will see later that is nothing but
  delta and the technique used to create a risk
  free portfolio is called delta hedging.
• Change in the value of the portfolio will be
• ?? - ?f ?s
• -

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The Black Scholes differential equation

• But ?s µS?t?S?z
• or ?? -
• -
• or ?? -
• -
• This equation does not have a ?s term.
• It is a riskless portfolio, with the stochastic
  or risky component having been eliminated.
• The total return depends only on the time. That
  means the return on the portfolio is the same as
  that on other short term risk free securities.
  Otherwise, arbitrage would be possible.
• So we could write the change in value of the
  portfolio as
• ?? r ? ?t where r is the risk free rate.
  (Because this is a risk free portfolio)

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The Black Scholes differential equation

• But ? -f
• and
• or -
• or
• or
• This is the Black Scholes differential equation.
• The portfolio used in deriving the Black Scholes
  differential equation is riskless only for a very
  short period of time when is constant.
• With change in stock price and passage of time,
  can change.
• So the portfolio will have to be continuously
  rebalanced to achieve what is called a perfectly
  hedged or zero delta position.
• This is also called dynamic hedging.
• Solving the equation with appropriate boundary
  conditions gives us the black Scholes formula.

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The Black Scholes formula

• Let C be the value of the call, P that of the
  put, K the strike price
• C S0 N(d1) Ke-rT N(d2)
• d1 ln(S0/k) (r?2/2)T / ??T
• d2 d1 - ??T
• C P S0 Ke-rT
• or P C S0 Ke-rT
• S0N(d1) Ke-rT N(d2) S0 Ke-rT
• Ke-rT 1 N(d2) S0 N (d1) 1
• Ke-rT N(-d2) S0 N(-d1)