Explain why margin accounts are only required when clients write options but not when they buy options?

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Chapter12-Black-ScholesFormula-Students.pptx
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Chapter 12 The Black-Scholes Formula

1

1

Introduction

We will lay the basic foundation that you need to begin working with the standard continuous-time option pricing model, the Black-Scholes model

2

Introduction

Our first concern is to develop a statistical model of how stock prices behave over time

The term stochastic process simply means that the variable moves over time is random

Stochastic processes can be classified in several ways

A stock follows a stochastic process

3

The Markov Property of Stocks

A Markov property is a particular type of stochastic process where only the present value of a variable is relevant for predicting the future

The past history of the variable and how it got to that point are irrelevant

4

Wiener Processes

A Wiener process is a particularly useful type of Markov process: it has a mean change of zero and a variance

Let’s begin by defining a variable, z, which follows a Weiner process

Define Δt as a small length of time (nearly instantaneous) and define Δz as the change in z during time Δt

5

Wiener Processes

A Wiener process, has two primary properties:

Property 1: Δz is related to Δt by the equation:

where ε is a random draw from a standard normal distribution (i.e. ε~N(0,1))

Property 2: The values of Δz for any two different short intervals are independent

It follows from property 1 that Δz itself is distributed normally with mean 0 and variance Δt  It’s standard deviation is (Δt).5

6

Wiener Processes

Thus, σ2 = N*Δt = T and hence σ = T.5

So far we have dealt with a discrete time case

We are interested in a continuous time case

Basically nothing changes except we use continuous time notation:

7

Generalized Wiener Process

We have dealt only with a simple Wiener process

It has a drift rate of 0 and a variance rate of 1.0

We can add non zero drift rates and non-unitary variance rates

8

Generalized Wiener Process

To get to such a process we build off of the basic Weiner process we already have:

dx = a dt + b dz

where a and b are constants, and dz is our previously defined Weiner process

To see this, consider the two components of this equation separately

The 1st portion (a dt) is not random

It merely says that dx is generally drifting upward at some constant rate

9

Generalized Wiener Process

The second term, however, does contain a random component

What we have is the white noise process (dz) multiplied by some non-random constant

Recall that the mean of the product of a random variable times a constant is the product of the mean of that random variable times the constant

In this case, i.e. for

b dz

since dz~N(0,1), the mean of this term is zero

Thus the drift of the entire process, is simply (a*dt)

10

Generalized Wiener Process

Similarly, the variance of the product of a random variable times a constant is the variance of the random variable times the square of the constant

So adding this all together we get: dx ~ N(a dt, b2 dt) and hence the standard deviation of dx is given by: b (dt).5

Similarly over a non-small length of time T:

Mean of change in x = aT

variance of change in x is = b2T

standard deviation of change in x is = b T.5

If a and b are allowed to be functions of both x and t, then we have the most general Markov, stochastic process, the Ito process:

dx = a(x,t) dt + b(x,t) dz

11

The Process for Stock Prices

It is at first tempting to say that stocks follow a generalized Weiner process, that is, they have a constant drift and variance rate

However, this would not take into account that investors demand a rate of return that is independent of a stock’s price

Because of this we cannot use the constant expected-drift rate, but rather have to have a constant proportional drift rate

This means that if S is the stock price, investors want a constant proportional return μS where μ is a constant

12

The Process for Stock Prices

This leads us to represent S by an Ito process of the form:

dS = μS dt + σS dz

or, in the more generally seen form:

dS/S = μ dt + σ dz

Generally μ is referred to as the stock’s expected return, and σ as it’s volatility

13

The Process for Stocks

14

Ito’s Lemma

In the early 1950's Ito developed a method for describing how functions based on random variables evolve

Assume that G is a function of time, t, and a random variable x, i.e. G(x,t)

Let x be a general Ito process:

dx = a(x,t)dt + b(x,t) dz

then Ito’s lemma states that

where dz is the typical Wiener process

15

Ito’s Lemma

Thus G itself follows an Ito process with drift of

and variance rate of

Now, applying this to stocks, if a stock follows a proportional Ito process:

dS = μS dt + σ S dz 

16

Ito’s Lemma

with constant μ and σ, a function G of S (such as a derivative) will follow:

This is the key to pricing all derivatives

Notice that both S and G have one common (and sole) source of uncertainty: dz

This is what allows us to price G in terms of S!

17

Derivation of Black-Scholes

We assume a stock price S which follows the process:

dS = μS dt + σS dz

working with a discrete time version of this:

ΔS = μS Δt + σS Δz.

Let f represent the value of any derivative written on S. From Ito’s lemma, it must be the case that:

18

Derivation Black-Scholes

which by substitution gives us:

19

Derivation Black-Scholes

Notice that we have managed to remove all of the Δz’s from the equation. As a result we know that it must grow at the risk-free rate. It follows, therefore, that

ΔΠ = r Π Δt

where r is the risk-free rate. Substituting back into previous equations gives:

which leads us to the general equation:

20

Derivation Black-Scholes

Notice that f is ANY derivative based on S, not just a call or a put

In fact, to price any particular derivative, you must first impose upon this equation appropriate boundary conditions

These literally specify the boundary values of S and t, and how the derivative behaves at those boundaries

Fortunately, these boundary conditions are basically our payoff conditions:

call option: f = max(S-K,0) when t=T

put option: f = max(K-S,0) when t=T

21

Risk-Neutral Valuation

Notice in the PDE that the drift rate of the stock does not appear

It has fallen out in all of the algebra

Thus, how investors feel about μ is irrelevant when it comes to pricing the option f

We can choose to work in a world with any set of investor attitudes about μ that we want to, and values obtained in those worlds are valid in any other world

