Optionsprisfasts忙ttelse Introduktion _amp; repetetion by yurtgc548

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									Option Pricing and Dynamic
Modeling of Stock Prices
Investments 2004
Motivation
  We must learn some basic skills and set
  up a general framework which can be
  used for option pricing.
  The ideas will be used for the
  remainder of the course.
  Important not to be lost in the
  beginning.
  Option models can be very
  mathematical. We/I shall try to also
  concentrate on intuition.             2
Overview/agenda
 Intuition behind Pricing by arbitrage
 Models of uncertainty
  n   The binomial-model. Examples and general results
  n   The transition from discrete to continuous time
 Pricing by arbitrage in continuous time
  n   The Black-Scholes model
  n   General principles
  n   Monte Carlo simulation, vol. estimation.
 Exercises along the way

                                                    3
Here is what it is all about!
   Options are contingent claims with
   future payments that depend on the
   development in key variables (contrary
   to e.g. fixed income securities).
Value(0)=?
                 ?

   0                                  T
                               Value(T)=[ST-X]+
                                            4
The need for model-building
  The Payoff at the maturity date is a well-
  specified function of the underlying variables.
  The challenge is to transform the future
  value(s) to a present value. This is
  straightforward for fixed income, but more
  demanding for derivatives.
  We need to specifiy a model for the
  uncertainty.
  Then pricing by arbitrage all the way home!

                                                5
Pricing by arbitrage - PCP
 Transaction Time 0 price Time T flow Time T flow
                                  ST>X         ST<X
 Long stock      -S0               ST           ST
 Long put(X)      -P                0          X-ST
 Loan(X)        PV(X)              -X           -X
 Sum          PV(X)-P-S0          ST-X           0
 Long call(X)     -C              ST-X           0

Therefore: C = P + S0 – PV(X) ..otherwise there is arbitrage!
                                                      6
Pricing by arbitrage
  So if we know price of
  n   underlying asset
  n   riskless borrowing/lending (the rate of interest)
  n   put option
  then we can uniquely determine price of
  otherwise identical call

  If we do not know the put price, then we
  need a little more structure......

                                                          7
The World’s simplest model of
undertainty – the binomial model

   Example: Stockprice today is $20
   In three months it will be either $22 or
   $18 (+-10%)
                             Stockprice = $22

Stockprice = $20
                             Stockprice = $18



                                                8
A call option

  Consider 3-month call option on the stock and with an
  exercise price of 21.

                                    Stockprice = $22
                                    Option payoff = $1
Stockprice = $20
Option price=?
                                   Stockprice = $18
                                   Option payoff = $0


                                                         9
Constructing a riskless portfolio
 Consider the portfolio:   long D stocks
                           short 1 call option

                                 22D – 1


                                 18D

 The portfolio is riskless if 22D – 1 = 18D ie.
 when D = 0.25.



                                                  10
Valuing the portfolio

  Suppose the rate of interest is 12%
  p.a. (continuously comp.)
  The riskless portfolio was:
           long 0.25 stocks
           short 1 call option
  Portfolio value in 3 months is
     22´0.25 – 1 = 4.50.
  So present value must be
     4.5e – 0.12´0.25 = 4.3670.
                                        11
Valuing the option
   The portfolio which was
       long 0.25 stocks
       short 1 option
   was worth 4.367.
   Value of stocks
       5.000 (= 0.25´20 ).
   Therefore option value must be
       0.633 (= 5.000 – 4.367 ),
   ...otherwise there are arbitrage
   opportunities.
                                      12
Generalization

  A contingent claim expires at time T and
  payoff depends on stock price

                           S0u
                           ƒu
            S0
            ƒ              S0d
                            ƒd

                                             13
Generalization
 Consider portfolio which is long D stocks and short 1
 claim
                                S0 uD – ƒu
    D S0–   f
                                S0dD – ƒd
 Portfolio is riskless when S0uD – ƒu = S0d D – ƒd or




 Note: D is the hedgeratio, i.e. the number of stocks
 needed to hedge the option.
                                                        14
Generalization
    Portfolio value at time T is
       S0u D – ƒu. Certain!
    Present value must thus be
    (S0u D – ƒu )e–rT
    but present value is also given
    as S0D – f
    We therefore have
       ƒ = S0D – (S0u D – ƒu )e–rT
                                      15
Generalization

  Plugging in the expression for D we get

         ƒ = [ q ƒu + (1 – q )ƒd ]e–rT


 where



                                         16
Risk-neutral pricing
 ƒ = [ q ƒu + (1 – q )ƒd ]e-rT = e-rT EQ{fT}
 The parameters q and (1 – q ) can be interpreted as
 risk-neutral probabilities for up- and down-
 movements.
 Value of contingent claim is expected payoff wrt. q-
 probabilities (Q-measure) discounted with riskless
 rate of interest.
                           q       S0u
                                    ƒu
               S0
               ƒ                   S0d
                     (1 –
                          q    )    ƒd
                                                        17
Back to the example
                                     S0u = 22
                                 q    ƒu = 1
                 S0
                  ƒ
                          (1 –       S0d = 18
                      )          q    ƒd = 0


   We can derive q by pricing the stock:
       20e0.12 ´0.25 = 22q + 18(1 – q ); q =
   0.6523
   This result corresponds to the result from
   using the formula

                                                18
Pricing the option

                               S0u = 22
                     0.6 523    ƒu = 1
                S0
                ƒ
                     0.34      S0d = 18
                          77
                                ƒd = 0
     Value of option is

   e–0.12´0.25 [0.6523´1 + 0.3477´0] = 0.633.


