Financial Options
1
Topics in Chapter
1.
2.
3.
4.
5.
6.
What is a financial option?
Options Terminology
Profit and Loss Diagrams
Where and How Options Trade?
The Binomial Model
Black-Scholes Option Pricing Model
2
1. What is a financial option?


An option is a contract which gives its
holder the right, but not the obligation,
to buy (or sell) an asset at some
predetermined price within a specified
period of time.
It does not obligate its owner to take
any action. It merely gives the owner
the right to buy or sell an asset.
3
2.Option Terminology


A call option gives you the right to
buy within a specified time period at a
specified price
The owner of the option pays a cash
premium to the option seller in
exchange for the right to buy
4
2. Option Terminology


A put option gives you the right to sell
within a specified time period at a
specified price
It is not necessary to own the asset
before acquiring the right to sell it
5
2. Option Terminology

Strike (or exercise) price: The price
stated in the option contract at which
the security can be bought or sold.

Expiration date: The last date the
option can be exercised.
6
2. Option Terminology

Exercise value: The value of a call
option if it were exercised today =



Max[0, Current stock price - Strike price]
Note: The exercise value is zero if the
stock price is less than the strike price.
Option price: The market price of the
option contract.
7
2. Option Terminology

The price of an option has two
components:

Intrinsic value:



For
the
For
the
a call option equals the stock price minus
striking price
a put option equals the striking price minus
stock price
Time value equals the option premium
minus the intrinsic value
8
3. Profit and Loss Diagrams

For the Disney JUN 22.50 Call buyer:
Breakeven Point = $22.75
Maximum profit
is unlimited
$0
-$0.25
Maximum loss
$22.50
9
3. Profit and Loss Diagrams

For the Disney JUN 22.50 Call writer:
Maximum profit
Breakeven Point = $22.75
$0.25
$0
Maximum loss
is unlimited
$22.50
10
3. Profit and Loss Diagrams

For the Disney JUN 22.50 Put buyer:
Maximum profit = $21.45
Breakeven Point = $21.45
$0
-$1.05
Maximum loss
$22.50
11
3. Profit and Loss Diagrams
Consider the following data:
Strike price = $25.
Stock Price
$25
30
35
40
45
50
Call Option Price
$3.00
7.50
12.00
16.50
21.00
25.50
12
3. Profit and Loss Diagrams
Exercise Value of Option
Price of
stock (a)
$25.00
30.00
35.00
40.00
45.00
50.00
Strike
price (b)
$25.00
25.00
25.00
25.00
25.00
25.00
Exercise value
of option (a)–(b)
$0.00
5.00
10.00
15.00
20.00
25.00
13
3. Profit and Loss Diagrams
Market Price of Option
Price of
Strike
Exer.
stock (a) price (b) val. (c)
$25.00 $25.00
$0.00
30.00
25.00
5.00
35.00
25.00
10.00
40.00
25.00
15.00
45.00
25.00
20.00
50.00
25.00
25.00
Mkt. Price
of opt. (d)
$3.00
7.50
12.00
16.50
21.00
25.50
14
Time Value of Option
Price of
Strike
Exer. Mkt. P of Time value
stock (a) price (b) Val. (c) opt. (d)
(d) – (c)
$25.00 $25.00
$0.00
$3.00
$3.00
30.00
25.00
5.00
7.50
2.50
35.00
25.00
10.00
12.00
2.00
40.00
25.00
15.00
16.50
1.50
45.00
25.00
20.00
21.00
1.00
50.00
25.00
25.00
25.50
0.50
15
Call Time Value Diagram
Option
value
30
25
20
15
Market price
10
5
Exercise value
5 10
40
15
20
25
30
35
Stock Price
16
4. Where and How Options
Trade?
17
4. Where and How Options
Trade?

Options trade on four principal
exchanges:

Chicago Board Options Exchange (CBOE)

American Stock Exchange (AMEX)

Philadelphia Stock Exchange

Pacific Stock Exchange
18
4. Where and How Options
Trade?

AMEX and Philadelphia Stock Exchange
options trade via the specialist
system


All orders to buy or sell a particular
security pass through a single individual
(the specialist)
The specialist:


Keeps an order book with standing orders
from investors and maintains the market in a
fair and orderly fashion
Executes trades close to the current market
price if no buyer or seller is available
19
4. Where and How Options
Trade?

CBOE and Pacific Stock Exchange
options trade via the marketmaker
system


Competing marketmakers trade in a
specific location on the exchange floor near
the order book official
Marketmakers compete against one
another for the public’s business
20
4. Where and How Options
Trade?