This allows us to make a hugely simplifying assumption that investors are risk-neutral

In such a world, all securities would grow at the risk-free rate because investors do not demand a premium to induce them to take risk

22

Risk-Neutral Valuation

This provides us with a rather simple way of valuing a derivative in this world: simply assume that the stock will grow at the risk free rate r until time T

At that point see what would be the payoffs to the derivatives, and then discount those payoffs at the risk-free rate back to today

This is the value of the derivative in both the risk-neutral and risk-averse world

23

Black-Scholes-Merton Assumptions

To get to this point we make a number of assumptions

Most of these assumption are not very binding

The stock price follows the process developed with μ and σ constant

The short selling of securities is allowed

There are not transactions costs or taxes, all securities are perfectly divisible

There are no dividends during the life of the option

There are no arbitrage opportunities

Securities trade continuously

The risk-free rate, r, is constant

24

24

The Black-Scholes-Merton Formulas

25

25

The N(x) Function

N(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x

See tables at the end of the book

26

26

Black-Scholes-Merton with Dividend Yield

27

Implied Volatility

The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price

Traders and brokers often quote implied volatilities rather than dollar prices

28

28

t=z 

dt =dz 

Stock Movements over 3 Years

For 4 Sample Paths

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Week

Stock Price

Path 1

Path 2

Path 3

Path 4

Chart1

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Path 1
Path 2
Path 3
Path 4
Week
Stock Price
Stock Movements over 3 Years For 4 Sample Paths
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52.3064556186
48.4787305121
49.8261155798
49.0047946436
51.3119240989
47.9042099101
48.1373151234
48.3757027587
50.3128692472
45.5924309369
48.1510765863
48.2727280319
48.3536675096
46.1476874418
50.1360804805
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49.224378885
47.1503731726
47.8621686812
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48.7003639601
44.8501995733
50.4279067869
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43.5891457264
32.7034937639
61.4287621092
92.3793644162
44.5356370395
34.8781144845
61.6712323321
95.9316471839
44.3858684811
34.9557428024
61.9685993286
94.2063518392
42.3537263509
36.2451879273
63.404516053
93.949590952
43.4033841833
34.851578065
63.4985995863
93.5723669336
41.636118003
31.7936456973
61.92198927
94.8131383213
42.9709721243
31.3680739495
62.5602997939
94.5146195287
40.5529814892
30.5971786341
63.5059933519
95.5814283416
43.0003502462
31.4092994335
63.1442257644

