Stock Price Modeling for Option Valuation

1
Stock Price ModelingforOption Valuation

• Zoe Oemcke
• University of Connecticut
• Department of Statistics

2
Outline of Discussion

• Modeling Stock Prices
• Geometric Brownian Motion
• Time Series Models
• Binomial Trees
• Valuing Assets
• Option Pricing
• European Call
• European Puts and American Options
• Estimating Volatility

3
Brownian Motion

• B(t) or W(t) also referred to as a Wiener process
  is a stochastic process with the following
  properties
• B(0)0
• B(t) t0 has independent increments
• i.e. B(tn)-B(tn-1), B(tn-1)-B(tn-2), . . .,
  B(t2)-B(t1),B(t1) are independent of one another
• B(t) t0 has stationary increments
• i.e. B(ts)-B(s) behaves in the same manner as
  B(t)-B(0)
• when s0
• For all t0 , B(t) is normal with mean 0 and
  variance t
• Brownian motion is the continuous version of a
  random walk processes.

4
Other Notes About B(t)

• Brownian motion was first realized by botanist
  Robert Brown when trying to describe the motion
  exhibited by particles when immersed in a gas or
  liquid.
• The particles were essentially being bombarded
  by the molecules present within the matter
  causing the displacement or movements.
• A Brownian motion process with drift µ and
  variance s2 can be written as
• X(t)sB(t) µt
• So this process X(t) is normal with mean µt and
  variance s2t.
• Due to independent increments argument B(t)-B(s)
  is independent of
• Fs s(B(r) rs) (filtration of the process)
• B(t) is a Continuous Time Martingale with respect
  Ft
• E(B(t) Fs)E(B(t)-B(s)B(s)Fs)B(s)
• B(t)2-t is also a Martingale with respect to Ft
• B(t) is continuous but not differentiable

5
A Deductive Approach

• Osbornes Contribution in 1959 was to finally
  make the connection between Brownian motion and
  stock prices via an analogy.
• It was clear that some kind of connection
  existed based upon movement of daily closing
  prices in relation to Brownian motion.

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Brownian Motion / Index Value
7
Brownian Motion / Index Value
8
A Deductive Approach

• Osbornes Contribution in 1959 was to finally
  make the connection between Brownian motion and
  stock prices via an analogy.
• It was clear that some kind of connection
  existed based upon movement of daily closing
  prices in relation to Brownian motion.
• Like the particle being bombarded, stock prices
  deviates from the steady state as a result of
  being jolted by trades. Essentially a
  macroscopic version of Brownian motion.

9
A Deductive Approach

• Osbornes Contribution in 1959 was to finally
  make the connection between Brownian motion and
  stock prices via an analogy.
• It was clear that some kind of connection
  existed based upon movement of daily closing
  prices in relation to Brownian motion.
• Like the particle being bombarded, stock prices
  deviates from the steady state as a result of
  being jolted by trades. Essentially a
  macroscopic version of Brownian motion.
• First, we determine the steady state of the
  prices by examining the closing prices on a
  single day.

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Lognormal Distribution
11
Lognormal Distribution
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A Deductive Approach

• Osbornes Contribution in 1959 was to finally
  make the connection between Brownian motion and
  stock prices via an analogy.
• It was clear that some kind of connection
  existed based upon movement of daily closing
  prices in relation to Brownian motion.
• Like the particle being bombarded, stock prices
  deviates from the steady state as a result of
  being jolted by trades. Essentially a
  macroscopic version of Brownian motion.
• First, we determine the steady state of the
  prices by examining their value on a single day.
• The distribution of prices appears to follow a
  lognormal distribution.
• So the log of the prices are well approximated
  by a normal.

13
A Deductive Approach

• Focusing now on an individual stock
• Weber-Fechner law states that equal ratios of
  physical stimulus corresponds to equal intervals
  of subjective sensation. This argument implies
  that one should not study the absolute level of
  the price, but rather focus on the change in the
  price.
• The focus then changes to log of the price
  ratios

Also referred to as the return on the stock over
the period from ti to ti1.
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Example of the Return Values
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A Deductive Approach

• For a buyer, the estimated values of the return
  is positive, and for the seller, the opposite,
  the estimated value must be negative. So for the
  market as a whole the expected value must balance
  out to be zero.
• Bachelier observed in the early 1900s that the
  long-run standard deviation of the return varies
  according to the square root of time elapsed
  multiplied by short-run volatility.
• Essentially, log(Sti1/Sti) follows a Brownian
  motion with drift of 0 and a variance of s2.
• This means that for t1

