Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options
Derivatives is an investment whose value to day or at some future date is derived
entirely from the value of other assets, the underlying asset.
Examples are:
Interest rate futures contracts
Options on futures
Mortgage-backed securities
Interest rate caps- and floor
Swap options
Commodity linked bonds
Zero-coupon treasury trips
Major Break-through in the valuation of derivatives came with two finance
professors at MIT, Black and Scholes, came out with a formula that related the
price of a call option to the price of the stock to which the option applies.
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Examples
Forwards and Futures
Represents the obligation to buy (sell) a security or commodity at a pre-
specified price, known as the forward price, at some future date
The most important financial forward market is the inter-bank forward market
for currencies, particularly dollars for yen and dollars for Euros.
Swaps
Is an agreement between two investors, or counterparties as they are
sometimes called, to periodically exchange the cash flow of one security for
the cash flow of another. The last date of exchange determines the swap
maturity.
Forwards, Futures and Swaps zero-cost instruments.
Options
Gives their buyers the right, but not the obligation, to buy (call option) or sell
(sell option) an underlying security at a pre-specified price, known as the strike
price.
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options
Values of Calls and Puts at Expiration:
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Examples
Options cont.
Warrants
Warrants are options, usually calls, that companies issue on their own stock. In
contrast too options, which are mere bets between investors on the value of the
company’s underlying stock – in which the corporation never gets involved –
warrants are contracts between a corporation and an investor.
Embedded Options
A number of corporate securities have option like components. Ex. Convertible
bonds, callable or refundable corporate bonds. Corporate equity and debt contain
option like characteristics
Real Assets
Many real assets may be viewed as derivatives.
Ex. Copper mine/Mortgage-back securities/Structured Notes
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Basics
Two basic components: 1. Concept of Perfect tracking
2. Principle of no arbitrage
A fair market price, is simply a no-arbitrage restriction between the tracking
portfolio and the derivative.
It is always possible to develop a portfolio consisting of the underlying asset and a
risk-free asset that perfectly tracks the future cash flows of a derivative. A perfect
tracking portfolio is a combination of securities that perfectly replicates the future
cash flows of another investment.
Note: In absence of tracking error, arbitrage exists if it costs more to buy the
tracking portfolio than the derivative, or vice versa.
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Forward Value
Ex. Obligation (forward) to buy Ekornes stock one year from now for 100 kroner! The price
sells to day for 97 and Ekornes will not pay dividends in the period. One-year zero
coupon bonds (par. 100) currently sell for 92.
Cost Today Cash Flow one Year from
today
Strategy 1(forward) ? S1 − 100
Strategy 2 (tracking 97-92=+5
portfolio) S1 − 100
Strategy 2 costs 5 kroner. If strategy 1 cost > 5 kroner, arbitrage exists. Go short
in strategy 1 and go long in strategy 2 (sell forward (short), buy the share and
short 200 kroner par of zero coupon bonds). If Strategy 1 cost < 5 kroner,
arbitrage exists. Go long in strategy 1 and go short in strategy 2 (buy forward
(long), selling short the share and go long 200 kroner par of zero coupon
bonds). kroner 5 is a fair value of the attractive obligation to buy Ekornes
for kroner 100 one year from now.
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Binomial Model
The derivative valuation models induce that the current price of the underlying
asset determines the price of the derivative to day. Hence, a call option has a
known value at expiration date of the call when the stock price to day is known.
Linear functions of the underlying asset’s future payoffs:
Static investment strategies (buy and hold) ex. Forward contracts
Non-linear functions of the underlying asset’s future payoff:
Dynamic investment strategies (continuous rebalancing) ex. Option contracts
The ability to perfectly track a derivative’s payoff with a dynamic strategy requires
that the following conditions are met:
1. The price of the underlying security must change smoothly; that is, it does not
make large jumps.
2. It must be possible to trade both the derivative and the underlying security
continuously.
3. Markets must be frictionless.
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Binomial Model
If the price of the underlying security follows a binomial process, the investor can
still perfectly track the derivative’s future cash flows.
UP Give high degree of accuracy, when the
binomial periods are small and numerous.
Down
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Binomial Model
Binomial Model Tracking of a Structured Bond
OBX
1.500 (up state)
1.275
750 (down state)
1,10 (up state)
1
1,10 (down state)
331,75 (up state)
=100,00+6,75%(100,00) +225,00
106,75 (down state)
=100,00+6,75%(100,00) +0,00
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Binomial Model
The binomial process allow perfect tracking of the value of the derivative applying
a tracking portfolio consisting of the underlying asset and a risk-free bond.
Two steps: Identification of the tracking portfolio (TP) and valuation of the TP.
Identification:
Find the perfect tracking portfolio is the major task in valuing a derivative. With
binomial processes, the tracking portfolio is identified by solving two equations
in two unknowns, where each equation corresponds to one of the two future
nodes to which one can move (up or down).
Up node: ∆Su + B (1 + rf ) = Vu Solved simultaneously yields a
unique solution for ∆ and B.
Down node: ∆Sd + B(1 + rf ) = Vd
Valuation:
The fair market value of the derivative equals the amount it costs to buy the tracking
portfolio. Buying ∆ shares and B dollars of the risk free asset. Thus,
V = ∆S0 + B 10

Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Binomial Model
Example structured bond
Next period Value
Today’s value Up State Down State
TP portfolio ∆(1,275)+B ∆(1,500)+1,1B 0,3(750)+1,1B
Derivative ? 331,75 106,75
Two equations in two unknowns
The Solution :
∆ ⋅1.500 + 1,1⋅ B = 331, 75
∆ ⋅ 750 + 1,1⋅ B = 106, 75 ∆ = 0,3 and B = -107,50
Valuation: V = ∆S0 + B
V = 0,3 ⋅1.275 − 107,50 = 275
The value of the derivative equals the value of the tracking portfolio.
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Wall Street Approach
Based on our binomial results:
Important 1: The value of the derivative in relation to the value of the underlying
asset does not depend on the probabilities of up and down.
Important 2: The grade of investor risk aversion was not necessary for calculation
fair market values.
Why not?
This information is already captured by the price of the underlying asset on which
we base out valuation of the derivative. Note: Once the stock price is known, risk
aversion and mean return information are superfluous, not that they are irrelevant.
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Wall Street Approach
The risk-neutral valuation method:
Step 1. Identify risk-neutral probabilities that are consistent with investors being risk
neutral, given the current value of the underlying asset and its possible future values.
Step 2. Multiply each risk-neutral probability by the corresponding future value for
the derivative and sum the products together.
Step 3. Discount the sum of the products in step 2 (the probability weighted average
of the derivative’s possible future values) at the risk free rate.
Fewer steps than the tracking portfolio approach.
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Wall Street Approach
A General formula for risk-neutral probabilities:
π u + (1 − π )d = 1 + rf For our example:
where rf = 10%
rf − is the risk − free rate u − up node = 1,1765
u − rate of return at the up node d − down node = 0,5882
d − rate of return at the down node rearraging the Expression :
rearraging the Expression : 1 + 0,1 − 0,5882
π= = 0,87
1 + rf − d 1,1765 − 0,5882
π=
u−d
Valuation of the derivative:
331, 75 ⋅ 0,87 + 106, 75 ⋅ (1 − 0,87) = 302,5
302,5
= 275
1.1
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Wall Street Approach
Risk-Neutral Probabilities and Zero-cost Forward and Future Prices
Because futures contracts are one class of popularly traded financial instruments
with known terms (the future price) and known market values, it is often useful to
infer risk-neutral probabilities from them.
To use the prices of zero-cost forwards and futures to obtain risk-neutral
probabilities, it is necessary to slightly modify the risk-neutral valuation formulas.
For futures, the expected cash flows at the end of the period is
π ( Fu − F ) + (1 − π )( Fd − F ) /1 + rf = 0
The no-arbitrage futures price is the same as a weighted average of the expected
futures prices at the end of the period, where the weights are the risk-neutral
probabilities: F = π ⋅ F + (1 − π ) ⋅ F
u d
At the end of the period (maturity): Fu = Su and Fd = Sd. Substitution gives us:
F = π ⋅ Su + (1 − π ) ⋅ S d
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Wall Street Approach
Using Risk-Neutral Probabilities to obtain Future Prices
OBX
1.500 (up state)
1,275
750 (down state)
1,10 (up state)
1
1,10 (down state)
1.500-F (up state)
750 -F (down state)
Using 0,87 and 0,13 we can derive future prices:
0.87(1.500) + 0,13(750) = 1402,50 consistent (1275 *1,1=1402,75)
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Wall Street Approach
It is possible to rearrange to identify the risk neutral probabilities π and 1 –π
from futures prices. This yields: F − Fd
π=
Fu − Fd
and at maturity
F − Sd
π=
Su − S d
If future prices can appreciate 10% (up state) or depreciate 10% (down state), we can
calculate risk neutral probabilities.
