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									           A Markovian Options
              Pricing Model
Hope Breskman
Ruby Chiu
Walter Creighton
Philip Larson
Vince Lim
Mike McMakin
Deep Shah




Professor James W. Lark, III
SYS 360 Group 12

Thursday, February 03, 2011
                                                                  A Markovian Options Pricing Model




APPROVAL AND PLEDGE
     ―We, the undersigned, have read, understood, and approve the contents of this document and
     this project. All members have contributed fairly to the project and this document‘s contents. We
     have also neither given nor received unauthorized assistance in the development of this project.‖


     Hope Breskman           ______________________________________________________


     Ruby Chiu               ______________________________________________________


     Walter Creighton        ______________________________________________________


     Philip Larson           ______________________________________________________


     Vince Lim               ______________________________________________________


     Mike McMakin            ______________________________________________________


     Deep Shah               ______________________________________________________




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TABLE OF CONTENTS
MARKOVIAN OPTIONS PRICING MODELS ...................................................................................................... I

APPROVAL AND PLEDGE ...................................................................................................................................... I

TABLE OF CONTENTS ...........................................................................................................................................II

1      GOALS AND OBJECTIVES .............................................................................................................................1
    1.1       GOALS ............................................................................................................................................................1
    1.2       OBJECTIVES ....................................................................................................................................................1
2      A BACKGROUND ON OPTIONS ....................................................................................................................2

3      DEFINITION OF VARIABLES ........................................................................................................................3
    3.1     INTRODUCTION ...............................................................................................................................................3
       3.1.1    Spreadsheet User Inputs ........................................................................................................................3
       3.1.2    Asset-Related Variables .........................................................................................................................3
       3.1.3    Spreadsheet Parameters ........................................................................................................................3
       3.1.4    Expected Value of a Call Option Using the Markovian Model..............................................................3
       3.1.5    Expected Value of the Put Option Using the Markovian Model ............................................................4
4      DESCRIPTIVE SCENARIO—AN INTRODUCTION TO OPTIONS PRICING THEORY .....................5
    4.1     INTRODUCTION ...............................................................................................................................................5
    4.2     BLACK-SCHOLES ANALYSIS ...........................................................................................................................5
       4.2.1    Assumptions ...........................................................................................................................................5
       4.2.2    Black-Scholes Differential Equation and Pricing Formulas .................................................................5
    4.3     COX-ROSS-RUBENSTEIN PRICING THEORY ....................................................................................................6
       4.3.1    Description of the Binomial Tree ...........................................................................................................6
       4.3.2    Variable Definitions ...............................................................................................................................6
       4.3.3    Pricing Options Using the Cox-Ross-Rubenstein Model .......................................................................7
5      METHODOLOGY ..............................................................................................................................................8
    5.1     INTRODUCTION ...............................................................................................................................................8
    5.2     OVERVIEW OF PROJECT DESIGN .....................................................................................................................8
    5.3     ADAPTING THE COX-ROSS-RUBENSTEIN MODEL ...........................................................................................8
       5.3.1    Design of the Markovian Model ............................................................................................................8
       5.3.2    Adding the Time Element to the Markovian Model ...............................................................................9
       5.3.3    The Transition Matrix ............................................................................................................................9
    5.4     CALCULATING THE VALUE OF THE OPTION .................................................................................................. 10
       5.4.1    Calculation of Options Using Markovian Model Versus Cox-Ross-Rubenstein Model ....................... 11
6      IMPLEMENTATION ....................................................................................................................................... 12
    6.1     INTRODUCTION ............................................................................................................................................. 12
    6.2     COLLECTING ASSET DATA ........................................................................................................................... 12
       6.2.1    Determining Volatility ......................................................................................................................... 12
    6.3     INFORMATION ON THE UNDERLYING ............................................................................................................ 13
       6.3.1    The Risk-Free Interest Rate ................................................................................................................. 13
       6.3.2    Time Step Interval ................................................................................................................................ 13
       6.3.3    Choice of Volatility .............................................................................................................................. 14
    6.4     FORMATION OF THE TRANSITION MATRIX .................................................................................................... 14
       6.4.1    Matrix Values....................................................................................................................................... 14
       6.4.2    Computing Probabilities ...................................................................................................................... 14



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6.5     STOCK VALUES AND PROBABILITIES ............................................................................................................ 15
   6.5.1    Calculating Option Values................................................................................................................... 15
6.6     BLACK-SCHOLES PRICING FORMULAS CALCULATIONS ................................................................................ 15
6.7     OPTION VALUES ........................................................................................................................................... 16
6.8     STATISTICAL VALUES ................................................................................................................................... 16




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                                                                      A Markovian Options Pricing Model




1 GOALS AND OBJECTIVES
1.1 Goals
     The goal of this project was to develop a basic European options pricing model utilizing
     Markovian principles and concepts that were learned in SYS 360.

