Options Basic Concepts of Option Valuation

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Options Basic Concepts of Option Valuation
Professor Dr. Rainer Stachuletz Corporate
Finance Berlin School of Economics
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Topics Covered

• Simple Option Valuation Model
• Binomial Model
• Black-Scholes Model
• Black Scholes in Action
• Option Values at a Glance

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What DeterminesOption Values ?
A stock ist actually traded at 150 . The current
price is not expected to change in the future. A
call on that stock with a time to exercise of 1y
has a strike of 120 . The risk free interest
rate is at 10 / year. What is the fair price of
the call ?
The fair price of a call ( 41,42) equals the
difference between the stock price and the
present value of the strike
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What DeterminesOption Values ?
The example shows, how each factor affects the
options price
The higher S0, the higher the price (option deep
in the money), The smaller the strike (X), the
higher the price (option in the money), The
higher the interest rate r, the higher the
price, The longer the time to exercise (t), the
higher the price, et vice versa. And how about
the volatility ??
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Duplication of Cash Flowsfrom a Call Option

• Duplication of future cash flows is an
  appropriate measure to value seemingly unknown
  products. In general, this method is based on the
  translation of complex cash flows from unknown
  products into a combination of cash flows from
  well known products.
• Epistemically this method is similiar to the use
  of analogues in problem solving processes
• We often make use of such analogues. I.E. if you
  try to value a five years old car youll
  presumably look for the price of analogue models
  before negotiating the price. In this sense,
  benchmarking gives also an example for using
  analogues.

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Pricing by Duplicationof Cash Flows
Case The current stock price of Cyrus Clops Ltd.
is at 100 . Try to price a call, offering the
right to buy at a price of 150 (Strike) with a
time to exercise of 1y. It is assumed, that the
future price of Clopss stock will rise to 200
or drop to 50 . The interest rate is at 10.
downside change
upside change
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Pricing a Call by Duplicationof Cash Flows
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Pricing a Call by Duplicationof Cash Flows
To duplicate the future Call - Cash Flow, a Hedge
Ratio Dh informs about the proportion of shares
and credites to generate the Call Cash Flow
The hedge ratio D h determines the number of
shares and the amount at which a credit has to be
taken to duplicate the calls cash flow
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Pricing a Call by Duplicationof Cash Flows
Economically a Long Call Position is analogue to
a debt financed share purchase. The cash flow,
generated by a long call position can then be
duplicated by a long share and a long credit
position
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Duplication of Cash Flows from a Call Option
Thus, the duplicated portfolio will contain 0.333
shares (i.e. at a current price of 100 , that
needs an investment of 33,33 ). The loan needed
amounts to 15,15 in t 0, which will one year
after lead to a total cash flow out of 16,665
( 15,15 x 1,10). After adding up all the cash
flows from the duplication portfolio, the total
cash flow in t 0 (i.e. the present value of the
duplicated cash flows) is at 18,18 . As the
duplicated cash flows (from the share- and the
loan-component) equal the determined payoff from
the Call-position, 18,18 must also be the fair
price of the Call opiton in t0.
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Fair Call Price and Arbitrage Opportunities
The duplicated price of 18,18 / Call is a fair
price. There is no arbitrage opportunity. If the
price were to be at 20 per Call (i.e.
overvalued call), arbitrage would be possible
Sell three calls at 20 each, take a one year
credit of 40 and buy a share at the current
price of 100 As the market is imbalanced, a
risk-free (arbi-trage-) profit of 6 will result
(6 /1.10/31,82 20 18,18 )
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Fair Call Price and Risk Free Interest
The hedge ratio could also be used to set up a
risk free portfolio. D h indicates the number of
shares, that could be hedged by one option. In
our example, one share needs to be hedged by 3
calls.
A perfectly hedged (risk-free) portfolio will
allow to earn a risk-free rate, which is (in this
example) given to be at 10 per year. What ever
happens in t1, the return will be 110 .
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The Risk Free Rate and the Binominal Model to
Value Options
Creating a Hedge-Portfolio it will always be
possible to gain the risk-free rate. This leads
to a trick in option valuation Referring to
our example we can state, that the minimum
expected return would equal the risk free-rate
(10). Now we can interprete this return to be an
expected return, i.e. an minimum average return
of an upside (100) or downside (-50) movement
of the share under consideration. Although we do
not know the probabilities of an upside or
downside change of the share price, the
probabilities are determined in a binominal price
structure
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The Risk Free Rate and the Binominal Model to
Value Options
More general we can state
Knowing the probabilities of future positions it
is only one step to carry out the call valuation.
If there is an upside change (p40), the call
will have a value of 50 . In case of a
downside change (1-p60), the value is to be at
0. The average call-value in t1 will then be
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The Risk Free Rate and the Binominal Model to
Value Options
Example continued What is the price of the call
(S 100, X 150, r 10 p.a., s 69,50, t
1year, h 0,25 i.e. one interval is over 3
month)
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The Risk Free Rate and the Binominal Model to
Value Options
Using the figuresf from above, the share price
will change from interval to inter-val by
41,55 or by 29,35 respectively. After 4
intervals the share price is then between 24,91
(min.) and 401,51 (max.) The range is according
to the specified standard deviation (volatility)
of 69,5 on an annual basis.
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Binominal ValuationRoll Back - Method
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Binominal ValuationRoll Back - Method
S 100X 150n 1Intervalle 6r 10s
69.5
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Binominal ValuationRoll Back - Method
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Black-Scholes Option Pricing Model
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Black-Scholes Option Pricing Model
PCall - Call Option Price S - Stock Price N(d1)
- Cumulative normal density function of (d1) X x
e r x n - Present Value of Strike or Exercise
price (X) N(d2) - Cumulative normal density
function of (d2) r - discount rate (90 day
commercial paper rate or risk free rate) n - time
to maturity of option (as of year) s -
volatility - annualized standard deviation of
daily returns
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Black-Scholes Option Pricing Model
Example continued What is the price of the call
(S 100, X 150, r 10 p.a., s 69,50, t
1year, h 0,25 i.e. one interval is over 3
month)
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Black-Scholes Option Pricing Model
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Black-Scholes versus Binominal Option Pricing
Different valuation approaches resulted in
different call - option values Duplication
Portfolio 18,18 Discrete Binominal Valuation
Models 1-Step Binominal Model 18,18 4-Step
Binominal Model 18,42 6-Step Binominal
Model 17,63 Continous Black Scholes
Model Black-Scholes f. EuropeanCalls 17,07

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Black-Scholes Call Options
Reasoning behind Black and Scholes is similar to
that of the binomial model
Does a combination of option and shares exist
that provides a risk-free payoff?

• Stock prices can move at every moment in time.
• Movements of stock prices are random, and
  therefore unpredictable.
• Volatility (standard deviation) of stocks
  movements is known.
• Interest rates remain stable until expiration

Assumptions of the model
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Put-Call-Parity
P
Long Credit

L
Share Price Put Price Call Price PV
Exercise
Put Price Call Price PV Exercise
Share Price
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Put - Valuation
Put Price 17,07 150 x 2,718-0.10 100
52,79
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Put-Call-Parity

P
Long Credit

L