Sheet1

Parameters:
S 50
mu 0.1
sigma 0.2
dt 0.0192307692 Trial 1 Trial 2 Trial 2 Trial 3 Trial 3 Trial 4
Week Path 1 Path 2 Path 3 Path 4 Step Stock Price Random Numbers Step Stock Price Random Numbers Step Stock Price Random Numbers Step Stock Price Random Numbers
0 50 50 50 50 0 50 0 50 0 50 0 50
1 52.3064556186 48.4787305121 49.8261155798 49.0047946436 1 52.3064556186 1.593871275 1 48.4787305121 -1.166340553 1 49.8261155798 -0.1947273631 1 49.0047946436 -0.7869903129
2 51.3119240989 47.9042099101 48.1373151234 48.3757027587 2 51.3119240989 -0.7897028809 2 47.9042099101 -0.4815206012 2 48.1373151234 -1.2869077182 2 48.3757027587 -0.5216020327
3 50.3128692472 45.5924309369 48.1510765863 48.2727280319 3 50.3128692472 -0.7915855349 3 45.5924309369 -1.7334787117 3 48.1510765863 -0.0568309133 3 48.2727280319 -0.141341161
4 48.3536675096 46.1476874418 50.1360804805 47.8588261833 4 48.3536675096 -1.4825718608 4 46.1476874418 0.3371758339 4 50.1360804805 1.3646331354 4 47.8588261833 -0.3654110969
5 48.7661752837 49.224378885 47.1503731726 47.8621686812 5 48.7661752837 0.230409114 5 49.224378885 2.1546384232 5 47.1503731726 -2.2225503926 5 47.8621686812 -0.0639579412
6 48.2412335219 50.2819775182 47.6698409453 47.5907307637 6 48.2412335219 -0.4461674052 6 50.2819775182 0.6943832886 6 47.6698409453 0.309207735 6 47.5907307637 -0.2621095518
7 45.5915852581 51.8236031702 46.4958256137 48.7003639601 7 45.5915852581 -1.9775870896 7 51.8236031702 1.0419535101 7 46.4958256137 -0.9127006706 7 48.7003639601 0.7341714081
8 44.8501995733 50.4279067869 46.7292965966 50.3075280934 8 44.8501995733 -0.5978449735 8 50.4279067869 -1.0783173821 8 46.7292965966 0.1038802111 8 50.3075280934 1.0914072845
9 46.250535077 50.9917890958 45.973932846 50.6780953328 9 46.250535077 0.947600256 9 50.9917890958 0.3366903911 9 45.973932846 -0.6095024219 9 50.6780953328 0.1974558472
10 47.3751758438 50.5044667222 45.5836334631 48.6076607057 10 47.3751758438 0.746852038 10 50.5044667222 -0.4221260497 10 45.5836334631 -0.3452032615 10 48.6076607057 -1.5632895156
11 45.2980299944 50.3230075143 45.948471194 48.2421961559 11 45.2980299944 -1.5635487216 11 50.3230075143 -0.2008891897 11 45.948471194 0.1998751031 11 48.2421961559 -0.3309469321
12 45.5576006992 52.0037514128 47.9246736116 49.318807913 12 45.5576006992 0.1243620318 12 52.0037514128 1.1422162061 12 47.9246736116 1.3613407646 12 49.318807913 0.7094558896
13 45.3212235673 48.5606769719 46.1370053606 48.9121314165 13 45.3212235673 -0.233630999 13 48.5606769719 -2.5549525162 13 46.1370053606 -1.3555654732 13 48.9121314165 -0.3616514732
14 44.2563989565 49.9338138426 45.1271284849 49.0938243174 14 44.2563989565 -0.8307051758 14 49.9338138426 0.9228415365 14 45.1271284849 -0.7922130862 14 49.0938243174 0.0631916919
15 42.7823408602 48.7543051563 46.6416092437 50.5630841351 15 42.7823408602 -1.1243309928 15 48.7543051563 -0.9198015505 15 46.6416092437 1.0295275388 15 50.5630841351 0.991417437
16 43.3508770727 50.306201336 46.7914201802 52.2832021952 16 43.3508770727 0.3506488611 16 50.306201336 1.0514781934 16 46.7914201802 0.0433499281 16 52.2832021952 1.1702763913
17 41.2898044405 50.7441825738 46.7321198978 50.0964209797 17 41.2898044405 -1.5463774616 17 50.7441825738 0.2460706128 17 46.7321198978 -0.1076500666 17 50.0964209797 -1.6494141164
18 38.7114804592 51.0266356958 46.1467250021 50.0565949638 18 38.7114804592 -1.9165145204 18 51.0266356958 0.1333103228 18 46.1467250021 -0.4869400527 18 50.0565949638 -0.0981901849
19 39.5238847453 49.7179900305 46.8568874774 50.4356644137 19 39.5238847453 0.5321498975 19 49.7179900305 -1.0144390217 19 46.8568874774 0.4481114502 19 50.4356644137 0.2039348601
20 39.8837820751 49.2690223823 46.7588100773 49.7712472986 20 39.8837820751 0.2047158887 20 49.2690223823 -0.3927016223 20 46.7588100773 -0.1357034307 20 49.7712472986 -0.5490596777
21 41.3997639853 47.11087658 45.4731885655 47.6032836924 21 41.3997639853 1.0378812476 21 47.11087658 -1.624584911 21 45.4731885655 -0.9919176591 21 47.6032836924 -1.6323610907
22 44.4926883875 48.1979062261 46.0637433751 46.9148510527 22 44.4926883875 2.1729283617 22 48.1979062261 0.7185371942 22 46.0637433751 0.3627951628 22 46.9148510527 -0.5624497135
23 43.7947896521 48.7942684363 46.672012896 49.4683347391 23 43.7947896521 -0.5649621926 23 48.7942684363 0.3632044354 23 46.672012896 0.3747504707 23 49.4683347391 1.7762840798
24 42.551735841 52.3758419143 48.0580084625 49.593543626 24 42.551735841 -0.9571112969 24 52.3758419143 2.5150438887 24 48.0580084625 0.9347331797 24 49.593543626 0.0216891749
25 44.9970394365 52.139589143 48.5280909279 49.6812254009 25 44.9970394365 1.704324859 25 52.139589143 -0.2429965207 25 48.5280909279 0.2723368198 25 49.6812254009 -0.0055456439