1.2 Objectives
     The main objective of this model was to create a simple options pricing model that would be
     comparable to accepted methods in terms of understandability and accuracy. Current pricing
     methods such as Black-Scholes analysis and the Cox-Ross-Rubenstein pricing model are the
     most prevalent theories of pricing models, however they are somewhat difficult to understand and
     difficult to apply with regards to the Cox-Ross-Rubenstein model. As such, these two models
     were used as the standards for developing the Markovian model, in order to compare our model
     with the standard ones.
     There were many reasons for attempting to simulate the options market as a Markovian model.
     For one, the stock market exhibits the Markovian property of depending solely on the present
     state of stock prices. This idea is embodied in the Efficient Market Hypothesis, which states that
     a stock‘s current price reflects all relevant information for pricing that security at any given time.
     Hence any movement of stock price should only be dependent upon the present state.
     Also the transition matrices of Markov models allow for time dependency to be easily
     incorporated by self-multiplication for time steps in the future. This aspect of a Markovian model
     was very appealing for the case of options pricing, since the execution of options are dependent
     on the future values. The transition matrices allow for predicting option future values at the
     specific expiration date.




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2 A BACKGROUND ON OPTIONS
     An option gives the holder of the option the right to purchase or sell the specified underlying
     security at a particular price (the strike price) by a certain date. There are two types of options. A
     call option permits the holder to purchase the security at the strike price, while a put option
     permits the holder to sell the security at the strike price. The date of the option after which it is
     no longer valid is the expiration date. The following figure demonstrates the expiration date
     values of both a call and a put option.

      Value of Call Option                                     Value of Put Option




                185             195             205                      185            195          205
                             Strike Price                                            Strike Price

                 Price of Stock at Expiration                             Price of Stock at Expiration

                             Figure 1—Values of Call and Put Options at Expiration

     One can buy a call option if they expect the stock price to rise above the strike price before the
     expiration date. Then, they can buy the stock at the strike price for less money than the market
     would demand. The opposite bet is being made when a put option is purchased. The buyer of a
     put option is expecting the stock price to drop below the strike price before the expiration date;
     then, the holder may sell his stock at a price that is higher than the market price.
     The holder of the option is never obligated to buy or sell the underlying stock. However, the
     allowable time period to exercise the option depends on another category of option pricing. An
     option can either be an American option or a European option. With an American option, the
     holder can exercise their option at any time before or on the expiration date, while with a
     European option can only be executed on the expiration date.
     An important relationship can be shown between these two option types using the put-call parity
     theory. This relationship states how European and American call option prices are always equal
     while American put prices are more expensive than European puts.




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3 DEFINITION OF VARIABLES
3.1 Introduction
      This section describes the various variables that are used throughout this document.

3.1.1 Spreadsheet User Inputs
      r = risk-free interest rate
      T = end time (expiration time of option)
      L = number of days of past stock closing prices
      n = number of time steps
      X = strike price of option
       = significance level for statistical tests and intervals

3.1.2 Asset-Related Variables
       = volatility of stock
      u = percent amount the stock increases
      d = percent amount the stock decreases
      p = probability the stock value increases
      1 – p = probability the stock value decreases
      S = stock price at current time
      ST = stock price at end time (expiration of option)
      T – t = time from present until expiration
      cx = value of call at state i
      px = value of put at state i
      P(n) = n-step transition matrix
            (n)
      pij         = probability of going from state i to state j in n time steps

3.1.3 Spreadsheet Parameters
       = stdev(ln(S(i)/S(i – 1)) )/(1/252)                  for i = 1…L
                  t
      u=e
                  –t
      d=e
                  rt
      a=e
      p = (a – d)/(u – a)

3.1.4 Expected Value of a Call Option Using the Markovian Model
                          rnt
      PV(Si) = Sie                               for i = 1…2n + 1
      ci = max(0, PV(Si) – X)                    for i = 1…2n + 1
                    i = 0…n p0i [max(0,
                                (n)
      E[c] =                               PV(ST(i)) – X))]          for i = 0…2n + 1