16
Mathematical Approach

• We want to form a connection between Brownian
  motion and the price of a security.
• Brownian motion can be negative while the
  security price is strictly positive, so it is
  clear that the security will not be a multiple of
  Brownian motion.
• Profit or loss is a measure of the proportional
  increase, so the quantity ?St/St is of interest.
• Different stocks have different volatilities, and
  thus they have different risks. Depending upon
  the risk, one expects to be compensated a mean
  rate of return µr (risk-free rate of return,
  rate of return demanded of securities which have
  relatively no risk).

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Mathematical Approach

• We obtain the following stochastic differential
  equation

or
Recall that Brownian motion is not differentiable
with respect to t, the derivative is only
understood with respect to the stochastic
integral. The integral itself defined as the
limit in L2 of Riemann sums.
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Mathematical Approach

• For the stochastic differential equation

or
In order to solve this differential equation to
obtain St, we need to apply a Itos formula.
This formula is the fundamental theorem of
calculus expanded to stochastic integration.
Which results from Taylors theorem.
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Mathematical Approach

• In this formula

The first sum resembles a stochastic integral as
previously defined and the second sum is
approximately the quadratic variation. As a
result, so long as the first two derivatives of
the function f are continuous, then Itos formula
says
20
Mathematical Approach

• Similarly for a semimartingale, Xt, which has the
  form which is Btt

From the previous study, we already have a
feeling that in our case, Log(St/S0) is the Xt
and that the f(x)ex. Say XtsBt (µ-(s2/2))t,
a Brownian motion with variance s2 and drift
(µ-(s2/2)). We claim that by applying Itos
formula that we can verify that this forms the
solution to our SDE or that
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Mathematical Approach

• Ito Calculation

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Mathematical Approach
So the solution below states that the ratio St/S0
follows geometric Brownian motion
In comparison to Osbornes result,
we now have
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Why (µ-(s2/2))?

• Why not just µ as the drift parameter?
• Due to the jolts to the price, the rate of return
  is discounted by the amount of variability.
• If a positive return is followed by a negative of
  the same degree, the overall return will be
  depressed by the degree x2, since
  (1x)(1-x)1-x2.
• The Brownian motion determines the E(x2) s2.
• Divided over two periods of time, we have an
  average degree of depression s2/2.

24
The Model

• So the model for the stock price is

This model is fairly simple, and clearly defines
that the returns for non-overlapping time periods
are independently normal
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Problems

• Due to its simplicity and easy application to
  option evaluation, this model was highly
  regarded.
• Until . . .
• In 1987, there was a stock market crash, and the
  inadequacy of applying a model with a constant
  variance, or constant volatility, became apparent.

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Stochastic Volatility

• Johnson Shano, Wigins, Hull White (1987) all
  introduced a SDE for the volatility measure. The
  following is the equations as selected by Hull
  and White in 1987

• Stein Stein (1991)

The second equation expresses the volatility like
an AR(1) process. That means the volatility at
time ti has a value that depends on the
volatility at time ti-1.
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ARCH models

• The October 1987 stock crash suggested that the
  volatility of stocks could encounter sudden
  change, yet it was also noted, as the previous
  model attempted to introduce, that volatility has
  positive serial correlation.
• i.e. The degree of volatility has a tendency to
  cluster.
• In the early 80s, Engle developed a model called
  the ARCH (Autoregressive Conditional
  Heteroscedasticity)

The ARCH is able to capture that serial
autocorrelation observed for the volatilities.
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Other Issues Discovered

• Other factors that are true about the stocks that
  we want to include within our model
• Both the trading days as well as the non-trading
  days contribute to volatility. (It is observed
  that Monday tends to be the most volatile day of
  the week.)
• When the return is negative, or when the price of
  a stock drops, the volatility tends to rise. (If
  the equity of the firm drops, the firm becomes
  more leveraged.)
• Volatility tends to be high during times of
  financial crisis and recessions. (It is very
  difficult to distinguish this effect from the
  leverage effect above, since both occur
  simultaneously.
• High interest rates are often associated with
  high volatility.

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Generalized ARCH or GARCH

• The Generalized ARCH was first introduced by
  Bollerslev in 1986.

