bob mcdonald

8/12/2019 Bob Mcdonald

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Using Rto Teach MBA Derivatives

Robert L. McDonald

Kellogg School of Management

Northwestern University

May 19, 2014

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Outline

How I got started with R

Why use Rin an MBA class?

Binomial pricing example

The Radoption decision as an option exercise

problem

Examples of Rin action

Everyone in this audience made a decision to useR

; myfocus will be on persuadingothersto use R

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Context

Derivatives Markets II is an advanced elective

Topics include: The lognormal model Brownian motion and Its Lemma Black-Scholes-Merton analysis

Stochastic volatility Exotic options One-factor fixed income models Credit risk

Focus on computation. Originally we used Excel/VBA

Fully transitioned to Rthis year

All Kellogg students receive Stata

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Educational Goals

In addition to the obvious (for example, learning about

different kinds of exotic options), major themes include:

Simple data skills (e.g., historical volatility)

Monte Carlo valuation Appreciation of Jensens inequality

Implications of normality and why it is inadequate for

modeling stock returns

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Why R?

Vectorization Hard to overstate pedagogical importance set.seedpermits comparison of loops and vectors

High quality graphics

Integrated statistics

RStudio: Makes Rfeasible for students

knitr a terrific tool for the instructor

Free and cross-platform

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Example: Binomial Option Pricing

Major topic in an introductory derivatives class

Assume the stock moves discretely up or down, from

S touSordS

Probability of up move is the risk-neutral probability,p

uanddare calibrated to match volatility

The binomial model has great pedagogical value and

it can be used to price American options

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Binomial Stock Price Tree

0.0 0.2 0.4 0.6 0.8 1.0

60

80

100

140

00

118.9

84.1

141.40

100.00

70.72

168.

118.

84.1

59.

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American Options

American options may be exercised by the holder at

any time

When is it optimal to exercise the option: payingK inexchange for the stock, earning the intrinsic value,

S K?

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The Economics of Early Exercise

WhenS>K, three factors determine whether to exercisea standard American call prior to expiration

Dividend payments induce exercise (the holder of an

unexercised call forgoes dividends)

Interest leads to delay of exercise (defer payment ofthe strike price)

Volatility leads to delay of exercise (with higher

volatility, greater chance of later regret for having

exercised)

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Illustrating Early Exercise

An Rplot can be used to show simultaneously

The possible values of future stock prices

The probability of each price Whether the option is exercised at a given node

(green means exercise, red means not)

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Illustrating Early Exercise: ExampleS = 100,K = 95, = 0.3,r = 0.08, = 0.08,T = 1,

n= 20:

0.0 0.2 0.4 0.6 0.8 1.0

100

200

30

0

400

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Drawing the Stock Price Points

The circles are color-coded to denote early exercise(green) or not (red):

exer 0])*pointsize

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Comparative Statics for Early ExerciseS = 100,K = 95,T = 10,n= 20:

0 2 4 6 8 1 0

= 0.03,r =0.1, =0.3

0 2 4 6 8 1 0

0

200

400

600

800

= 0.1,r = 0.03, = 0.3

= 0.1,r = 0.03, =0.1

0

200

400

600

800

= 0.1,r = 0.03, = 0.5

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Application to the RAdoption Decision

Adopting software is analogous to exercising anAmerican call

The adoption cost is the strike price The value from adopting is the stock price The flow of benefits lost by not adopting is the

dividend yield The cost reduction by waiting is the interest rate Volatility measures uncertainty about the value of

adopting

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The Adoption Decision, cont.

Optimal to adopt when The flow of value is great (dividends are large) The cost reduction from waiting is small

interest rate is small

strike price is constant or increasing Likelihood of regret is small (volatility is low and/or

intrinsic value is great)

All of these must be assessed relative to

alternatives. . .

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Adoption Decision

Table :Comparison of Software for MBA Use

VBA/Excel Stata Matlab R SAS

Programming? Y Y/N Y Y Y

Ease of learning Medium Easy/Hard Med/Hard Med/Hard Med

Versatility Low Medium High High Low

Transferability

of Skills Medium Low High High Low

Future use likely High Low ? Med ?

Longevity ? High High High HighCost Low Medium High Very Low High

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Questions to Consider. . .

What backgrounds and capabilities do students

have?

Should students be active or passive users?

Is the software intended for class only, or for

long-term student use (after graduation)

Student believe that VBA and Matlab are important

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Adoption Issues Specific to R

Interface / RStudio

Necessity of programming Doessource('foo.R')count as programming?

Instructor friendly? Should one use base Ror packages?

I use base Runless there is a compelling reason to

do otherwise

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The Decision to Adopt

Large dividend flow from R Whether its greater than for alternatives depends on

context

Prior to RStudio, high gain to waiting

Volatility (relative to alternatives) was high, now

seems low For example, Octave looked like a contender, raising

volatility, delaying my adoption of R

IsS>K?