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3.1.5 Expected Value of the Put Option Using the Markovian Model
     PV(Si) = max(0, X – PV(Si))              for i = 1 … 2n + 1
              i = 0…n p0i
                          (n)
     E[p] =                     [max(0, X – PV(ST(i)))]       for i = 0…2n + 1




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4 DESCRIPTIVE SCENARIO—AN INTRODUCTION TO OPTIONS
  PRICING THEORY
4.1 Introduction
      This section discusses two of the current prevalent options pricing theories—the Black-Scholes
      pricing formulas and the Cox-Ross-Rubenstein binomial pricing model. Both models utilize key
      assumptions and underlying financial fundamentals which are crucial to a basic understanding of
      options pricing theory. Thus, a brief introduction to Black-Scholes analysis and Cox-Ross-
      Rubenstein pricing theory will be discussed in this section.

4.2 Black-Scholes Analysis
      In the early 1970s, Fischer Black and Myron Scholes made a major breakthrough by deriving a
      closed-form differential equation that must be satisfied by the price of any derivative with a stock
      underlying.

4.2.1 Assumptions
      One of the major assumptions Black-Scholes analysis relies upon is the idea that stock price
      changes are lognormally distributed. This assumption is seen in empirical data, although recently
      there have been several alternative models that better describe the distribution of stock prices.
      Nevertheless, the lognormality property is a safe basis assumption that, although not perfect, is a
      fairly good description of the movement of stock prices.
      The Black-Scholes analysis also assumes that stock prices move according to a geometric
      Brownian motion process:
                                                 dS = S dt + S dz
      where S is the price of the stock,  is the rate of return of the stock,  is the volatility of the stock,
      z is Wiener process, and t is time. Wiener processes are discussed in more detail within Hull
      (1993).

4.2.2 Black-Scholes Differential Equation and Pricing Formulas
      The Black-Scholes differential equation is as follows:
                                                                         2   2     2      2
                                     (f/t) + rS(f/S) + (1/2) S ( f/S ) = rf
      where f is the price of the derivative, r is the risk-free interest rate,  is the volatility of the
      underlying, and t is time.
      It can be shown that the solution to the Black-Scholes differential equation results in the following
      pricing formulas for European calls:
                                                                     –r(T – t)
                                             c = SN(d1) – Xe                     N(d2)
      where c is the price of a Eurpoean call option, X is the strike price, T – t is the time to expiration of
      the option and where:
                                     d1 = (ln(S/X) + (r +  /2)(T – t)) / (T – t)
                                                                 2


                                                  d2 = d1 – (T – t)
      N(x) is the cumulative probability distribution function for a variable that is normally distributed
      with a mean of zero and a standard deviation of one (i.e. a standard normal distribution).
      Alternatively, a Eurpoean put is valued as:
                                                   –r(T – t)
                                           p = Xe              N(–d2) – SN(–d1)



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      As discussed in the previous section, since American and European calls are always equal, the
      Black-Scholes call formula can be used to price both types of derivative securities. However,
      since American puts are always greater than Eurpoean puts, it is not possible to reliably
      determine the value of such securities using Black-Scholes analysis.

4.3 Cox-Ross-Rubenstein Pricing Theory
      Unlike the Black-Scholes analysis, which is a continuous pricing theory, the Cox-Ross-
      Rubenstein pricing theory is a discrete pricing theory based on a binomial model of stock price
      movement.
      The binomial model looks as follows:

                                                                                     Su3

                                                               Su2

                                             Su                                       S
                                 p                                                    u
                     S                                          S

                                          Sd                                          S
                               1–p                                                    d
                                                               Sd2

                                                                                     Sd3

                    t=0                  t=1                   t=2                  t=3

                              Figure 2—Cox-Ross-Rubenstein Binomial Model

      The model in the above figure is a so-called three-step binomial tree. Each ―step‖ of the tree is
      indicative one time step in the future where each node is a particular stock price value. The last
      set of nodes in the tree are the final stock values that can be reached upon expiration of an
      option.

