6. basics of derivatives pricing
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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
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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
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Page 3Investment AnalysisDenis Schweizer
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
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Page 4Investment AnalysisDenis Schweizer
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
/ = +
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Page 5Investment AnalysisDenis Schweizer
Payoff and P/L to Call Holder at Expiration
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Page 6Investment AnalysisDenis Schweizer
Payoff and P/L to Call Writers at Expiration
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Page 7Investment AnalysisDenis Schweizer
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
/ = +
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Page 8Investment AnalysisDenis Schweizer
Payoff and Profit to Put Holder at Expiration
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Page 9Investment AnalysisDenis Schweizer
Call Option Value Before Expiration
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Page 10Investment AnalysisDenis Schweizer
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?
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Page 11Investment AnalysisDenis Schweizer
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
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Page 12Investment AnalysisDenis Schweizer
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 = /
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Page 13Investment AnalysisDenis Schweizer
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
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Page 14Investment AnalysisDenis Schweizer
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
…
…
…
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Page 15Investment AnalysisDenis Schweizer
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
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Page 16Investment AnalysisDenis Schweizer
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“
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Page 17Investment AnalysisDenis Schweizer
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
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Page 18Investment AnalysisDenis Schweizer
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
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Delta for Put and Call Options
Delta
Δ
1 0
Δ
1
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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
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Gamma for Long and Short Options
Gamma
Gamma
Positive Gamma
Gamma
Negative Gamma