Any given option has two prices at any
given time:


The bid price is the highest price anyone
is willing to pay for a particular option
The asked price is the lowest price at
which anyone is willing to sell a particular
option
21
4. Where and How Options
Trade?
5. The Binomial Model
5. The Binomial Model
 The value of the replicating portfolio at time T, with stock price ST, is
ΔST + (1+r)T*B where Δ is the # of share s and B is cash invested
 At the prices ST = Sou and ST = Sod, a replicating portfolio will satisfy
(Δ * Sou ) + (B * (1+r)T )= Cu
(Δ * Sod ) + (B * (1+r)T )= Cd
Solving for Δ and B
Δ = (Cu - Cd) / (So(u –d) )
B = (Cu - (Δ * Sou) )/ (1+r)T
and substitute in C = ΔSo + B to get the option price
5. The Binomial Model


Stock assumptions:

Current price: So = $27

In next 6 months, stock can either

Go up by factor of u = 1.41

Go down by factor of 0.71
Call option assumptions



Expires in t = 6 months = 0.5 years
Exercise price: X = $25
Risk-free rate: 6% a year a day (r = 0.06/365 a
day)
25
5. The Binomial Model
Ending "up" price = fu = Sou = $38.07
Option payoff: Cu = MAX[0, fu−X] = $13.07
Current
stock price
So = $27
Ending “down" price = = fd = Sod = $19.17
Option payoff: Cd = MAX[0, fd −X] = $0.00
u = 1.41
d = 0.71
X = $25
26
5. Riskless Portfolio’s Payoffs
at Call’s Expiration: $13.26
Ending "up" price = fu = $38.07
Current
stock price
So = $27
Ending "up" stock value = Δ*fu = $26.33
Option payoff: Cu = MAX[0, fu −X] = $13.07
Portfolio's net payoff = Δ*fu - Cu = $13.26
Ending “down" stock price = fd = $19.17
u = 1.41
d = 0.71
X = $25
Δ = 0.6915
Ending “down" stock value = Δ*fd = $13.26
Option payoff: Cd = MAX[0, fd −X] = $0.00
Portfolio's net payoff = Δ*fd - Cd = $13.26
27
5. Create portfolio by writing 1 option
and buying Δ shares of stock.

Portfolio payoffs:

Stock is up: Δ * Sou − Cu

Stock is down: Δ * Sod − Cd
28
5. The Hedge Portfolio with a
Riskless Payoff

Set payoffs for up and down equal, solve for
number of shares:



Δ = (Cu − Cd) / (So(u − d))
B = (Cu − (Δ * Sou) )/ (1+r)T
= - (Portfolio's net payoff )/ (1+r)T
In our example:


Δ = ($13.07 − $0) / $27(1.41 − 0.71) =0.6915
B = - $13.26/(1 + 0.06 / 365 )365*0.5 = -$12.87
29
5. The Hedge Portfolio with a
Riskless Payoff

Option Price C = ΔSo + B
= 0.6915($27) + (-$12.87)
= $18.67 − $12.87
= $5.80
If the call option’s price is not the same as the
cost of the replicating portfolio, then there will be
an opportunity for arbitrage.
Option Premium C + (Commission)
30
5. Arbitrage Example

Suppose the option sells for $6.



You can write option, receiving $6.
Create replicating portfolio for $5.80, netting
$6.00 −$5.80 = $0.20.
Arbitrage:



You invested none of your own money.
You have no risk (the replicating portfolio’s
payoffs exactly equal the payoffs you will owe
because you wrote the option.
You have cash ($0.20) in your pocket.
31
5. Arbitrage and Equilibrium
Prices



If you could make a sure arbitrage profit, you
would want to repeat it (and so would other
investors).
With so many trying to write (sell) options,
the extra “supply” would drive the option’s
price down until it reached $5.80 and there
were no more arbitrage profits available.
The opposite would occur if the option sold
for less than $5.80.
32
5. Multi-Period Binomial
Pricing



If you divided time into smaller periods and
allowed the stock price to go up or down
each period, you would have a more
reasonable outcome of possible stock prices
when the option expires.
This type of problem can be solved with a
binomial lattice.
As time periods get smaller, the binomial
option price converges to the Black-Scholes
price, which we discuss in later slides.
33
6. Black-Scholes


Fischer Black and Myron Scholes
published the derivation and equation
of the Black-Scholes Option Pricing
Model in 1973 under several
assumptions.
Bottom line: options of traded stocks
are fundamentally priced
6. Black-Scholes Option Pricing Model



The stock underlying the call option
provides no dividends during the call
option’s life.
There are no transactions costs for the
sale/purchase of either the stock or the
option.
Risk-free rate, rRF, is known and
constant during the option’s life.
(More...)
35
6. Black-Scholes Option Pricing Model