4.3.1 Description of the Binomial Tree
      Assume that at time 0 the price of a stock is S. Then at each consecutive time step it is assumed
      that a stock price can only go either up or down by a percentage u or d, respectively. Thus, for
      instance, at time 1, the price of a stock can go up from S to Su or go down to Sd. Likewise, from
                                        2
      Su, the stock can go up to Su or down to S. Note that an upward movement followed by a
      downward movement effectively cancels the price movement of the stock such that it returns to
      its previous level.
      Also, prices can only move up with probability p or go down with probability 1 – p. These
                                                                              2
      probabilities stay consistent with each time step. Therefore to reach Su , for instance, has a
                      2
      probability of p .

4.3.2 Variable Definitions
      The variables discussed in the previous section for the Cox-Ross-Rubenstein Binomial Model are
      all selected such that the prices of the stock nodes are indeed lognormally distributed with the
      appropriate probabilities. Specifically:
                                                  u = et



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                                                                     A Markovian Options Pricing Model


                                                   d = 1/u
                                             p = (a – d) / (u – d)
     where
                                                         rt
                                                   a=e

4.3.3 Pricing Options Using the Cox-Ross-Rubenstein Model
     The pricing of options using the Cox-Ross-Rubenstein model is not entirely difficult, but requires a
     bit of calculation and is best done using a computer. A full discussion of how options are priced is
     not given here (see Hull (1993) for an in-depth discussion), but essentially the method to price
     call and put options is a recursive method, and hence is quite tedious, especially with many-step
     trees.
     The advantage of this particular model, however, is that it can price American put options, which
     the Black-Scholes analysis cannot price.




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                                                                        A Markovian Options Pricing Model




5 METHODOLOGY
5.1 Introduction
     This section discusses the methodology used by this project team in adapting the Cox-Ross-
     Rubenstein model to accommodate a so-called ―Markovian‖ market. The actual implementation
     of the methodology is given in the next section, only an overview of the mathematical concepts is
     given here.

5.2 Overview of Project Design
     We implemented our model in an Excel spreadsheet. We made the spreadsheet very dynamic,
     so the user can enter values for certain variables and the spreadsheet updates itself by
     calculating the expected value of the option. The user must input or accept the default interest
     rate, month of expiration of the option, number of past stock closing prices, number of time steps,
     strike price of the option, and the significance level of statistical tests and intervals into the model.

5.3 Adapting the Cox-Ross-Rubenstein Model
     After studying the Black-Scholes model and the Cox-Ross-Rubenstein model, we decided to
     base our model on the Cox-Ross-Rubenstein model. This model used a binomial tree to
     represent discrete times steps, finite states, and stationary probabilities and was easily
     implemented using Markovian principles.
     Our model closely resembles the Cox-Ross-Rubenstein, except for fundamental differences in
     the definition of states and how to calculate the expected value of the option. Both models use
     the same tree, with nodes representing the value of the stock and edges representing the
     probabilities of traversing between the nodes.

5.3.1 Design of the Markovian Model
     The first step in generating the model was gathering one year of past stock closing prices for one
     stock. We computed the standard deviation based on the number of days of past stock closing
     prices the user entered. We
     calculated u, d, p, and 1 – p                                                                 7
     using volatility and delta-time,
     the time until expiration
     divided by the number of time                                             4
     steps.
     The      Cox-Ross-Rubenstein                         2                                          8
     binomial model uses states to              p
     represent the stock value at a
                                        1                                       5
     given time.      This leads to
     unique states throughout the
     tree. Different states have the                       3
     same stock value but at                  1-p                                                    9
     different times, such as state
     1 which has a value of S at                                                6
     time 0 and state 5 which has a
     value of S at time 2. This
     quickly leads to an abundance                                                                   10
     of redundant states. Overall
     the binomial model uses                        Figure 3—Cox-Ross-Rubenstein States
     (1/2)n(n+1) states.




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                                                                     A Markovian Options Pricing Model


      We designed our model to accommodate and solve for this problem of abundant states found in
      the Cox-Ross-Rubenstein binomial model. Similar to the model, we also use states to represent
      the value of the stock at any time, but we wanted to simplify the binomial model by setting up the
      states so that a traverse-up followed by a traverse-down, or the reverse, would lead to the same
      state. This leads to states that are non-unique through the tree but unique within each time step.
      Although state 1 exists twice in the example tree, in is unique at time 0 and at time 2. By reusing
      states we only use 2n+1 states.