26 44.7470333655 56.2085328149 46.9569906542 48.0665280696 26 44.7470333655 -0.2426816081 26 56.2085328149 2.8618524084 26 46.9569906542 -1.2002328731 26 48.0665280696 -1.2332702681
27 45.9289314245 56.5701405639 45.2047942441 48.6845706318 27 45.9289314245 0.7902258403 27 56.5701405639 0.1828118457 27 45.2047942441 -1.32864443 27 48.6845706318 0.3790205483
28 46.0655870946 57.8303110157 47.8778658835 46.9572700392 28 46.0655870946 0.034851837 28 57.8303110157 0.8302731658 28 47.8778658835 1.8648916011 28 46.9572700392 -1.3130875232
29 44.0058693512 58.7830491888 47.9365725395 47.1357348896 29 44.0058693512 -1.5491650629 29 58.7830491888 0.6068330549 29 47.9365725395 -0.0240606823 29 47.1357348896 0.0635748165
30 44.7204896669 58.0714681234 47.0589244713 47.3353881224 30 44.7204896669 0.4542948773 30 58.0714681234 -0.5946458259 30 47.0589244713 -0.6993570878 30 47.3353881224 0.0786064902
31 43.6846471815 58.7007814806 46.0749007842 46.6188206707 31 43.6846471815 -0.8089727999 31 58.7007814806 0.3732736786 31 46.0749007842 -0.7748485586 31 46.6188206707 -0.5823665106
32 43.9360923298 60.4490289504 46.3296592601 45.7655517969 32 43.9360923298 0.1207399691 32 60.4490289504 1.1792758414 32 46.3296592601 0.1198145583 32 45.7655517969 -0.6799496077
33 47.1218028696 60.53297194 46.1163151796 46.4055971589 33 47.1218028696 2.2363201424 33 60.53297194 -0.0232955699 33 46.1163151796 -0.218092282 33 46.4055971589 0.3980778729
34 49.7506212471 59.3367909535 47.769369581 44.9935883791 34 49.7506212471 1.8303217075 34 59.3367909535 -0.9465225048 34 47.769369581 1.1280826584 34 44.9935883791 -1.082566996
35 50.3730438742 60.2601320964 44.6163602094 43.7247268609 35 50.3730438742 0.379843641 35 60.2601320964 0.5835454431 35 44.6163602094 -2.3399115889 35 43.7247268609 -0.9773839338
36 51.2742471158 60.1168915978 44.8548594982 43.3813558127 36 51.2742471158 0.5800120562 36 60.1168915978 -0.1868579602 36 44.8548594982 0.1101125235 36 43.3813558127 -0.3082436706
37 52.2729085723 59.8015928693 43.4740986445 42.2467778097 37 52.2729085723 0.6490404303 37 59.8015928693 -0.3107322755 37 43.4740986445 -1.0578833098 37 42.2467778097 -0.8783149497
38 53.1602562047 59.7510831154 41.2749173173 44.3334629643 38 53.1602562047 0.567385996 38 59.7510831154 -0.1193529897 38 41.2749173173 -1.6461399355 38 44.3334629643 1.4461443243
39 55.2443719662 59.9424613137 42.2920068331 42.6977640687 39 55.2443719662 1.429157237 39 59.9424613137 0.0551449375 39 42.2920068331 0.6761956683 39 42.6977640687 -1.2409986994
40 55.9243029341 60.7002215259 41.5722674997 41.646220193 40 55.9243029341 0.4136950338 40 60.7002215259 0.4633034223 40 41.5722674997 -0.5776598755 40 41.646220193 -0.8174902177
41 56.467435438 62.4698800238 42.1990856432 39.3176265161 41 56.467435438 0.3141053639 41 62.4698800238 1.1919428289 41 42.1990856432 0.394354629 41 39.3176265161 -1.7369256966
42 58.565305192 62.3759910624 42.4498117479 39.2100828937 42 58.565305192 1.4344891497 42 62.3759910624 -0.1543344297 42 42.4498117479 0.1222815627 42 39.2100828937 -0.1320745469
43 59.9545979952 61.2472173435 42.937151364 37.300651093 43 59.9545979952 0.920617822 43 61.2472173435 -0.9004702406 43 42.937151364 0.2925582976 43 37.300651093 -1.4312854546
44 58.4668343608 61.1921125228 42.33428811 37.0936911498 44 58.4668343608 -1.1559836821 44 61.1921125228 -0.12467126 44 42.33428811 -0.4942739906 44 37.0936911498 -0.2009676336
45 58.7642443653 61.0885709293 42.7111919134 38.7805661013 45 58.7642443653 0.133386493 45 61.0885709293 -0.159523097 45 42.7111919134 0.2130821031 45 38.7805661013 1.1649831322
46 60.0530708155 62.1133203909 40.4951878338 42.1549573681 46 60.0530708155 0.8478946256 46 62.1133203909 0.65424274 46 40.4951878338 -1.6572130335 46 42.1549573681 2.3795291781
47 59.4715918842 62.7986112188 39.397093558 43.5216340174 47 59.4715918842 -0.5025890459 47 62.7986112188 0.4080345661 47 39.397093558 -0.848003765 47 43.5216340174 0.9270661394
48 61.8899850122 59.7125937501 39.2002239272 43.3309651403 48 61.8899850122 1.6614558263 48 59.7125937501 -2.3124448489 48 39.2002239272 -0.1965986485 48 43.3309651403 -0.19784693
49 61.2084354422 59.313994447 38.6768626054 40.5103845984 49 61.2084354422 -0.5772983513 49 59.313994447 -0.3702405138 49 38.6768626054 -0.431762146 49 40.5103845984 -2.0940387913
50 62.3132583515 59.5790494338 38.9452107821 40.7000636231 50 62.3132583515 0.7118183021 50 59.5790494338 0.1088801582 50 38.9452107821 0.1398734639 50 40.7000636231 0.0806016942
51 62.926918321 59.7922063742 38.7182081733 39.2753642209 51 62.926918321 0.3561035555 51 59.7922063742 0.0710883796 51 38.7182081733 -0.2177011993 51 39.2753642209 -1.0838061826
52 62.5955965612 60.9490689034 39.0876140662 38.1355386888 52 62.5955965612 -0.3261834536 52 60.9490689034 0.751308562 52 39.0876140662 0.2126898835 52 38.1355386888 -0.8764050108