                                                19
Two-period example
                              24.2
                22

     20                       19.8

                18
                              16.2
     Each step represents 3
     months, dt=0.25

                                     20
Pricing a call option, X=21
                                                 24.2
                                                  3.2
                               22
                                      B
              20             2.0257              19.8
                    A
          1.2823                                  0.0
                                18
                                      C
                                0.0              16.2
 Value in node B                                  0.0
     = e–0.12´0.25(0.6523´3.2 + 0.3477´0) = 2.0257
 Value in node A
     = e–0.12´0.25(0.6523´2.0257 + 0.3477´0)
     = 1.2823

f = e-2rdt[q2fuu + 2q(1-q)fud + (1-q)2fdd] = e-2rdt EQ{fT} 21
General formula




                  22
Put option; X=52
 u=1.2, d=0.8, r=0.05, dt=1, q=0.6282
                                        72
                                         0
                       60
         50          1.4147             48
     4.1923                              4
                        40
                      9.4636            32
                                        20



                                             23
American put option – early
exercise

                                           72
                                            0
                       60
                              B
          50         1.4147                48
               A
      5.0894                                4
                        40
                             C
                       12.0                32
                                           20

 Node C: max(52-40, exp(-0.05)*(q*4+(1-q)*20))

                                  9.4636         24
Delta

 Delta (D) is the hedge ratio,- the change in
 the option value relative to the change in the
 underlying asset/stock price
 D changes when moving around in the
 binomial lattice
 It is an instructive exercise to determine the
 self-financing hedge portfolio everywhere in
 the lattice for a given problem.



                                              25
How are u and d chosen?
  There are different ways. The following
  is the most common and the most
  simple




  where s is p.a. volatility and dt is length
  of time steps measured in years. Note
  u=1/d. This is Cox, Ross, and
  Rubinstein’s approach.                    26
Few steps => few states.
A coarse model




                           27
Many steps => many states.
A ”fine” model




                             28
Call, S=100, s=0.15, r=0.05, T=0.5, X=105




                                            29
30
Alternative intertemporal
models of uncertainty

  Discrete time; discrete variable (binomial)
  Discrete time; continuous variable
  Continuous time; discrete variable
  Continuous time; continuous variable

 All can be used, but we will work towards the
 last type which often possess the nicest
 analytical properties

                                                 31
The Wiener Process – the key
element/the basic building block
Consider a variable z, which takes on
continuous values.
The change in z is dz over time interval
of length dt.
z is a Wiener proces, if
1.
2. Realization/value of dz for two non-
overlapping periods are independent.
                                           32
Properties of the Wiener process

  Mean of [z (T ) – z (0)] is 0.
  Variance of [z (T ) – z (0)] is T.
  Standarddeviation of [z (T ) – z (0)] is


A continuous time model is obtained by letting dt
approach zero. When we write dz and dt it is to be
understood as the limits of the corresponding
expressions with dt and dz, when dt goes to zero.

                                              33
The generalized Wiener-process

   The drift of the standard Wiener-process
   (the expected change per unit of time) is
   zero, and the variance rate is 1.
   The generalized Wienerprocess has
   arbitrary constant drift and diffusion
   coefficients, i.e.
                    dx=adt+bdz.
   This model is of course more general but it
   is still not a good model for the dynamics of
   stock prices.
                                               34
35
Ito Processes
 The drift and volatility of Ito processes
 are general functions
          dx=a(x,t)dt+b(x,t)dz.
 Note: What we really mean is

  where we let dt go to zero.
 We will see processes of this type many
 times! (Stock prices, interest rates,
 temperatures etc.)                    36
A good model for stock prices




where m is the expected return and s is
the volatility. This is the Geometric
Brownian Motion (GBM).
The discrete time parallel:

                                          37
The Lognormal distribution
  A consequence of the GBM specification is

                                    We will show
                                    this shortly!!




  The Log of ST is normal distributed, ie. ST
  follows a log-normal distribution.