Depends on course, student, and instructor RStsudio made the adoption decision much easier

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D E l


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Data: Example

Even students with some statistics background often

have little or no experience dealing with primarydata

Usingread.csvto obtain web data gets their attention

Following example is not trivial, but It illustrates the power of a script It demonstrates time variation in volatility The same exercise could be done in Excel (students

can compare the difficulty and reproducibility) Deciphering the code is a worthwhile exercise

The example is easy to generalize (looping across

tickers, for example)

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D t E l


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Data: Example

1 ## Yahoo format is date, open, high, low, close,volume, adj close

2 URL


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Data: Result

1960 1970 1980 1990 2000 2010

0.1

5

0.2

0

0.2

5

0.3

0

0.3

5

0.4

0

0.4

5

Annual Volatility for IBM

Year

Volatility

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Val ing a E ropean Call


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Valuing a European Call

Some cases display a perfect correspondence between

equations in the book and equations in R:

1 s0


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Valuing a European Call (jump version)

Merton: lognormal jumps occur as a Poisson process

Letgbe the jump intensity,J = ea

1 the mean

jump, andvJthe jump volatility.

On paths when no jumps occur, the price is

st0


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Simulating the Merton Jump Model

1 s0


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Arithmetic Asian Options

The payoff of an Arithmetic Asian call is

max

0,

n

i=1

Si

n K

where theSis are sampled at regular intervals

Because each subsequent stock price is based on

the previous, it is tempting to write a loop.

This is not necessary, but the correct code is tricky

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The Problem


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The Problem

Rcan be too lenient:

1 > 1:3 + matrix(1, nrow=6, ncol=2)2 [,1] [,2]3 [1,] 2 24 [2,] 3 35 [3,] 4 4

6 [4,] 2 27 [5,] 3 38 [6,] 4 49 > 1:3 + matrix(1, nrow=2, ncol=6)

10 [,1] [,2] [,3] [,4] [,5] [,6]

11 [1,] 2 4 3 2 4 312 [2,] 3 2 4 3 2 4

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Arithmetic Asian Call: Code


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Arithmetic Asian Call: Code

1 s0


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Black-Scholes Pricing

bsOpt(s=100, k=95, v=0.3, r=0.08, tt=1, d=0,greeks=TRUE)

## Call Put## Price 18.38706 6.083114

## Delta 0.7216145 -0.2783855## Gamma 0.0111875 0.01119105## Vega 0.3356785 0.3356785## Rho 0.5377439 -0.3392167## Theta -0.02558117 -0.00636012## Psi -0.7216145 0.2783855## Elasticity 3.924577 -4.576365

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Code to show off R


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Code to show off R

bsOptreturns a data frame:

columns are calls and puts rows are price and Greeks

1 k


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Output

0 100 200

0

100

Call Price

S

Price

0 100 200

0.0

0.6

Call Delta

S

De

lta

0 100 200

0.0

00

0.0

10

Call Gamma

S

Gamma

0 100 200

0.0

0.3

Call Vega

S

Vega

0 100 200

0.0

1.0

Call Rho

S

Rho

0 100 200

0

.020

0.0

00

Call Theta

S

The

ta

0 100 200

5

2

0

Call Psi

S

Ps

i

0 100 200

5

20

Call Elasticity

S

Elas

tic

ity

0 100 200

0

40

80

Put Price

S

Price

0 100 200

1

.0

0

.2

Put Delta

S

De

lta

0 100 200

0.0

00

0.0

10

Put Gamma

S

Gamma

0 100 200

0.0

0.3

Put Vega

S

Vega

0 100 200

1

.5

0.0

Put Rho

S

Rho

0 100 200

0.0

00

Put Theta

S

The

ta

0 100 200

0.0

0.6

Put Psi

S

Ps

i

0 100 200

7

3

0

Put Elasticity

S

Elas

tic

ity

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The Batman: Expiration


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The Batman: Expiration

40 60 80 100 120 140 160

30

20

10

0

10

s

profit

Sell two 85 puts, buy one 95 put, buy one 105 call, sell two 115 calls

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The Batman: Price


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The Batman: Price

StockPrice

4060

80100

120140

160

Tim

e

0.2

0.4

0.6

0.8

1.0

Positio

nP

rofit

30

20

10

0

10

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The Batman: Delta


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The Batman: Delta

StockPrice

4060

80100

120140

160

Matu

rity

0.5

1.0

1.5

2.0

Delta

1.0

0.5

0.0

0.5

1.0

Delta

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The Batman: Gamma


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The Batman: Gamma

StockPrice

4060

80100

120140

160

Matu

rity

0.5

1.0

1.5

2.0

Gamma

0.5

0.0

Gamma

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The Batman: Theta


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The Batman: Theta

StockPrice

4060

80100

120140

160

Matu

rity

0.5

1.0

1.5

2.0

Theta

0.5

0.0

0.5

1.0

Theta

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The Batman: Rho


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StockPrice

4060

80100

120140

160

Matu

rity

0.5

1.0

1.5

2.0

Rho

1.5

1.0

0.5

0.0

0.5

1.0

Rho

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The Quincunx


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A quincunx, or Galton board, is a physical device that

illustrates the Central Limit Theorem

Balls hit a series of pegs, randomly falling left or right

Rcan (very crudely) simulate the appearance of aquincunx

With R, unlike with a physical quincunx, it is possible

to alter the probability from 50-50.

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The Quincunx


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0 1 2 3 4 5 6 7 8

Number of Rightward Bounces

0

100

200

300

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Thank You


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Thanks to all the fantastic folks who push this great

software forward: R Core Team

the RStudio team

the Revolution Analytics team

Hadley Wickham

Yihui Xie

Dirk Eddelbuettel

R/Finance organizers and UIC etc.

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