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Advanced Option Pricing Concepts

This document discusses key concepts in option pricing such as replicating portfolios, risk neutrality, lognormal distributions, and dynamic hedging. It also covers risk neutral valuation and the assumptions of lognormal and normal distributions as they apply to modeling asset returns. Examples are provided to illustrate calculating probabilities of returns given mean and standard deviation parameters under the lognormal distribution. The benefits of working with log returns rather than simple returns are explained.

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cristiano
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43 views22 pages

Advanced Option Pricing Concepts

This document discusses key concepts in option pricing such as replicating portfolios, risk neutrality, lognormal distributions, and dynamic hedging. It also covers risk neutral valuation and the assumptions of lognormal and normal distributions as they apply to modeling asset returns. Examples are provided to illustrate calculating probabilities of returns given mean and standard deviation parameters under the lognormal distribution. The benefits of working with log returns rather than simple returns are explained.

Uploaded by

cristiano
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Options Pricing

Important ideas in Option Pricing


•  Replicating portfolio/Risk less hedge
•  Risk neutrality
•  Lognormal distribution
•  Dynamic hedging
•  GBM

2
Risk Neutral valuation
•  Risk aversion
•  Risk neutrality

3
Risk Neutrality

4
Normal Distribution
•  Random variables – are variables that take values
determined by the outcome of a random process.
–  Discrete
–  Continuous

•  Although we don’t know the future with certainty but past


experience allows to assess the likelihood of the values these
variables can take on.

5
Examples
Consider the following values data of Sensex returns:
Annual mean = 15% and volatility = 25%
1.  What is the prob that Sensex returns 5% or less in a given
year?
2.  What is the prob that Sensex returns 30% or more in a
given year?
3.  What is the prob that Sensex returns will be between 5%
and 20% in a given year?

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Answers

7
Log normality
•  A random variable X is said to have a lognormal
distribution if the (natural) log of X has a normal
distribution:
X is lognormal ~ ln X is normal.

8
•  In describing the evolution of an asset’s prices, we are
asking:
•  Given the current (time-0) price S0, what does the time-T
price ST look like?
•  Equivalently, we are asking the question:
•  What is the behavior of the returns ST /S0 ?
•  The lognormal returns distribution assumption is that the log
of these returns is normally distributed.

9
•  To put this in notational terms, let
S0 be the current (time-0) price of the asset.
ST be the time-T price of the asset.
•  N (m, v) denote a normal distribution with mean m and
variance v.
•  Returns on the asset over the interval [0, T ]: ST /S0.
•  The asset has lognormal returns if

where µ and σ are the two parameters of the distribution.


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•  Taking T = 1,
•  µ is the expected annual log-return.
•  σ is the standard deviation of annual log-returns.
•  σ = Volatility.
•  So volatility is the standard deviation of log-returns
expressed in annualized terms.
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Why Log returns?
•  Log returns ~ continuous compounded returns

12
•  If X ~ Ln (m, v ), then it is the case that
•  So if returns are lognormal over [0, T ], then we have

13
Sensex (ln/CC) returns – 1997-2020
mean = 0.000393318, sd = 0.0149498
40
Test statistic for normality: Ln_Rets
N(0.00039332,0.01495)
Chi-square(2) = 4413.735 [0.0000]

35

30

25
Density

20

15

10

0
-0.15 -0.1 -0.05 0 0.05 0.1 0.15

14 Ln_Rets
Sensex (gross (P1/P0)) returns –
1997-2020
45
Test statistic for normality: Simple_rets
N(1.0005,0.015001)
Chi-square(2) = 4723.034 [0.0000]
40

35

30

25
Density

20

15

10

0
0.9 0.95 1 1.05 1.1 1.15

15 Simple_rets
Example
•  Suppose returns are lognormal over a one-year horizon with
µ = 0.10 and σ = 0.40.
•  The expected log-returns = 10%.
•  Variance of log-returns = 40%
•  Expectation and variance of simple returns:

16
Working with lognormal dist
•  Almost all of the properties of normal distribution are
preserved in the lognormal.
•  To construct the CIs for a lognormal distribution all we need
to do is construct confidence intervals using the underlying
normal distribution for log-returns and then exponentiate.

17
Example 13.2

18
Exercise no. 1

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Exercise no 2
•  Repeat Q. No. 1 with mu =0.1 and sigma = 0.1.

20
Exercise no 5

21
Exercise no. 6

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