53 63.1737941763 60.7168325745 38.6907090497 40.6330531858 53 63.1737941763 0.3301397555 53 60.7168325745 -0.2519891495 53 38.6907090497 -0.3404170457 53 40.6330531858 1.748098839
54 62.6925724206 58.3744319323 36.50424298 41.6970783411 54 62.6925724206 -0.4346202331 54 58.3744319323 -1.773328222 54 36.50424298 -1.630337465 54 41.6970783411 0.7109315447
55 63.8468712582 57.5557644167 37.1437461154 43.4929643584 55 63.8468712582 0.7454377737 55 57.5557644167 -0.6713003131 55 37.1437461154 0.4105299922 55 43.4929643584 1.2372083802
56 63.6108917758 58.384681882 35.1911411817 44.7413665978 56 63.6108917758 -0.2587069048 56 58.384681882 0.5179254003 56 35.1911411817 -1.4595525499 56 44.7413665978 0.8399217677
57 63.4304544884 57.3974848202 37.6612937033 47.2896425941 57 63.4304544884 -0.2183276138 57 57.3974848202 -0.7928429113 57 37.6612937033 1.7324509827 57 47.2896425941 1.7755428416
58 66.8006141301 56.6063664944 39.7247793048 50.002593286 58 66.8006141301 2.3422944651 58 56.6063664944 -0.650079528 58 39.7247793048 1.435773811 58 50.002593286 1.8907576305
59 69.6692517051 56.2320083113 40.3502233823 50.1008205565 59 69.6692517051 1.9759681891 59 56.2320083113 -0.3484524314 59 40.3502233823 0.395925781 59 50.1008205565 0.0014915713
60 70.1881653151 57.4483593406 40.7006692256 50.586844839 60 70.1881653151 0.2775800567 60 57.4483593406 0.7991434359 60 40.7006692256 0.1967543994 60 50.586844839 0.2809997568
61 69.2800006438 57.3005739611 43.6771734744 50.9675835399 61 69.2800006438 -0.7522203305 61 57.3005739611 -0.1862360932 61 43.6771734744 2.0899460651 61 50.9675835399 0.2044032499
62 69.292620397 58.9783464139 42.6114322399 49.0054913814 62 69.292620397 -0.0869738415 62 58.9783464139 1.1303973224 62 42.6114322399 -0.8290862752 62 49.0054913814 -1.4855640984
63 67.2579829335 59.757730583 44.1779634519 45.7779233915 63 67.2579829335 -1.5632895156 63 59.757730583 0.4802336662 63 44.1779634519 1.0705502973 63 45.7779233915 -2.3953907657
64 65.6399371694 60.3041139543 44.5409634987 46.529541537 64 65.6399371694 -1.2600594346 64 60.3041139543 0.31113359 64 44.5409634987 0.2004992439 64 46.529541537 0.4785169949
65 66.5265896732 61.1792723953 46.6646037514 46.8503458609 65 66.5265896732 0.5483479981 65 61.1792723953 0.547458967 65 46.6646037514 1.4696115613 65 46.8503458609 0.1668104233
66 66.7558725414 59.8596415818 45.2078838336 46.9951427551 66 66.7558725414 0.0730824468 66 59.8596415818 -1.0364396985 66 45.2078838336 -1.1151678336 66 46.9951427551 0.0394447852
67 65.1895621867 57.5032260153 46.8995383904 46.7565135362 67 65.1895621867 -1.2220561985 67 57.5032260153 -1.7822458176 67 46.8995383904 1.1571773939 67 46.7565135362 -0.2372485142
68 67.0347377072 56.8817116341 47.9285102349 46.6060639419 68 67.0347377072 1.2401733329 68 56.8817116341 -0.5279230209 68 47.9285102349 0.6769641914 68 46.6060639419 -0.1733303634
69 67.1214763009 57.1735066978 49.2185326366 44.0031910885 69 67.1214763009 -0.030412366 69 57.1735066978 0.1315356712 69 49.2185326366 0.8637834981 69 44.0031910885 -1.9415892893
70 68.5083803204 58.2361571346 47.6247460834 43.9280683994 70 68.5083803204 0.9070299711 70 58.2361571346 0.6870027391 70 47.6247460834 -1.2175496522 70 43.9280683994 -0.1151931883
71 67.3116796899 58.4706762734 48.687791344 43.5352977027 71 67.3116796899 -0.957957127 71 58.4706762734 0.0883551365 71 48.687791344 0.7005291991 71 43.5352977027 -0.3441482477
72 67.5291820067 59.3677008318 49.4887578822 41.2725917834 72 67.5291820067 0.0634986463 72 59.3677008318 0.5657693691 72 49.4887578822 0.5100673661 72 41.2725917834 -1.6920330381
73 67.4047762328 57.3849029411 47.4040731853 40.4851734339 73 67.4047762328 -0.1833564056 73 57.3849029411 -1.5121440811 73 47.4040731853 -1.5719160729 73 40.4851734339 -0.6250502338
74 68.2907583048 57.4197416455 47.6415310797 41.8221339549 74 68.2907583048 0.5454171514 74 57.4197416455 -0.0544559953 74 47.6415310797 0.1054957011 74 41.8221339549 0.9079531083
75 69.2132733925 59.8072172427 48.0525605784 39.3535275391 75 69.2132733925 0.5705328476 75 59.8072172427 1.6420062821 75 48.0525605784 0.2303306701 75 39.3535275391 -1.838134267
76 68.302247007 60.1932421215 49.2156284132 40.3001266864 76 68.302247007 -0.7529320101 76 60.1932421215 0.1954288109 76 49.2156284132 0.7720632311 76 40.3001266864 0.6280288289
77 67.8782315405 60.4766474024 48.0781631709 39.2435273816 77 67.8782315405 -0.4004800758 77 60.4766474024 0.1208934464 77 48.0781631709 -0.8884876479 77 39.2435273816 -0.8178108146
78 69.0663245121 61.0692946105 48.5161258718 40.0562201613 78 69.0663245121 0.7626158549 78 61.0692946105 0.343497959 78 48.5161258718 0.2491469786 78 40.0562201613 0.5316201168
79 68.8881157283 60.0228801561 47.8964048944 37.8545650281 79 68.8881157283 -0.2242859409 79 60.0228801561 -0.8392680684 79 47.8964048944 -0.5141669135 79 37.8545650281 -1.6431840777