In the up state, Fu=1.1F, in the down state Fd=0,9F. Applying the above equation:
F − 0.9 ⋅ F 0,1F
π= = = 0,5
1,1 ⋅ F − 0,9 ⋅ F 0, 2 F
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Multiperiod Binomial
Risk neutral valuation can be applied whenever perfect tracking is possible.
However, when large jumps in the value of the tracking portfolio or the
derivative can occur, perfect tracking is in general not possible. The exception is
a binomial price process.
Numerical example in a multiperiod setting:
121
110 38=121-83
38=121 - 83
100 99
90
16=99 - 83
81
0
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Multiperiod Binomial
Risk neutral valuation method(working backward through the tree diagram)
121 π + 99 (1-π) = 110 π = 0,5
Value of the option at node U is:
0,5 (38) + 0,5 (16) = 27
Value of the option at node D is (the same π)
0,5 (16) + 0,5 (0) = 8
The value at the derivative is
0,5 (27) + 0,5 (8) = 17,5
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Multiperiod Binomial
Algebraic Representation of Two-Period Binomial Valuation
In two steps: In one step:
Node u : π V + (1 − π )Vud π V + (1 − π )Vdd
V = π uu + (1 − π ) ud
1 + rf 1 + rf
π Vuu + (1 − π )Vud
Vu = π 2Vuu + 2π (1 − π )Vud + (1 − π ) 2 Vdd
1 + rf V=
1 + rf
π Vud + (1 − π )Vdd
Vd =
1 + rf
To day ' s value V :
π V + (1 − π )Vd
V= u
1 + rf
The discounted expected future value of the derivative with an expected value that
is computed with risk-neutral probabilities rather than true probabilities.
Note: π can vary along the nodes of the three diagram.
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Valuation Techniques
Numerical Techniques:
The techniques for valuing virtually all the new financial instruments developed by
Wall Street firms consists of Numerical Methods; that is, no algebraic formula is
used to compute the value of the derivative as a function of the value of the
underlying security. Instead, a computer is fed a number corresponding to the price
of the underlying security along with some important parameter values. The
computer derives the numerical value of the derivative + sometimes, the number of
shares of the underlying share in the tracking portfolio.
Simulation:
Generate random numbers to generate outcomes and then averages the outcomes of
some variable to obtain values (together with the standard deviation). Exclude
mortgages and American put options.
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Valuation Techniques
Binomial-Like Numerical Methods:
Simplification. First, the binomial method can be used to approximate many kinds
of continuous distributions if the time periods are cut to extremely small intervals.
One popular continuous distribution is the lognormal distribution. The natural
logarithm of the return of a security is normally distributed when the price
movements of the security are determined with by the lognormal distribution. Once
the annualized standard deviation, σ, of the normal distribution is known, u and d
are estimated as follows:
1
u = eσ T/N
and d =
u
where
T = number of years to Expiration
N = number of binomial periods
e = Exponential ConsTant (2.718281828)
Thus
T / N = square root of the number of years per binomioal period
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Valuation Techniques
All derivative valuation procedures make use of a short term risk-free return. The
most common used input for the risk free rate is LIBOR.
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options
Summary and Conclusions
The price movements of a derivative are perfectly correlated over short time
intervals with the price movements of the underlying asset on which it is based.
Hence, a portfolio of the underlying asset and a riskless security can be formed
that perfectly tracks the future cash flows of the derivative. To prevent
arbitrage, the tracking portfolio and the derivative must have the same value.
24

Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Put-Call Parity
Profiles: Buy a call and sell a put
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Put-Call Parity
26

Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Corporate Securities
It is possible to view equity as a call option on the assets of the firm and to view
risky corporate debt as riskless debt worth PV(K) plus a short position in a put
option on the asset of the firm(-p0) with a strike price of K.
1. Equity
The call option characteristic of equity arises
because of the limited liability of
corporate equity holders.
K V0 E0 = Max(Vo – Debt(K), 0)
2. Debt
The put option characteristic of debt arises
because of the put-call formula.
Debt are assets less a call option:
K S0 D0 = S0 – c0
The put-call: c0-p0=S0 – PV(Debt), substitution
D0 = PV(Debt(K)) – p0
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Corporate Securities
Hence, any characteristics of the assets of the firm that affect option values will
alter the values of debt and equity (for example the variance of the asset return).
The options result implies that the more debt a firm has, the less in the money is the
implicit option in equity. Thus, knowing how option risk is affected by the
degree to which options is in or out of money may shed light on how the mix of
debt and equity affects the risk of the firm’s debt and equity securities.
Finally, because stock is an option on the assets of the firm, a call option on the
stock of a firm is really an option on an option, or a compound option
(Geske,79).
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Binomial Valuation
Assume that the one-period risk-free rate is constant, and that the ratio of price in
the next period to price in this period is always u or d.
The hypothetical probabilities that
would exist in a risk-neutral world
must make the expected return on
the stock equal the risk free rate. The
risk neutral probabilities satisfy:
pu +(1-p)d=1+rf , giving the
relationship
1 + rf − d
π= and
u−d
u − 1 − rf
1−π =
u−d
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Binomial Valuation
The proper no-arbitrage call value c0, as a function of the stock price S0 becomes:
π max [uS0 − K , 0] + (1 − π ) max [ dS0 − K , 0]
c0 =
1 + rf
and the GENERAL binomial formula :
N
1 N!
c0 =
(1 + rf ) N
∑
j =0 j !( N − 1)!
π j (1 − π ) N − j π max 0, u j d N − j S0 − K 
 
Ex. Find the value of a three month at-the-money call option on DNB, trading at 32
kroner. Assume rf =0, u=2, d=0.5 and the number of years 3.
1 + 0 − 0,5 0,5 1 1 2
π= = = , and 1 − π = 1 − =
2 − 0,5 1,5 3 3 3
2 2 2 3
1 1  2  1  2   2
  (256 − 32) + 3     (64 − 32) + 3     (0) +   (0) = 15, 41
 3  3  3  3 3   3
Since the discount rate is 0, 15,41 is the value of the call option.
30

Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Binomial Valuation
256
224
128
64 64
32 32 32
16 16
8
0
4
0
31

Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Black&Scholes Valuation
Black-Scholes Formula
If a stock that pays no dividends before expiration of an option has return that is
lognormally distributed, can be continuously traded in frictionless market, and
has a constant variance, then, for a constant risk-free rate, the value of a
European call option on that stock with a strike price of K and T years to
expiration is given by
c0 = S0 N (d1 ) − Ke − r ⋅T N (d1 − σ T )
where
ln( S0 / PV ( K )) σ T
d1 = +
σ T 2
The Greek letter σ is the annualized standard deviation of the natural logarithm of
the stock return, ln() represents the natural logarithm, and N(z) is the probability
that a normally distributed variable with a mean of zero and variance of 1 is less
than z.
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Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Black&Scholes Valuation
Ex. Black-Scholes Formula
The non-dividend paying stock Ekornes has a current price of 150 kroner and a
volatility of 20 percent per year. What is the price of a 3 month European option
with strike price of 150 when the risk-free rate is 5%:
3
−0,05⋅
ln(30 / 28e 12
) 0, 2 3 /12 0, 0815 0, 2 ⋅ 0,5
d1 = + = + = 0,865
0, 2 3 /12 2 0, 2 ⋅ 0,5 2
c0 = 30 ⋅ N (0, 74) − 27.652 ⋅ N (0, 74 − 0, 2 ⋅ 0,5)
c0 = 30 ⋅ N (0, 74) − 27.652 ⋅ N (0, 74 − 0, 2 ⋅ 0,5)
c0 = 23.1105 − 20, 4326 = 2, 678
33

Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Estimating Volatility
Using Historical Data
Note that B&S is based on the volatility of instantaneous volatility:
Procedure for calculation of historical volatility:
1. Obtain historical returns for the stock the option is written on.
2. Covert returns to gross returns (1+return in decimal form).
3. Take the natural logarithm of the decimal version of the gross return.
4. Compute the unbiased sample variance of the logged return series and
annualize it by multiplying it by the square root of ratio of 365 to the number of
days in the return interval.
34

Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Estimating Volatility
The implied volatility approach
Look at other options on the same security. If market values of the options exist,
there is a unique implied volatility that maked the B&S model consistent with
the market price of a particular option.
The EXCEL-book: NewtRaphImp-vol.xls shows a particular form for implementation.
Method 2: New ton Raphson technique.
S 100 Price
K 125 Strike Markør 0
r 12 % Interest rate Initial 1.24304
tau 0.25 Maturity
sigma 0.471234 Volatility Derivative 25.5193
d(1) -0.70193 N'(d1) 0.510386
d(2) -0.93754
C (call price) 3 Mål kjøp 3
35

Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: B&S Greeks
The Greeks of B&S Formula
Delta: The sensitivity to Stock price Changes (Gamma measures Delta changes)
Vega: The sensitivity to Volatility Changes
Theta: The sensitivity to Expiration Changes
Rho: The sensitivity to Risk-free Interest rate Changes
36

Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options: Complex Assets
The Forward Price Version of the Black-Scholes Model
c0 = e − r ⋅T  F0 N (d1 ) − KN (d1 − σ T ) 
 
where
ln( F0 / K ) σ T
d1 = +
σ T 2
US $.40 ⋅1, 06
Forward Prices i) = US $0,3926
1, 08
ii ) $102 ⋅1, 06 − $4 ⋅ 1, 06 − $4 = $100, 0017
iii ) ($800 − $20) ⋅1, 06 = $826.80
iv) ($18 + $1 − $1) ⋅ (1, 06 = $19, 08
Call Option Prices
0,3926 ⋅ N (d1 ) − $0,5 ⋅ N (d1 − 0, 25) ln(0,3926 / 0,5
i ) c0 = where d1 = + 0,125 = −0,84; c0 = $0, 01
1, 06 0, 25
$100, 0017 ⋅ N (d1 ) − $100 ⋅ N (d1 − 0, 25) ln(100, 0017 /100
ii ) c0 = where d1 = + 0,125 = 0,13; c0 = $9,39
1, 06 0, 25
$826,80 ⋅ N (d1 ) − $850 ⋅ N (d1 − 0, 25) ln(826,80 / 850
iii ) c0 = where d1 = + 0,125 = 0,13; c0 = $68, 22
1, 06 0, 25
$19, 08 ⋅ N (d1 ) − $20 ⋅ N (d1 − 0, 25) ln($19, 08 / 20
iv) c0 = where d1 = + 0,125 = −0, 06; c0 = $1, 43
1, 06 0, 25 37

Styring og Fondsmegling Dr.oecon Per B Solibakke
Pricing Derivatives and Options
Summary and Conclusions
Despite a few biases in the Black & Scholes option pricing formula, it appears
that the formula work reasonably well when properly implemented.
The Excel-book: Black&ScholesImpl.xls shows various implementations.
Black-Scholes Option-Pricing Formula
S 25 Current stock price
X 25 Exercise price
r 6.00 % Risk-free rate of interest
T 0.5 Time to maturity of option (in years)
Sigma 30 % Stock volatility
d1 0.2475 <-- (LN(S/X)+(r+0.5*sigma^2)*T)/(sigma*SQRT(T))
d2 0.0354 <-- d1-sigma*SQRT(T)
N(d1) 0.5977 <-- Uses formula NormSDist(d1)
N(d2) 0.5141 <-- Uses formula NormSDist(d2)
Call price 2.47 <-- S*N(d1)-X*exp(-r*T)*N(d2)
Put price 1.73 <-- call price - S + X*Exp(-r*T): by Put-Call parity
1.73 <-- X*exp(-r*T)*N(-d2) - S*N(-d1): direct formula
38