                                                                           6


                                                              4


                                             2                              2
                                     p
                               1                             1


                                              3
                                    1-p                                        3


                                                             5


                                                                            7


                           Figure 4—Definition of States in Our Markovian Model


5.3.2 Adding the Time Element to the Markovian Model
      While the Cox-Ross-Rubenstein model is Markovian, it is not aperiodic and therefore not ergodic.
      Our model, however, has states that can return to themselves and are therefore aperiodic. It is
      exactly this property of aperiodicity that allows our model to have significantly fewer states as
      discussed in the previous paragraph.
      However, the problem with our model is that by itself, it does not incorporate a time element that
      is crucial for pricing options. The Cox-Ross-Rubenstein model does (simply because it identifies
      every node as a separate state, thus it is possible to know what state one is by knowing which
      states are in which time period).
      Thus in order to accommodate this time element, we are only interested in our transition matrix
      after n time steps. That is, we can keep track of time by the number of times one multiplies the
      transition matrix by itself. So the transition matrix of P(2), for instance, describes all of the
      probabilities from stepping from one state to another in only two time steps. So, given this
      property, and given that the model incorporates only a finite number of states representing
      possible values of the stock and stationary transition probabilities that are independent of the
      current state, we can form a transition matrix for our model and analyze our model as a Markov
      chain with distinct Markovian properties.

5.3.3 The Transition Matrix
      To form the transition matrix from the new binomial tree with repeatable states, we map the
      probability of exiting each state as a new row. The probability of going from state S to Su is p and
      from S to Sd is 1 – p.



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                                                                           A Markovian Options Pricing Model


     Each tree node maps as a row in the transition matrix, using the probabilities from the branches.
     The first row of the transition matrix shows that the probability of going from S to Su is p, S to Sd
     is 1 – p, and from S to any other state is 0. Note that the transition probabilities between states
     are not time dependent; the probability of the stock price going up or down is always the same,
     independent of the current state. This is a key property that allowed us to define states
     regardless of the time step. A sample one-step transition matrix for a three step tree is as
     follows:

                                             S     Su    Sd    Su^2 Sd^2 Su^3 Sd^3
                                             0      1     2     3     4      5    6
                                 S     0     0      p    1-p    0     0      0    0
                                Su     1     1-p    0     0     p     0      0    0
                                Sd     2     p      0     0     0    1-p     0    0
                                Su^2   3     0     1-p    0     0     0      p    0
                                Sd^2   4     0      0     p     0     0      0    1-p
                                Su^3   5     0      0     0     0     0      1    0
                                Sd^3   6     0      0     0     0     0      0    1

                                 Figure 5—Sample One-Step Transition Matrix

     We continue with this logic throughout the tree except for the end nodes. Rows have already
                                                     n        n
     been defined for all end states except for Su and Sd . We set these probabilities to 1 since we
     need to limit the range of the value of the stock. Once they reach the limits they stay there, but
     since we only look at the probabilities for n time steps, these probabilities are exercised only once
     (at the n-th time step). Note all rows within the matrix sum to 1 as required by the law of
     probability.

5.4 Calculating the Value of the Option
     We can calculate the expected value of the option using standard Decision Ttheory. Knowing the
     call value of each state and the probability of getting to each state, we can calculate the expected
     value of the option.
     The value of the stock at each state is S times the percent amount it has increased or decreased
             2   2   3
     (u, d, u , d , u , etc.). To get the present value of the stock, we need to discount it back to current
     dollars using the interest rate and time duration. The value of the call at each state is the
     discounted value of the stock at each state minus the strike price.
     If the discounted value of the stock is less than the strike, the value of the call at that state is 0,
     because the holder will not exercise his option. For puts, the value at each state is the strike
     price minus the discounted value of the stock at that state, or 0 if the strike if less then the
     discounted value of the stock at that state.
     Thus, formally:
                                              c = max(0, PV(ST) – X)
                                              p = max(0, X – PV(ST))
     where c is the price of a call, p is the price of a put, ST is the final stock price, X is the strike price,
     and PV(x) is the present value of x based on continuous compounding.
     The probability of getting from state 0 (i.e. the starting stock price at the purchase of the option) to
     every other state in n time steps can be found by multiplying the transition matrix by itself n times
     and examining the first row of the resulting matrix. The probability of getting to non-end nodes is
     0 as expected. The value of the option can be calculated by summing over every state the value




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                                                                        A Markovian Options Pricing Model


     of the call or put at that state times the probability of getting to that state in ‗n‘ time steps. This
     process is the same used in Decision Theory. Thus:
                      E[c] = i = 0…n p0i [max(0, PV(ST(i)) – X))]
                                        (n)
                                                                          for i = 0…2n + 1
                      E[p] = i = 0…n p0i
                                        (n)
                                              [max(0, X – PV(ST(i)))]     for i = 0…2n + 1