80 68.3605753589 58.0581608906 47.4883519959 37.1945332399 80 68.3605753589 -0.4759453986 80 58.0581608906 -1.5000159692 80 47.4883519959 -0.3606714927 80 37.1945332399 -0.5284505278
81 69.9915761803 56.7433746228 46.4871593575 38.3837898128 81 69.9915761803 1.081332357 81 56.7433746228 -1.0286180441 81 46.4871593575 -0.7878247743 81 38.3837898128 0.8060055734
82 71.8565650122 55.7394596939 46.464039169 39.5004660791 82 71.8565650122 1.2478017197 82 55.7394596939 -0.8026222531 82 46.464039169 -0.0811382961 82 39.5004660791 0.7520179679
83 74.5760721116 55.8329128816 44.582052576 39.386636784 83 74.5760721116 1.8614173314 83 55.8329128816 -0.009906671 83 44.582052576 -1.4215538613 83 39.386636784 -0.1368607627
84 73.649115503 56.064621856 43.2817956003 37.3783036656 84 73.649115503 -0.7718563211 84 56.064621856 0.0896613983 84 43.2817956003 -0.9994528227 84 37.3783036656 -1.5028490452
85 73.8907128996 55.7676190964 43.8940578734 36.62036331 85 73.8907128996 0.0720854132 85 55.7676190964 -0.2919193776 85 43.8940578734 0.3814875527 85 36.62036331 -0.5983929441
86 73.8399128855 55.7909540726 42.03244214 32.8093898162 86 73.8399128855 -0.1391003934 86 55.7909540726 -0.0605086825 86 42.03244214 -1.4033003026 86 32.8093898162 -2.798915375
87 74.2590896153 54.4793500202 41.3003921104 33.0924139094 87 74.2590896153 0.1998751031 87 54.4793500202 -1.0231792658 87 41.3003921104 -0.5861772934 87 33.0924139094 0.1585931386
88 75.5955919535 53.2789813946 39.6900414471 34.0645070222 88 75.5955919535 0.860786713 88 53.2789813946 -0.9411473911 88 39.6900414471 -1.2185137166 88 34.0645070222 0.6550953913
89 77.5778412666 54.3481610671 37.9817485404 35.4826695966 89 77.5778412666 1.3245880837 89 54.3481610671 0.6971117728 89 37.9817485404 -1.2869077182 89 35.4826695966 0.975412604
90 78.6706183334 53.1384245269 39.5969112317 35.5452299097 90 78.6706183334 0.6804316399 90 53.1384245269 -0.9477207641 90 39.5969112317 1.1120391719 90 35.5452299097 -0.0040927262
91 79.9468446076 53.7115632278 38.989062255 36.2557176448 91 79.9468446076 0.8112033356 91 53.7115632278 0.3396064585 91 38.989062255 -0.4932371667 91 36.2557176448 0.4630476269
92 79.6730195429 51.1382464481 38.7819927755 35.0841728787 92 79.6730195429 -0.3083243882 92 51.1382464481 -1.9301296561 92 38.7819927755 -0.2033880264 92 35.0841728787 -0.8950905794
93 80.6440358055 50.648538196 37.8004345174 34.6578156616 93 80.6440358055 0.5897231858 93 50.648538196 -0.424049631 93 37.8004345174 -0.7615926734 93 34.6578156616 -0.3561035555
94 79.6403857276 53.2734560351 38.1262116785 35.5832735152 94 79.6403857276 -0.8355755199 94 53.2734560351 1.8226182874 94 38.1262116785 0.1825014806 94 35.5832735152 0.6192954061
95 78.8659344972 54.637897652 37.4697481876 35.5187348494 95 78.8659344972 -0.6689060683 95 54.637897652 0.9100358511 95 37.4697481876 -0.5262540981 95 35.5187348494 -0.0958846158
96 77.1189824679 53.979092873 35.9535545943 36.0574267023 96 77.1189824679 -1.3691123968 96 53.979092873 -0.5508400136 96 35.9535545943 -1.1453039406 96 36.0574267023 0.3392005965
97 78.3682170813 54.0974850872 34.4514940857 37.6849380819 97 78.3682170813 0.793891104 97 54.0974850872 0.0105183062 97 34.4514940857 -1.1330098459 97 37.6849380819 1.1236124919
98 76.8629709071 53.5025067408 35.0781736349 39.7258393147 98 76.8629709071 -1.1941256162 98 53.5025067408 -0.5040647011 98 35.0781736349 0.4041294233 98 39.7258393147 1.4194552023
99 75.9097103919 51.5513433973 32.0286985285 41.9901561888 99 75.9097103919 -0.7939956959 99 51.5513433973 -1.4811985238 99 32.0286985285 -2.2476524464 99 41.9901561888 1.5777322915
100 72.7011898863 50.8062744199 34.9438905482 41.88271969 100 72.7011898863 -2.4189648684 100 50.8062744199 -0.6087657312 100 34.9438905482 2.0577590476 100 41.88271969 -0.1357034307
101 74.1347821401 51.6115577531 36.8672668248 41.7449455059 101 74.1347821401 0.9329596651 101 51.6115577531 0.5102424439 101 36.8672668248 1.3385079001 101 41.7449455059 -0.1574312591
102 76.0609977818 51.1665752698 36.2774352979 43.3756409351 102 76.0609977818 1.2862074072 102 51.1665752698 -0.392453785 102 36.2774352979 -0.4764592632 102 43.3756409351 1.1180213733
103 75.1313142261 49.3223441947 38.3305597495 41.9086389413 103 75.1313142261 -0.775881972 103 49.3223441947 -1.4008492144 103 38.3305597495 1.4302213458 103 41.9086389413 -1.1180213733
104 76.1887272112 51.4508155121 39.9742666846 43.4910566558 104 76.1887272112 0.6583229606 104 51.4508155121 1.4664647097 104 39.9742666846 1.1321390048 104 43.4910566558 1.0829808161
105 77.8619083379 51.1436110141 39.9287193745 41.3529835949 105 77.8619083379 1.1008933143 105 51.1436110141 -0.2928777576 105 39.9287193745 -0.0882789664 105 41.3529835949 -1.6020976545