                                           38
Lognormal-density




                    39
 Monte Carlo Simulation
     The model is best illustrated by sampling a
     series of values of e and plugging in……
     Suppose e.g., that m= 0.14, s= 0.20, and dt =
     0.01, so that we have




Methods for sampling e’s…
                                               40
Monte Carlo Simulation – One path




    You MUST go home and try this…
                                     41
A sample path:




                 42
Moving further: Ito’s Lemma

  We need to be able to analyze functions of S
  since derivates are functions of eg. a stock
  price. The tool for this is Ito’s lemma.
  More generally: If we know the stochastic
  process for x, then Ito’s lemma provides the
  stochastic process for G(t, x).




                                            43
Ito’s lemma in brief
  Let G(t,x) and dx=a(x,t)dt + b(x,t)dz




                                          44
Why the extra term?
  Because

  so

  But              has expected value of 1 and variance
                   of term is of order (dt)2


   So it is deterministic in the limit…..


                                                     45
Ito’s lemma
 Substituting the expression for dx we get:




 THIS IS ITO’S LEMMA!

 The option price/the price of the contingent claim is also
 a diffusion process!


                                                       46
Application of Ito’s lemma to
functions of GBM




                                47
Examples




           Integrate!

                        48
The Black-Scholes model
  We consider a stock price which evolves as a
  GBM, ie.
              dS = mSdt + sSdz.

  For the sake of simplicity there are no
  dividends.
  The goal is to determine option prices in this
  setup.

                                               49
Pre-Nobel prize methodology
  Calculate expected payoff. See note…
  Discount using r… or m…. or something
  else…??




                                     50
The idea behind the Black-
Scholes derivation
   The option and the stock is affected by the same
   uncertainty generating factor.
   By constructing a clever portfolio we can get rid of
   this uncertainty.
   When the portfolio is riskless the return must
   equal the riskless rate of interest.
   This leads to the Black-Scholes differential
   equation which we will then find a solution to.

   Let’s do it! ......

                                                      51
Derivation of the Black-Scholes equation




                                           52
Derivation of the Black-Scholes
differential equation




                 The uncertainty/risk of these terms
                 cancel, cf. previous slide.
                                             53
Derivation of the Black-Scholes
differential equation




                                  54
The differential equation
 Any asset the value of which depends on the
 stock price must satisfy the BS-differential
 equation.
 There are therefore many solutions.
 To determine the pricing functional of a particular
 derivative we must impose specific conditions.
 Boundary/terminal conditions.
 Eg: For a forward contrakt the boundary condition
 is        ƒ = S – K when t =T
 The solution to the pde is thus
               –r (T – t )
                               Check the pde!   55
    ƒ=S –Ke
Risk-neutral pricing
 The parameter m does not appear in the BS-
 differential equation!
 The equation contains no parameters with
 relation to the investors’ preferences for risk.
 The solution to the equation is therefore the
 same in ”the real World” as in a World where
 all investors are risk-neutral.
 This observation leads to the concept of risk-
 neutral pricing!


                                                    56
Risk-neutral pricing in practice

 w Assume the expected stock return is
   equal to the riskless rate of interest,
   ie. use m=r in the GBM.
 w Calculate the expected risk-neutral
   payoff for the option.
 w Perform discounting with riskless rate
   of interest, i.e.

                                         57
Black-Scholes formulas




                         58
The Monte Carlo idea
  General pricing relation:

  For example:

  These expressions are the basis of
  Monte Carlo simulation. The expectation
  is approximated by:

                                       59
The market price of risk
  The fundamental pde. holds for all derivatives written
  on a GBM-stock.
  If the underlying is not traded (eg. a ”rate of
  interest”, a temperature, a snow depth, a Richter-
  number etc.) we can derive a similar pde, but there
  will be a term for the market price of risk of this
  factor.
  For example we can use Ito’s lemma to show that
  derivatives will follow


  where

                                                      60
The market price of risk

 The market price of risk can not be
 determined from arbitrage arguments
 alone. It must be estimated using
 market data.
 When simulating the risk neutralized
 underlying variable the drift must be
 adjusted with a term which includes the
 market price of risk.
                                       61
Example of a non-priced
underlying variable




                          62
Historical volatility
 • Observe S0, S1, . . . , Sn with interval
   length t years.
 • Calculate continuous returns in every
   interval:


 3. Estimate standard deviation, s , of the
    ui´s.
 4. The historical annual volatility:

                                              63
Implied volatility

  The implied volatility is the volatility which
  – when plugged into the BS-formula –
  creates correspondence between model-
  and market price of the option.
  The BS-formula is inverted. This is done
  numerically.
  In the market volatility is often quoted in
  stead of price.

                                              64
Exercises/homework!
  Simulate a GBM and show the result graphically using
  a spread sheet.
  Compare the Black-Scholes price with the price of
  options found using the binomial approximation. How
  big must N be in order to obtain a ”good result”?
  Try to estimate the volatility using a series of stock
  prices which you have simulated (so that you know
  the true volatility).
  Try to determine some implied volatilities by inverting
  the BS formula.
  Try to determine a call price using Monte Carlo
  simulation and compare your result with the exact
  price obtained from the BS formula.
                                                      65

								
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