5.4.1 Calculation of Options Using Markovian Model Versus Cox-Ross-
      Rubenstein Model
     The Cox-Ross-Rubenstein binomial model calculates expected value of options recursively as
     discussed in the previous section. It doesn‘t use a transition matrix. Instead, it calculates the call
     or put value for the end nodes (in the same manner we did), then finds the option value for the
     parent node recursively (at one time step earlier) by taking the weighted average. This is the sum
     of p times the option value of the state reached with probability p and 1 – p times the option value
     of the state reached with probability 1 – p.
     Thus unlike the Cox-Ross-Rubenstein model, the value of an option can be found very quickly
     using simple matrix multiplication and basic Markovian theory regarding the properties of
     Markovian chains. As such, the Markovian model is much faster, but an analysis of its accuracy
     must be done as described in the next section.
     One note should be made about the Markovian model developed in this section. Unlike the Cox-
     Ross-Rubenstein model, the Markovian model cannot price American put options, similar to the
     Black-Scholes analysis.




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                                                                                    A Markovian Options Pricing Model




6 IMPLEMENTATION
6.1 Introduction
      This section discusses the actual implementation of the model‘s methodology discussed in the
      previous sections. In particular, it describes the Microsoft Excel spreadsheet that was developed
      for quickly calculating the prices for call and put options for both the Markovian options pricing
      model and the Black-Scholes model.
      Since there is a significant amount of information contained within the spreadsheet, the
      discussion will follow the order in which the spreadsheet was created.

6.2 Collecting Asset Data
      The first part in creating the spreadsheet was to first collect data on past closing prices for various
      stocks. The figure below shows the portion of the Excel spreadsheet containing this information
      for IBM stock. Approximately one year‘s worth of data of closing prices from April 24, 1998 to
      April 23, 1999, was retrieved from http://investor.msn.com and was set in columns A and B on the
      spreadsheet.

      Table 1—Asset Data Table Sample
                               International Business Machines Corporation
                               (IBM)
                               Daily prices (4/24/98 to 4/23/99)
                               DATE        CLOSE       S(I)/S(I-1)    LN(S(I)/S(I-1))
                                 4/24/98    117.375
                                 4/27/98    115.313       0.9824324            -0.0177238
                                 4/28/98    115.688        1.003252            0.0032467
                                 4/29/98    115.563       0.9989195            -0.0010811
                                 4/30/98    115.875       1.0026998            0.0026962

6.2.1 Determining Volatility
      The reason why this data had to be collected was to determine the volatility (standard deviation)
      of the underlying asset. One simple means of estimating volatility from historical data is to first
      calculate natural log of the percent changes in stock prices each day (column D) and estimating
      the standard deviation of this sample. The actual annual volatility of the asset can then be
      estimated by dividing by the square root of the length of time interval in years (which in this case
      is 1/252 since there are 252 trading days in one year).
      A somewhat more forma description of this procedure is as follows:
      Define:
                                                        ui = ln(Si / Si – 1)
      It follows that the standard deviation, s, of the ui‘s is given by:
                                      s = [(ui ) / (n – 1) – (ui) / (n(n – 1))]
                                                   2                       2


      Since s is an estimate of  (where  is the length of time interval in years), it follows that  itself
                            *
      can be estimated by s where:
                                                            s = s / 
                                                             *


      An excellent treatment of this procedure is given by Hull (1997).




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                                                                                   A Markovian Options Pricing Model


6.3 Information on the Underlying
      This section of the spreadsheet contains the vital information on the stock asset needed for
      calculating the necessary probabilities and stock price distribution for the Cox-Ross-Rubenstein
      binomial tree model for stock prices.
      The following figure depicts the section of the spreadsheet which contains this needed
      information. The boxes with the shaded cells indicate cell values which can and should be
      changed by the user to reflect the desired attributes of the stock that should be used for pricing
      purposes.