This model is obviously more parsimonious, and
the intercept term can made time dependent to
incorporate any seasonal or non-trading days
effects. Unfortunately, it still fails to
capture the leverage effects of stocks prices.
30
Exponential ARCH model

• Nelson introduced in 1988 his exponential ARCH
  model

We can let ?t?ln(1Ntd), Ntof non-trading
days between ti-1 and ti, and d contribution of
a non-trading day. The zt measure the shock or
effect of the trades. Here the parameter ?0
indicates that if the zt rises then so will the
volatility. For the parameter ?will increase the volatility to a greater degree
than a positive zt. This model performs well for
extended periods of high volatility, but it is
not as adequate when the periods of large
fluctuation are short.
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Exponential GARCH model

• Nelson introduced in 1991 his exponential GARCH
  model

• Nelson, under certain conditions, showed the weak
  convergence of the AR(1)-EGARCH model to

32
Binomial Trees

• This very basic approach was introduced in 1979
  by Cox, Ross, and Rubenstein, and it connects
  very nicely to the asset valuation techniques.
• Binomial model of stock price changes is a
  simplified discrete process, which is much more
  flexible than the continuous geometric Brownian
  motion model.
• There are two types of trees
• Standard trees directly connected to the
  geometric Brownian motion, still subject to
  restriction
• Flexible trees free from many restrictions, can
  adjust to control the level of volatility

33
General Binomial Tree

• We set a period of time in which we are
  interested in modeling the stock, an initial
  starting time to a termination date. This time
  span is then divided up into smaller intervals.
• At the end of each period, we can state what the
  possible values of the stock price.
• From each state, the stock only has two possible
  outcomes, to move up or to move down.
• The degree to which a stock moves up or down
  should be a measure determined by the volatility
  of the stock.
• There is an up-transitional probability and a
  down-transitional probability
• u(Su/S0)up ratio d(Sd/S0)down ratio

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General Binomial Trees

• Recombination tree the result of a down-up move
  is the same as a up-down move

Su
S0
Sud
Sd

• Each place where two lines cross is referred
  to as a node of the tree.

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Standard Binomial Trees

• The up ratio and down ratio are fixed as are the
  transitional probabilities.
• The length of each period of time is also fixed.
• The tree is guaranteed to be a recombination
  tree.
• The tree will be centered
• d1/u
• Centering condition
• ud e2r?t
• This way an up-down movement will take a forward
  value.

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Determining Parameters

• Expected rate of return influences the expected
  value of the stock at later periods in time. We
  should have

Where p is the up transitional probability.
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Determining Parameters

• The volatility should be the means of
  determination.

In a standard tree, this s is the same at every
node, since
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Cox, Ross, Rubinstein Method

• From the volatility equation under a standard
  tree

• Under the centering assumption
• d1/u
• Due to the expected value of the stock at one
  period ahead

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Equal Probability Method

• Assume that p.5, thus under the expected value

• The volatility further causes

• So together,

40
Option Evaluation

• The models previously introduced were not created
  with the purpose of predicting the value of the
  stock at some point in the future. Rather, we
  employ this type of modeling to find
  probabilistic distributions of the future stock
  prices.

41
Option Evaluation

• The models previously introduced were not created
  with the purpose of predicting the value of the
  stock at some point in the future. Rather, we
  employ this type of modeling to find
  probabilistic distributions of the future stock
  prices.
• Our intension is to use this distribution to
  value assets being sold today. Specifically, we
  would like to determine the fair value of
  financial contracts which involve the trading of
  stocks.
• Derivative Security-an instrument whose value
  depends on the underlying asset
• Long position in a security means that the
  individual owns that security. This person
  benefits if the price of the security increases.
• Short position in a security means that the
  individual borrowed the security and sold it in
  the market. Eventually, they will need to buy it
  back an return it to the owner. Any dividends,
  payments generated by the security must be paid
  to the original owner. This person benefits if
  the price of the security decreases.

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Different Types of Financial Contracts

• Call Option affords the buyer the right to
  purchase an underlying asset for a fixed price in
  the future.
• The fixed price of the underlying asset is
  referred to as the strike price.
• Put Option affords the buyer the right to sell
  the underlying asset for a fixed price in the
  future.
• European Option - can only be exercised on one
  day, the expiration date.
• American Option can be exercised at any point
  prior to expiration.
• Other options are just variations and
  complications of the above forms.