Security buyers may borrow any fraction of
the purchase price at the short-term risk-free
rate.
No penalty for short selling and sellers
receive immediately full cash proceeds at
today’s price.
Call option can be exercised only on its
expiration date.
Security trading takes place in continuous
time, and stock prices move randomly in
continuous time.
36
6. Black Scholes Equation
★Integration of statistical and mathematical models
★For example in the standard Black-Scholes model, the stock price evolves as
dS = μ(t)Sdt + σ(t)SdWt.
where μ is the drift parameter and σ is the implied volatility
★To sample a path following this distribution from time 0 to T, we divide the time interval
into M units of length δt, and approximate the Brownian motion over the interval dt
by a single normal variable of mean 0 and variance δt.
★The price f of any derivative (or option) of the stock S is a solution of the
following partial-differential equation:
6. Black-Scholes
Option Pricing Model

The Black-Scholes OPM:
C  S  N (d1 )   Ke
d1 
 rt
 N (d 2 ) 
ln( S / K )   R  ( 2 / 2)  t
 t
d 2  d1   t
38
6. Black-Scholes
Option Pricing Model

Variable definitions:





C = theoretical call premium
S = current stock price
t = time in years until option expiration
K = option striking price
R = risk-free interest rate
39
6. Black-Scholes
Option Pricing Model

Variable definitions (cont’d):




= standard deviation of stock returns
N(x) = probability that a value less than
“x” will occur in a standard normal
distribution
ln = natural logarithm
e = base of natural logarithm (2.7183)
40
6. Black-Scholes
Option Pricing Model
Example
Stock ABC currently trades for $30. A call option
on ABC stock has a striking price of $25 and
expires in three months. The current risk-free
rate is 5%, and ABC stock has a standard
deviation of 0.45.
According to the Black-Scholes OPM, but should
be the call option premium for this option?
41
6. Black-Scholes
Option Pricing Model
Example (cont’d)
Solution: We must first determine d1 and d2:
ln( S / K )   R  ( 2 / 2)  t
d1 
 t

ln(30 / 25)  0.05  (0.452 / 2)  0.25
0.45 0.25
0.1823  0.0378

 0.978
0.225
42
6. Black-Scholes
Option Pricing Model
Example (cont’d)
Solution (cont’d):
d 2  d1   t
 0.978  (0.45) 0.25
 0.978  0.225
 0.753
43
6. What is the value of the following
call option according to the OPM?






Assume:
N = P = $27
X = $25
rRF = 6%
t = 0.5 years
σ = 0.49
44
6. First, find d1 and d2.
d1 = {ln($27/$25) + [(0.06 + 0.492/2)](0.5)}
÷ {(0.49)(0.7071)}
d1 = 0.4819
d2 = 0.4819 - (0.49)(0.7071)
d2 = 0.1355
45
6. Second, find N(d1) and N(d2)




N(d1) = N(0.4819) = 0.6851
N(d2) = N(0.1355) = 0.5539
Note: Values obtained from Excel using
NORMSDIST function. For example:
N(d1) = NORMSDIST(0.4819)
46
6. Third, find value of option.
VC = $27(0.6851) - $25e-(0.06)(0.5)(0.5539)
= $19.3536 - $25(0.97045)(0.6327)
= $5.06
47
What impact do the following parameters
have on a call option’s value?


Current stock price: Call option value
increases as the current stock price
increases.
Strike price: As the exercise price
increases, a call option’s value
decreases.
48
6. Impact on Call Value



Option period: As the expiration date is
lengthened, a call option’s value increases
(more chance of becoming in the money.)
Risk-free rate: Call option’s value tends to
increase as rRF increases (reduces the PV of
the exercise price).
Stock return variance: Option value increases
with variance of the underlying stock (more
chance of becoming in the money).
49
6. Negative Affects