106 78.0066041144 49.3253142136 40.5616769333 38.4101433726 106 78.0066041144 -0.0036334313 106 49.3253142136 -1.3821158973 106 40.5616769333 0.4010610155 106 38.4101433726 -2.1794585336
107 79.6599670957 47.5004842479 41.9352660078 40.8936501389 107 79.6599670957 1.0840813047 107 47.5004842479 -1.3843055058 107 41.9352660078 0.9342602425 107 40.8936501389 1.7376169126
108 80.7484745016 47.2693695336 42.4340330425 42.1438596755 108 80.7484745016 0.6744653547 108 47.2693695336 -0.2325305104 108 42.4340330425 0.3015122729 108 42.1438596755 0.8448296285
109 80.3594110388 45.7916872852 41.6004669822 41.5263744366 109 80.3594110388 -0.3925356395 109 45.7916872852 -1.1311226444 109 41.6004669822 -0.6599384506 109 41.5263744366 -0.5037179562
110 81.1712337168 45.4438282256 39.2026818569 38.8205145367 110 81.1712337168 0.4739752058 110 45.4438282256 -0.31434638 110 39.2026818569 -1.7867569113 110 38.8205145367 -2.0088100428
111 78.7655474878 45.2140966113 40.9953664293 39.5440406593 111 78.7655474878 -1.8473292585 111 45.2140966113 -0.2286810741 111 40.9953664293 1.238358891 111 39.5440406593 0.4679077392
112 77.7987045925 45.1669073804 40.7056702712 38.8195181194 112 77.7987045925 -0.8064284884 112 45.1669073804 -0.096729309 112 40.7056702712 -0.265753215 112 38.8195181194 -0.5772983513
113 78.356017188 47.2003617597 39.3732667445 38.1652424018 113 78.356017188 0.2939964361 113 47.2003617597 1.4037095752 113 39.3732667445 -1.0172584552 113 38.1652424018 -0.5256379154
114 77.4156327196 43.7017372165 38.0066090393 38.7062158518 114 77.4156327196 -0.7867811291 114 43.7017372165 -2.5883491617 114 38.0066090393 -1.0401117834 114 38.7062158518 0.3371758339
115 77.8005677606 46.26147138 38.4840750008 38.4760372369 115 77.8005677606 0.170224439 115 46.26147138 1.7852471501 115 38.4840750008 0.2915999175 115 38.4760372369 -0.2196600235
116 78.8695793686 45.0241670282 38.7971560305 38.3040621577 116 78.8695793686 0.6629852578 116 45.0241670282 -0.9563859749 116 38.7971560305 0.1723981313 116 38.3040621577 -0.1773696567
117 81.1145668866 45.6338521772 36.7146066146 39.1711586863 117 81.1145668866 1.5095110939 117 45.6338521772 0.3772129276 117 36.7146066146 -1.5555497157 117 39.1711586863 0.5721540219
118 80.974413611 45.3863808614 38.0334630563 37.472497047 118 80.974413611 -0.2135516297 118 45.3863808614 -0.2417368705 118 38.0334630563 0.9001269063 118 37.472497047 -1.2792429516
119 81.7848097914 45.8823584464 38.073556632 39.8949916311 119 81.7848097914 0.4720936886 119 45.8823584464 0.2947149369 119 38.073556632 -0.0238310349 119 39.8949916311 1.6949206838
120 82.749685316 47.5150794474 37.1502076201 40.5145734937 120 82.749685316 0.5823665106 120 47.5150794474 1.1137444744 120 37.1502076201 -0.7186349649 120 40.5145734937 0.3914624358
121 83.6440888586 49.8099771957 36.8695761368 40.3658929336 121 83.6440888586 0.5302104 121 49.8099771957 1.5889827409 121 36.8695761368 -0.2538843091 121 40.3658929336 -0.1633986813
122 82.3302890082 49.0054467705 35.0040592974 43.6217996052 122 82.3302890082 -1.0633880265 122 49.0054467705 -0.6492291504 122 35.0040592974 -1.3963722267 122 43.6217996052 2.2918902687
123 80.8296289155 47.0894066461 36.8197024605 43.8433459639 123 80.8296289155 -1.1963129509 123 47.0894066461 -1.4496345102 123 36.8197024605 1.2607370081 123 43.8433459639 0.0992667992
124 80.7069193026 47.839386352 35.9262236853 47.3295835163 124 80.7069193026 -0.2005776878 124 47.839386352 0.4755167993 124 35.9262236853 -0.695356448 124 47.3295835163 2.4531618692
125 77.3818531242 46.7610280627 34.8968557339 47.746333516 125 77.3818531242 -2.5096596801 125 46.7610280627 -0.8439565136 125 34.8968557339 -0.7921084944 125 47.746333516 0.2348883754
126 78.5342505832 47.1439491446 35.31856017 48.1474415125 126 78.5342505832 0.7236963029 126 47.1439491446 0.2112824404 126 35.31856017 0.2557021617 126 48.1474415125 0.2230308382
127 78.3134464385 46.208703548 32.9144331068 48.9305061078 127 78.3134464385 -0.2681315436 127 46.208703548 -0.7397920854 127 32.9144331068 -1.7826187104 127 48.9305061078 0.497907422
128 80.4409392004 46.6181757893 35.3627895885 47.8318275452 128 80.4409392004 1.425555638 128 46.6181757893 0.2311946901 128 35.3627895885 1.7198908608 128 47.8318275452 -0.8601227819
129 79.3355671988 43.7463537112 37.7492860835 47.9747975459 129 79.3355671988 -0.908646598 129 43.7463537112 -2.1355481294 129 37.7492860835 1.6718877305 129 47.9747975459 0.0367663233
130 77.6871195389 43.7691975548 39.3965165561 49.0344547824 130 77.6871195389 -1.2987311493 130 43.7691975548 -0.0441923476 130 39.3965165561 1.1354859453 130 49.0344547824 0.6976006262
131 76.0438512108 44.0230487346 39.0613663714 50.4378581771 131 76.0438512108 -1.2927102944 131 44.0230487346 0.1223577328 131 39.0613663714 -0.2963133738 131 50.4378581771 0.9440100257