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Assumptions

• In order to determine the value of the previously
  listed options, we must make the following
  assumptions
• Arbitrage there are no opportunities to make
  risk-free profit
• Liquidity - there are enough buyers to satisfy
  sellers, and enough sellers to satisfy buyers
• No bid-ask spread the price at which a person
  can sell a security is the same price at which
  someone can buy that security
• Constant interest rate
• No market impact
• Be able to borrow money at the risk-free rate of
  interest
• No payouts from underlying asset, no dividends
• For the following equations, we are working with
  the European Option

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Determining the Value

• Agree to pay K at time T for stock.

Value at T
0
ST
K
45
Determining the Value

• Call option with strike price K Value max
  (ST-K,0)

Value at T
0
ST
K
46
Determining the Value

• Agree to sell the stock at T for price K

Value at T
0
ST
K
47
Determining the Value

• Put option with strike price K Valuemax(K-ST,0)

Value at T
0
ST
K
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Breakdown of Value

• The cost of the option at any point in time is
  broken into two pieces
• Intrinsic value the value if the option were to
  exercised immediately
• Time value whatever is not explained above
• (The spot price is the current value of the
  underlying asset.)
• Intrinsic value is
• Positive if the option is in the money
• Negative if the option is out of the money
• 0 if the option is at money or the strike and
  spot prices are equal

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Put-Call Parity

• Assume that we have two investments
• Buy the call option and sell the put option.
  Current Value C-P
• Go long on the stock and sell a riskless,
  zero-coupon bond which would mature at T to a
  value of K. Current Value S-e-r(T-t)K
• At time T, either the call or the put will be in
  the money. If the call, then C is worth ST-K,
  and the 1st investment is also worth ST-K. If
  the put, then P is worth K-ST, and the 1st
  investment is worth (K-ST) ST-K
• At time T , the 2nd investment is worth ST-K.
• Both investments have the same value at T and
  cost nothing to maintain, therefore they must
  also be equal at time t. i.e.
• C-PS-e-r(T-t)K

50
Expectations of the Value

• The European call options value is greater if
• The price of the stock is higher, especially
  compared to K
• The volatility of the stock is larger
• The farther the expiration date
• As the call option nears the expiration date, the
  value of the option should be close to
  max(ST-K,0).
• If the expiration date is far away, then the
  value of the option should be close to ST.

51
Hedging Strategy

• The hedging portfolio is created to defray the
  risks associated with the selling of the option.
  It will be a combination of stocks and riskless
  bonds which are intended replicate the payoff at
  the time at which the option is exercised.
• The hedging portfolios value at any point in
  time should be the same value as the option.
  Therefore the hedge must be dynamic, constantly
  adjusting to rebalance.
• There are costs incurred with hedging portfolio
• Set-up costs or the original investment
• Maintenance costs
• Infusion of funds / additional monies needed to
  rebalance
• Transaction costs / fees or taxes

52
Hedging Strategy

• The hedging portfolio with only set-up costs is
  set to be self-financing. We call ? the rate of
  change in the value of the option with respect to
  the change in value of the underlying asset.

• This same ? is the number of shares of the
  underlying stock that need to be held in the
  hedging portfolio.
• Further, the value of the bond e-r(T-t)Bt in
  the hedging portfolio should be adjusted along
  with the delta. Bt is the value at expiration.

53
Portfolio

• Sold a Call Option, short the call
• Purchased ? shares of the underlying, long that
  of shares
• Loaned out the money, short the bond
• The idea The investor can hold other assets to
  decrease risk, and therefore the risk influencing
  the discount rate will only be the risk which can
  not be diversified away.

54
Black Scholes

• In 1973, Black Scholes were the first to use the
  idea of a hedging strategy used by an investor in
  order to create a riskless portfolio.
• As a result, the following option value is
  produced when applying geometric Brownian motion
  model.

• This calculation is done under the risk neutral
  assumption, meaning that µr

55
Black Scholes Calculation

• Samuelson and Merton 1969 recognize discounting
  method

56
Under GARCH EGARCH Model

• Closed form solutions to the option valuation
  formula can not be determined, so most often
  simulation is implemented.
• For a particular GARCH process, Heston and
  Nandi(2000) have developed a closed form solution
  for the European Call Options

• The option valuation will not only be a function
  of the current or spot price, but also past
  prices.
• The average return is allowed to depend upon risk.