Myron Scholes was a partner of Long-Term Capital Management
(LTCM) established in 1994.
LTCM would use models and experienced investing knowledge
to pick differences in short and long securities, profiting from
the small margin.
In 1998 the firm had $5B in equity and $125B in liability
August 17, 1998 Russia devalues the rouble and suspended
$13.5B of it’s Treasury Bonds, causing LTCM’s equity to
decrease drastically, ever increasing it’s leverage ratio
In order to prevent a stock market crash due to the outflux of
investors and banks from LTCM’s funds the Federal Reserve
organized a $3.5 B rescue effort to take over LTCM’s
management
Had LTCM lost leverage, all banks and creditors would pull out
of other firms – including Salomon Brothers, Merrill Lynch.
7. Monte Carlo method
★In the field of mathematical finance, many problems, for instance the problem
of finding the arbitrage-free value of a particular derivative, boil down to the
computation of a particular integral.
★When the number of dimensions (or degrees of freedom) in the problem is large,
PDE’s and numerical integrals become intractable, and in these cases Monte
Carlo methods often give better results.
★Monte Carlo methods converge to the solution more quickly than numerical
integration methods, require less memory , have less data dependencies and
are easier to program.
★The idea is to use the result of Central Limit Theorem to allow us to generate a
random set of samples as a valid representation of the previous value of the stock.
“The sum of large number of independent and identically distributed random
variables will be approximately normal.”
7. Monte Carlo method
St = S0e(a – 0.5 σ 2)t + σ√tZ
If V(St,t) is the option payoff at time t, then the time-0
Monte Carlo price V(S0,0) is
1  rT n
V ( S0 ,0)  e V ( STi , T )
n
i 1
where ST1, … , STn are n randomly drawn time-T
stock prices
7. Option Pricing Sensitivities
The Greeks
8. Parallel Computing
8. MATLAB program for Monte Carlo
drift = mu*delt;
sigma_sqrt_delt = sigma*sqrt(delt);
S_old = zeros(N_sim,1);
S_new = zeros(N_sim,1);
S_old(1:N_sim,1) = S_init;
for i=1:N % timestep loop
% now, for each timestep, generate info for
% all simulations
S_new(:,1) = S_old(:,1) +...
S_old(:,1).*( drift + sigma_sqrt_delt*randn(N_sim,1) );
S_new(:,1) = max(0.0, S_new(:,1) );
% check to make sure that S_new cannot be < 0
S_old(:,1) = S_new(:,1);
%
% end of generation of all data for all simulations
% for this timestep
Peter Forsyth 2008
end % timestep loop
8. MATLAB program for Asian Options
function [Pmean, width] = Asian(S, K, r, q, v, T, nn, nSimulations, CallPut)
dt = T/nn;
Drift = (r - q - v ^ 2 / 2) * dt;
vSqrdt = v * sqrt(dt);
pathSt = zeros(nSimulations,nn);
Epsilon = randn(nSimulations,nn);
St = S*ones(nSimulations,1);
% for each time step
for j = 1:nn;
St = St .* exp(Drift + vSqrdt * Epsilon(:,j));
pathSt(:,j)=St;
end
SS = cumsum(pathSt,2);
Pvals = exp(-r*T) * max(CallPut * (SS(:,nn)/nn - K), 0); % Pvals dimension: nSimulations x 1
Pmean = mean(Pvals);
width = 1.96*std(Pvals)/sqrt(nSimulations);
Elapsed time is 115.923847 seconds.
price = 6.1268
www.fbe.hku.hk/doc/courses/tpg/mfin/20072008/mfin7017/Chapter_2.ppt
2009 Financial Derivatives Proposal
8. Monte Carlo Parallelism
★Iterations are independent - Embarrassingly parallel
★Parallel Random Number Generation: Master Processor
★SPRNG (Scalable Pseudorandom Number Generation) library
function MC (S, r, T, sigma, nSamples, nP)
dataRand[nSamples] = rand();
start = idP*p/nSamples; end = (idP+1)*p/nSamples;
for each idP
mean_value[idP] = findMC(S,E,r,T,sigma,dataRand[start,end));
end
reduce(mean_value[idP]);
endfunction
8. Binomial Parallelism
function Bin (S, E, r, T, sigma, nTime )
…
for step = nTime:-1:1
for i=1:nStep+1
value(i) = (pUp*value(i+1)+pDown*value(i))*exp(-r*dt);
price(i) = price(i+1)*exp(-sigma*sqrt(dt));
value(i) = max(value(i), price(i)-E)
endfunction
8. Binomial Parallelism
8. Finite Difference Parallelism
★Careful: dt and dS are not independent
★Use Ghost points to minimize the communication
★Communication comes from one side only
Explicit Finite Differences
8. Types of options
Standard options
★ Call, put
★ European, American
Exotic options (non standard)
★ More complex payoff (ex: Asian)
★ Exercise opportunities (ex: Bermudian)
2009 Financial Derivatives Proposal
Bibliography










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
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
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FIR 7155: Global Financial Management (Spring 2013)
https://umdrive.memphis.edu/cjiang/www/teaching/fir7155/powerpoint/FM12%20Ch%2008%20Show.ppt
Grid Enablement of Scientific Applications
http://users.cis.fiu.edu/~sadjadi/Teaching/Autonomic%20Grid%20Computing/CEN%205082-Spring2009/Presentations/FinancialDerivatives_05012009.ppt
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Peter Forsyth, “An Introduction to Computational Finance Without Agonizing Pain”
Guangwu Liu , L. Jeff Hong, "Pathwise Estimation of The Greeks of Financial Options”
John Hull, “Options, Futures and Other Derivatives”
Kun-Lung Wu and Philip S. Yu, “Efficient Query Monitoring Using Adaptive Multiple Key Hashing”
Denis Belomestny, Christian Bender, John Schoenmakers, “True upper bounds for Bermudan products via
non-nested Monte Carlo”
Desmond J. Higham, “ An Introduction to Financial Option Valuation”
Alexandros V. Gerbessiotis, “Architecture independent parallel binomial tree option price valuations”
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