132 77.8861196419 43.7408225401 39.4299139636 50.5751192162 132 77.8861196419 1.2230248103 132 43.7408225401 -0.2645651875 132 39.4299139636 0.2115950792 132 50.5751192162 0.0290356184
133 78.1352266194 45.6037513956 40.0407060698 50.3552554692 133 78.1352266194 0.0716249815 133 45.6037513956 1.2827194951 133 40.0407060698 0.385768999 133 50.3552554692 -0.2286810741
134 79.9423049056 44.5283491042 39.5141825283 50.3176314655 134 79.9423049056 1.19474862 134 44.5283491042 -0.8387246453 134 39.5141825283 -0.4352079941 134 50.3176314655 -0.0969612302
135 80.3144651887 44.6287796457 39.5836475655 51.4712446035 135 80.3144651887 0.1575085662 135 44.6287796457 0.0106717835 135 39.5836475655 -0.0047043613 135 51.4712446035 0.7621042641
136 78.6956473539 44.8334037801 38.9590710816 52.6959022165 136 78.6956473539 -1.2787222659 136 44.8334037801 0.0856675797 136 38.9590710816 -0.5052811503 136 52.6959022165 0.81173539
137 80.2296597624 44.7647886158 39.5120283999 51.8188539066 137 80.2296597624 0.9970608517 137 44.7647886158 -0.1116518433 137 39.5120283999 0.3447166819 137 51.8188539066 -0.7055245987
138 81.7157562959 45.4368384257 41.0682794652 53.0255514514 138 81.7157562959 0.9603809303 138 45.4368384257 0.4225444172 138 41.0682794652 1.0674352779 138 53.0255514514 0.7983021533
139 83.6986100091 45.8972577267 41.7397223073 54.5823476523 139 83.6986100091 1.3165367818 139 45.8972577267 0.2690035217 139 41.7397223073 0.4272328624 139 54.5823476523 1.0490884961
140 87.7573118431 44.403589706 39.2030395137 56.2504783991 140 87.7573118431 2.8107024264 140 44.403589706 -1.1407473721 140 39.2030395137 -1.8871105567 140 56.2504783991 1.127214091
141 89.7750065548 46.0839985906 39.0225412692 56.979836263 141 89.7750065548 1.333282853 141 46.0839985906 1.1501833797 141 39.0225412692 -0.1845239694 141 56.979836263 0.4479420568
142 90.7819183858 44.6340385822 39.804920788 57.8996506413 142 90.7819183858 0.6015989129 142 44.6340385822 -1.1094880392 142 39.804920788 0.5100673661 142 57.8996506413 0.5842707651
143 92.0908949384 42.3685520063 38.8253296992 58.5578675985 143 92.0908949384 0.8180245459 143 42.3685520063 -1.6955618776 143 38.8253296992 -0.7615926734 143 58.5578675985 0.394354629
144 89.2031742093 41.0951562975 35.1018385122 59.8343342879 144 89.2031742093 -2.2100721253 144 41.0951562975 -0.9770133147 144 35.1018385122 -2.7388887247 144 59.8343342879 0.8392680684
145 89.9050017619 45.7465416535 38.0656965279 59.9957147064 145 89.9050017619 0.3823924999 145 45.7465416535 3.2971729524 145 38.0656965279 2.088590918 145 59.9957147064 0.0333977823
146 90.7053025168 44.4222303326 37.627626877 59.9650077002 146 90.7053025168 0.4524292763 146 44.4222303326 -1.0184135135 146 37.627626877 -0.368684141 146 59.9650077002 -0.1053422238
147 92.2780182886 44.4714253626 34.28060868 59.0845805932 147 92.2780182886 1.0083158486 147 44.4714253626 -0.026127509 147 34.28060868 -2.4657492759 147 59.0845805932 -0.7180415196
148 95.2573382288 43.5891457264 32.7034937639 61.4287621092 148 95.2573382288 2.0204515749 148 43.5891457264 -0.6978916645 148 32.7034937639 -1.1848123904 148 61.4287621092 1.6084777599
149 92.3793644162 44.5356370395 34.8781144845 61.6712323321 149 92.3793644162 -2.2074345907 149 44.5356370395 0.622077323 149 34.8781144845 1.5227897165 149 61.6712323321 0.0896613983
150 95.9316471839 44.3858684811 34.9557428024 61.9685993286 150 95.9316471839 2.4334804039 150 44.3858684811 -0.1697594598 150 34.9557428024 0.0076113338 150 61.9685993286 0.128911779
151 94.2063518392 42.3537263509 36.2451879273 63.404516053 151 94.2063518392 -1.3771614249 151 42.3537263509 -1.5269506548 151 36.2451879273 0.8813572094 151 63.404516053 0.9495192899
152 93.949590952 43.4033841833 34.851578065 63.4985995863 152 93.949590952 -0.3157936135 152 43.4033841833 0.6981849765 152 34.851578065 -1.0552093954 152 63.4985995863 -0.020081643
153 93.5723669336 41.636118003 31.7936456973 61.92198927 153 93.5723669336 -0.4023047495 153 41.636118003 -1.3345834304 153 31.7936456973 -2.2534368327 153 61.92198927 -1.2249665815
154 94.8131383213 42.9709721243 31.3680739495 62.5602997939 154 94.8131383213 0.7649714462 154 42.9709721243 0.9048380889 154 31.3680739495 -0.3509740054 154 62.5602997939 0.3744219157
155 94.5146195287 40.5529814892 30.5971786341 63.5059933519 155 94.5146195287 -0.3467471288 155 40.5529814892 -1.8032278604 155 30.5971786341 -0.5994002095 155 63.5059933519 0.5951937965
156 95.5814283416 43.0003502462 31.4092994335 63.1442257644 156 95.5814283416 0.6382185802 156 43.0003502462 1.7085858417 156 31.4092994335 0.5431979844 156 63.1442257644 -0.3489412848

Sheet1
Path 1
Path 2
Path 3
Path 4
Week
Stock Price
Stock Movements over 3 Years For 4 Sample Paths

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