6. Basics of Derivatives Pricing

8/10/2019 6. Basics of Derivatives Pricing

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Dr. Denis Schweizer

Associate Professor of Finance John Molson School of Business, Concordia University

Mailing address: 1455 de Maisonneuve Boulevard West, Montreal, Quebec H3G 1M8

Office: MB 11.305

Phone: +1(514)-848-2424, ext. 2926Fax: +1(514)-848-4500

E-mail: [email protected]

6. Basics of Derivatives Pricing

Investment Analysis



Page 2Investment AnalysisDenis Schweizer

Options – Some Basics

Underlying for Exchange-Traded Options

− Stocks

− Interest Rates

− Foreign Exchange (FX)

− Stock Indices

−

Futures− …

Specification of Exchange-Traded Options

− Expiration date

− Strike price

− European or American

− Call or Put (option class)

Moneyness :

− At-the-money option

− In-the-money option

− Out-of-the-money option

Option class

Option series

Intrinsic value

Time value “Difficult” to evaluate




Put Call

L o n g

S h o r t

Summary

XS T

-X

-X+ P0

0

P0

P/L T, P T

P/L T

P T

0

X-P0

X

X S T

P T

P/L T

P0 = Put Purchase Price-P0

P/L T, P T

C0

0X S T

P/L T, C T

C T

P/L T

-C0

0X

S T

P/L T, C T

C0

= Call

Purchase Price

C T

P/L T




Payoffs and P/L on Options at Expiration – Call

Holder and Call Writer

Notation

Time = = 0, … , = , Stock Price at Maturity = , Call Price at Maturity = , andExercise (Strike) Price =

Payoff (= value at expiration) to Call Holder = ( ; 0)

, if >

0, if ≤

Profit to Call Holder

/ =

Payoff (= value at expiration) to Call Writer = ( ; 0)

, if >

0, if ≤

Profit to Call Writer

/ = +




Payoff and P/L to Call Holder at Expiration




Payoff and P/L to Call Writers at Expiration




Payoffs and P/L on Options at Expiration – Call

Holder and Call Writer

Notation

Time = = 0, … , = , Stock Price at Maturity = , Put Price at Maturity = , andExercise (Strike) Price =

Payoff (= value at expiration) to Put Holder = ( ; 0)

, if <

0, if ≥

Profit to Put Holder

/ =

Payoff (= value at expiration) to Put Writer = ( ; 0)

, if <

0, if ≥

Profit to Put Writer

/ = +




Payoff and Profit to Put Holder at Expiration




Call Option Value Before Expiration




The One-Step Binomial Model

Generalization

A derivative expires at and is dependenton a stock

0 = Stock price in = 0

0 = Price of Call-Option in = 0

= Up-Movement (>1)

= Down-movement (<1)

Example

A stock price is currently 0 = $20 In three months it will be either $22 or

$18 ( = 1.1, = 0.9)

A 3-month call option on the stock has a

strike price of = 21.

Risk-Free Rate is

= 5%

0 = Option Price

= ( –

; 0) 0 ∙

0 ∙

0

0

= $22

= $1

= $18 = $0

0 = $20

0 =?

How can we calculate the price of the Call-Option?




Pricing a Derivative in a Two-Period Model

Generalization

The value of a derivative is equal to theexpected pay-offs in the different states,

weighted by risk neutral probabilities,

discounted to = 0

Risk neutral probability:

=1 +

Example

The value of the option then is:

−

1 = 0.75 ∙ 1 + 0.25 ∙ 0 = 0.75

− 0 = 0.75

1+5% .= 0.7409

0 ∙

0 ∙

0

0

0 ∙ = 22

= 1

0 ∙ = 18

= 0

0

0




Binomial Model – Cox Ross Rubinstein

S0

c0

Su

cu

Suu

cuu

Suuu

cuuu

Suuuu

cuuuu

Sd

cd

Sdd

cdd

Sddd

cdddSdddd

cdddd

S0

c0

Su

cu

Sd

cd

Suu

cuu

S0

c0

Sdd

cdd

Assumption = /




Binomial Model – Cox Ross Rubinstein

S0

c0

Su

cu

Suu

cuu

Suuu

cuuu

Suuuu

cuuuu

Sd

cd

Sdd

cdd

Sddd

cdddSdddd

cdddd

S0

c0

Su

cu

Sd

cd

Suu

cuu

S0

c0

Sdd

cdd

R




Simulation Based Option Pricing

T

Stock price

S 0

X

Lognormal probability

distribution at expirationPotential (random) stock price path

with unfavorable outcome (below

X ) at maturity

time

Potential (random) stock price

path with favorable outcome

(above X ) at maturity

Shaded area below curve is

propability for the option being

in-the-money at expiration

…

…

…

From the probability distribution

and potential outcomes the

expected option value in T canbe determined

The value of the option in = 0

is the discounted value of the

expected option value atexpiration

Discounting with risk-free interest rate for maturity T

…

…

…




The Black-Scholes Framework

0 = 0 ∙ 1 − ∙ 2 ,

with 1 =

+ +

∙

∙ and 2 = 1 ∙

where

0 = Current call option value0 = Current stock price

() = probability that a random draw from anormal distribution will be less than

= Exercise price

= 2.71828, the base of the natural log

= Risk-free continuously compounded rate

= time to maturity of the option in years

ln = Natural log function

= Volatility of a continuously compounded rate of return on the stock




Simulation Based vs Black-Scholes Option

Pricing – Example of a BMW Option

Call Option on BMW Stock: Strike: 65 €, Maturity: March 14th 2013 (70/252 days)

Own Calculations

Source: Onvista.de

Option QuoteStock Data Underlying

Volatility

Risk Measures

Stock Price S = 66,56 € Strike Price X = 65,00 €

Volatility sigma = 26,13%

Risk-free interest rate r = 2,00%

Duration t = 0,277777778

sigma*t = 0,137717192

d1 = 0,281410724

d2 = 0,143693531

N(d1)= 0,610802303

N primed of d1 0,383454415

Gamma 0,041832351exchange ratio 10

Call Option Price C = 0,46 €

Drift 0% Trials 500

Volatility (daily) 1,65% Timeperiods 70

Start Price 66,49 Expected Value (Option) 0,47

Shock in Period 10 No Discounted 0,46

Interest Rate 2,0%

Strike 65,0

Volatility (annual) 26,1%

Maturity 07.03.2013

70 64

Input Statistics

Days to maturity

Open files „Option Pricing Black Scholes“ and „Empirical Derivative Valuation.xlsm“




Call Option Example

Calculate the price of a call option for the given input factors: 0 = $100 = $95

= 0.10 = 0.25 ()

= 0.50

with 1 =

$

$9 + 0.10+

.

∙0.25

0.5∙ 0.25 = 0.43 and 2 = 0.43 0.5 ∙ 0.25

Using a statistical table or the NORMDIST function in Excel, we find that

0.43 = 0.6664 and (0.18) = 0.5714

Therefore: 0 = 0 ∙ 1 − ∙ 2 , 0 = $100 ∙ 0.6664 $95−0.1∙0.25 ∙ 0.5714,

0 = $13.70




Risk Management with Options – The Greeks

Delta (Δ) measures the rate of change of the theoretical option value with respect to changesin the underlying asset's price

Delta is the first derivative of the value of the option with respect to the underlying price

Δ =

Properties of Delta for Put and Call Options:

− Delta is always lower than the price sensitivity of the underlying (= 1)

− Long Calls & Short Puts: positive delta

− Long Puts & Short Calls: negative delta

Delta




Delta for Put and Call Options

Delta

Δ

1 0

Δ

1




Risk Management with Options – The Greeks

Gamma (Γ) measures the rate of change in the delta with respect to changes in theunderlying price

Gamma is the second derivative of the value function with respect to the underlying price

Γ =Δ

Properties of Gammafor Put and Call Options:

− High Gamma means that small changes in the underlying value induce high changes in

the Delta

− Long Calls & Puts: positive gamma− Short Calls & Puts: negative gamma

Gamma




Gamma for Long and Short Options

Gamma

Gamma

Positive Gamma

Gamma

Negative Gamma

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6. Basics of Derivatives Pricing