57
One Step Binomial Tree

• Our hedge portfolio has two possible values at
  time t1
• ?SuBCu or ?SdBCd
• Since B is a common feature, we can set
• ?Su-Cu ?Sd-Cd
• Thus we can solve for the ? of the hedge
• ?(Cu-Cd)/(Su-Sd)
• We can also solve for B
• B(SuCd-SdCu)/(Su-Sd)
• Since this hedge is created with the intention of
  mirroring the option
• C0 ?S0 e-r(? t)B

58
One Step Binomial Tree

• C0 ((Cu-Cd)/(Su-Sd))S0 e-r(? t)
  ((SuCd-SdCu)/(Su-Sd))
• It would appear that the option value does not
  depend on p, the transitional probability,
  however, we can manipulate the above formula to
  read
• C0 e-r(? t) ((S0-e-r(? t)Sd)/(Su-Sd))Cu ((Su
  e-r(? t)-S0)/(Su-Sd))Cd
• C0 e-r(? t) pCu (1-p)Cd
• C0 e-r(? t) C1
• In order to find the current value, we can find
  the transitional probabilities and the values of
  Cu and Cd, then compute the expected value
  discounted back one time period.
• Multiple Step Binomial Tree can be computed by
  following this process repeatedly working
  backwards from time tn to tn-1 then to tn-2, etc.
  At each node one calculates the present value of
  the option.

59
Valuing the Other Options

• By the put-call parity, C-PS-e-r(T)K, the
  European put can be calculated.
• Therefore, under geometric Brownian motion, the
  value of put is

• Under the Multiple Step Binomial Tree follow the
  same process as for the call, but compute the
  current value of the put at each node by using
  the transitional probabilities.
• Under the assumption of no dividends the American
  Call Option should have a value the same as the
  European Call

60
Dividend Effects

• So far we have assumed that the underlying asset
  does not produce dividends.
• There are two kinds of dividend payments that can
  be made
• Lumpy dividends - dividends are awarded according
  to a schedule
• Continuous dividends
• For lumpy dividends in the geometric Brownian
  motion model, we can calculate the present value
  of the payment, D, and reduce initial value of
  the stock by that amount, St0-e-r(t1-t0)D. This
  also requires that the volatility measure be
  adjusted st ((St0)/St0-e-r(t1-t0)D) s.
• For continuous, we assume a µr-q, and we make
  the adjustment

• When we have a multiple step binomial model and
  the dividends are lumpy, if the dividend is
  percentage of the spot price, then at the
  ex-dividend date, alter the tree to have Su(1-q)
  and Sd(1-q). If the lump payments act as a fixed
  dollar amount, the nodes will need to be shifted
  on that date by a fixed amount.

61
Estimating Volatility

• Use observed stock prices to estimate volatility,
  based upon

Where the tii/N, so
From the likelihood, we create a MLE for the
constant volatility.
62
Estimating Volatility

• Use observed option prices, we could also attempt
  to compute the value of the measure of the
  volatility that created it. Since the formula
  created from the geometric Brownian motion is not
  invertible, then iterative procedures are
  required to produce estimates.
• Bisections method
• Guess at the value of s0.
• Compute the option value using the formula.
• If at the kth attempt, the formula value exceeds
  market value, then
• sksk-1- s0/(2k) is the new guess, otherwise
• sksk-1 s0/(2k)
• Newton-Raphson method
• Guess at the value of s0.
• Compute V(s)rate in change of option per rate in
  change in volatility S(T(1/2))f(d1)
• sksk-1 (C(sk-1)-C)/V(sk-1)

63
Estimating Volatility

• The GARCH/EGARCH models assist in the estimation
  of unobserved volatility. Such calculations can
  be less burdensome than the implied volatility
  methods.
• These estimates of volatilities will be based
  solely upon the underlying asset returns rather
  than other options.
• Can produce out-of-sample estimates.
• Nelson has established the consistency of such
  estimators.

64
Goals to Estimating Volatility

• We would like to produce an estimator that is
  consistent and is asymptotically normal. So from
  the option formula under geometric Brownian
  motion, which is essentially a nonlinear
  stochastic regression, we would like to show that
  the least squares estimator will carry these two
  properties.
• Further, we would like to combine the information
  from the stock prices, as well as the option
  prices, to obtain the form of MLE for the
  volatility which can be shown to have the
  properties desired.

65
References

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