      Table 2—Information on Underlying Table Sample
              Information on Underlying


              r=                     4.50%
              delta_t =        0.0069444


                   Lookup          Data Set     STDEV             u           d           a           p
                          0 1-month           0.5172911   1.0440502    0.9578083   1.0003125   0.4928489
                          1 3-months          0.4259911   1.0361369    0.9651234   1.0003125   0.4955274
                          2 6-months          0.3777533   1.0319802    0.9690109   1.0003125   0.4970943
                          3 12-months         0.3527109   1.0298288    0.9710352   1.0003125   0.4979684


                   Choice:               1




                   Lookup      Expiration          T-t       delta_t
                          0 May                    1/12   0.0069444
                          1 June                    1/6   0.0138889
                          2 July                    1/4   0.0208333
                          3 October                 1/2   0.0416667
                          4 January                 3/4      0.0625


                   Choice:               0

6.3.1 The Risk-Free Interest Rate
      The first shaded box indicates the current risk-free interest rate which is approximately 4.5%.
      This is the current repurchase rate which is used instead of the usual 90-day Treasury Bill rate
      since technically this is the relevant risk-free rate of interest for many arbitrageurs operating in the
      futures and options markets. This rate is more commonly known as the repo rate, and is the rate
      at which banks ―lend‖ money to one another for an overnight loan.

6.3.2 Time Step Interval
      The row labeled ‗delta_t‘ is actually related to the last lookup table in this section. ‗delta_t‘ is the
      time interval in years between steps of the Cox-Ross-Rubenstein binomial tree. However, this is
      dependent on exactly when the option the user wants to price expires.
      In this example, it is desired to price IBM May option thus Choice 0 is chosen. Since these
      options expire in one month, and there are twelve time steps within the tree in this particular
      implementation of the model, ‗delta_t‘ then equals (1/12) / 12 = 1/144 = 0.0069444 years.




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                                                                          A Markovian Options Pricing Model


6.3.3 Choice of Volatility
      It has already been discussed how the volatility can be calculated. However, sometimes it is
      better to use the volatility from a smaller or larger data set to more accurately price an option.
      This section allows the user to choose whether to use a data set as small as the last one month‘s
      worth of data or as long as the last twelve month‘s worth of data.
      A general rule of thumb is to use the data set equal to the expiration length of the option. Thus,
      for instance, an option with an expiration date three months from now should use a data set
      consisting of the last three months of data. However, most people generally use the last 90 to
      180 days of data simply because it gives a larger set of data to work from and is a good
      compromise of using either too much or too little data. Data that are too old may not be relevant
      for predicting the future. In this example, Choice 1 for the last three-months of data is chosen.
      This choice also automatically calculates the appropriate values for u, d, a, and p as defined by
      the Cox-Ross-Rubenstein binomial model.

6.4 Formation of the Transition Matrix
      The one-step transition matrix is conveniently automatically generated by the spreadsheet once
      the appropriate choices are made for the data set choice and the time the expiration of the
      desired options to price. A snippet of the matrix is given in the figure below.

      Table 3—Transition Matrix Sample
                         Transition Matrix
                          n=           12
                                                     S          Su          Sd
                          p=       0.4955274         0           1          2
                           S            0                 0   0.4955274   0.5044726
                          Su            1         0.5044726          0           0
                          Sd            2         0.4955274          0           0


      The ‗n‘ value is the number of time steps that are to be simulated in the binomial tree model.
      Unfortunately, this value must stay fixed since currently it has not been possible to add or
      subtract states and thus simulate any other step intervals other than twelve.
      The ‗p‘ in the upper corner of the matrix simply reminds the user of what the probability of an
      upward movement is just so one can verify the calculated values within the transition matrix.

6.4.1 Matrix Values
      The matrix value simply correspond to the upward and downward probabilities of the Cox-Ross-
      Rubenstein binomial model for stock prices. Thus, for instance, starting at state 0 (stock price
      ‗S‘), one can only move up to state 1 (stock price ‗Su‘) with probability ‗p‘ = 0.4955274, or down to
      state 2 (stock price ‗Sd‘) with probability 1 – p = 0.5044726.
      The matrix continues for all twelve steps of the binomial tree ranging from the lowest stock value
            12‘                                12
      of ‗Sd to the highest stock value of ‗Su ‘.

6.4.2 Computing Probabilities
      As discussed in a previous section, the transition needs to be multiplied by itself ‗n‘ = 12 times in
      order to determine the n-step transition matrix from moving from the initial ‗S‘ state to any of the
      terminating states upon expiration of the option.
      This multiplication is done on the ‗Probabilities‘ tab of the Excel workbook. Here the initial one-
      step transition matrix is multiplied by itself to get the two-step transition matrix, P(2). This in turn



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                                                                         A Markovian Options Pricing Model


      is then multiplied by itself to get P(4), and the step is repeated again to get P(8).            P(8) is
      multiplied with P(4) to get P(12).

6.5 Stock Values and Probabilities
      The stock values and probabilities section of the spreadsheet determines the necessary values
      for the final stock prices and the probabilities associated with reaching these end states. The
      figure below gives a portion of this information on the spreadsheet.

      Table 4—Stock Value and Probabilities Table Sample
        Stock Values and Probabilities                                      Option Table
          S=           199.75                                                   X=              200
          u=       1.0361369
          d=       0.9651234


                            u            Value   Discount Probability         Call Value   Put Value
               S            0        199.75      199.00234   0.2254777                0     0.22495
              Su            1     206.96834      206.19366          0                 0           0
          Su^2              2     214.44753      213.64486   0.1898396        2.5903345           0


      The user needs to enter in the current stock price in the shaded box indicated by ‗S‘. In this
      example, IBM stock is currently trading at $199.75. The calculated values for ‗u‘ and ‗d‘ from the
      ‗Information on the Underlying‘ section of the spreadsheet are repeated for the user‘s reference.
      The nominal stock values are calculated for each node and are given under the ‗Value‘ column.
      These nominal values are then discounted to the present value for the purposes of applying
      Decision Theory concepts. These discounted values are given under the ‗Discount‘ column.
      The probabilities for reaching the various states after ‗n‘ steps are from the calculated
      probabilities as discussed in the previous section. These probabilities are listed under the
      ‗Probability‘ column.

6.5.1 Calculating Option Values
      The expected value for the various options are calculated based on the value entered in for the
      strike price (the shaded box) under the ‗Option Table‘ heading. The strike in this example is for
      $200.00.
      These expected option values are calculated by multiplying each profit margin by the probability
      of reaching the appropriate state and then summing these values to calculate the overall
      expected value. These overall values for the call and put options at the given strike are given at
      the bottom of the ‗Call Value‘ and ‗Put Value‘ columns.

6.6 Black-Scholes Pricing Formulas Calculations
      In order to compare the option values obtained by the Markovian model developed in this paper
      to a standard options pricing theory, the values of call and put options were priced using Black-
      Scholes pricing theory. A sample of such a computation is given in the figure below.




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                                                                                        A Markovian Options Pricing Model


     Table 5—Black-Scholes Table Sample
                             Black-Scholes Values
                              sigma =        0.4259911                   call value =   10.032511
                               T-t=                  1/12                put value =    9.5339159
                               d1 =          0.0818098
                               d2 =         -0.0411632


     The values for ‗d1‘ and ‗d2‘ are obtained using the standard Black-Scholes theory as described in
     the section discussing Black-Scholes pricing theory. The values for the call and put are given in
     the table above as well. The values shown in this example are for a strike price of $200.00.

6.7 Option Values
     The option value for various strike prices are calculated under the ‗Option Value‘ segment of the
     spreadsheet. These value are obtained by using the ‗Table‘ feature within Excel which replicate
     multiple calculations and displays only their results within the table. A sample of this table is
     given below. Market options values were retrieved from http://www.cboe.com.

     Table 6—Option Values Table Sample
             Option Values
              Expiration =       May


                                        Ours                   Black-Scholes                  Market
              Strike Price Call Value        Put Value      Call Value     Put Value    Call Value   Put Value
                               9.5777519     9.8277519      10.032511     9.5339159
                       140 59.754209         0.0042094      60.284345     0.0103277            66         1/16
                       145 54.771579         0.0215793      55.319988     0.0272562            51          1/8
                       150 49.792354         0.0423537      50.376556     0.0651097        46 1/8          1/8


     As such, multiple strike prices can be used to calculate various call and put option values for the
     Markovian model discussed in this paper, the Black-Scholes theoretical values, and the actual
     market values.
     The expiration of the options being priced is given in this section for the user‘s reference.

6.8 Statistical Values
     Finally, statistical values for the 95% confidence intervals are also calculated as shown in the
     figure below. Confidence intervals for the difference between the market and the Markovian
     model, and the difference between the market and the Black-Scholes model are given. Half-
     widths are also calculated.

     Table 7—Confidence Interval Table Sample
                                           Confidence Interval
                                           half-width =     1.6301533       0.672244
                                           lower =          -2.2838357    0.4865736
                                           upper =          0.9764708     1.8310616




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                                      A Markovian Options Pricing Model


   Bibliography!




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