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A novel way to speed up calculation of
American Style Options pricing
Bhooma Lalitha Shivakumar
Adlai Stevenson High School
Grade 10
Shivakumar
Table of Contents
HYPOTHESIS ............................................................................................................................................................. 3
REVIEW OF THE LITERATURE ........................................................................................................................... 4
MATERIALS ............................................................................................................................................................. 14
METHODS OF PROCEDURE ................................................................................................................................ 15
CONCLUSIONS ........................................................................................................................................................ 20
REFERENCE LIST .................................................................................................................................................. 22
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Acknowledgements
I would like to thank the Science Fair team, including Mrs. Ragusa and Mrs. Palffy, for helping
me out with the paper and guiding in my research. I would also like to thank them for making the
WikiSpace available to facilitate the entire process. Lastly, I would like to thank my father for
reviewing the write up, code, user interface and helping me with testing the program.
Thank you all!
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Purpose
The purpose of this experiment was to determine if slight changes to the Binomial model could
be used to speed up the pricing of American style options, which can be exercised at any time.
The binomial model breaks down the time to expiration into potentially a very large number of
time intervals, or steps making it useful for American style options.
Hypothesis
Changing the Binomial model, by skipping select nodes, to calculate options pricing will
improve the speed of the calculation by 20% on a standard personal computer.
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Review of the Literature
The material wealth of a nation depends on the productive capacity of its economy. The
economy in turn entails the goods and services that can be provided to its citizens and those
living in that nation. This wealth, in turn, depends on the real assets, which include land,
building, machines, etc. Financial assets on the other hand, do not directly represent a nation’s
wealth. They, however, contribute indirectly by separating the ownership from management of
firms and enterprises and facilitate the transfer of funds to enterprises with attractive investment
opportunities. Real assets generate income, whereas financial assets define the allocation of this
income or wealth among investors, who collectively are the owner of the enterprise.
Investment, as per the Oxford dictionary, means an act or process of investing money for
profit. However, many invest not only with money but with time, devotion, energy, etc.
However, the end goal of investment is reaping a reward at a specific time in the future.
Financial assets such as stocks and bonds are claims to the income generated by real
assets or claims on the income from the government (Zvi Bodi, Alex Kane, Alan J. Marcus,
1999).
A relatively recent, but extremely important class of financial assets is derivative
securities or derivatives. Like the name suggests derivatives’ price is derived from the price of
other, underlying securities. Derivatives are also called “contingent claims” because their payoffs
is dependent or contingent upon the price of other securities. Because of this property,
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derivatives are used as a powerful tool to hedge, whereby one invests to reduce the overall risk of
a set of financial assets, and speculation (Philip M. Johnson, 1999).
An option, as the name suggests, gives the holder of the option to buy or sell an asset for
a specified price on a specified date for European style options or on or before a specific
expiration date for US style options. A CALL option gives its holder the right to purchase an
asset for a specified – previously determined – price called the EXERCISE or STRIKE price on
or before a specific date depending on whether the option is a European or American style.
Likewise, a PUT option gives the holder the right to sell an asset for a specified – previously
determined – price. The purchase price of the option itself is called the PREMIUM. This
PREMIUM is the compensation the purchaser of the CALL must pay for the right to exercise the
option.
Holder of a CALL option makes money only when the market value of the asset to be
purchased as part of the options is higher than the STRIKE price. Likewise the holder of a PUT
option makes money only when the market value of the asset to be sold as part of the option is
lower than the STRIKE price. And there is no obligation to buy or sell the asset. One can simply
let the options expire without exercising it on or before the expiration date.
An option is in the money if the exercise price would produce a profit for its holder. For a
CALL, the exercise price should be lower than the market price and for a PUT, the exercise price
must be higher than the market price. An option is out of the money when exercising a CALL or
a PUT produces a loss. An option is at the money when exercising the options breaks even, i.e.,
the asset price and the exercise price are equal (Zvi Bodi, Alex Kane, Alan J. Marcus, 1999).
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In 1973, the trading of standardized options contracts on a national exchanged started.
The Chicago Board of Options Exchange or CBOE for short began listing CALL options. These
were very successful and the volume of trade – number of contracts traded – increased rapidly
from 1973 to the late 1980s. Today options are traded in several exchanges worldwide. Options
are now written on common stock, stock indexes, equity indexes, foreign exchange, commodities
including agricultural and metals and interest rate futures.
An American style option allows its holder to sell (if a PUT) or buy (if a CALL) the
underlying asset on or before the expiration date. European options allow for exercise only on
the expiration date. Since an American option offers more flexibility, they are generally more
valuable than their European counterpart. There are some exceptions to this rule. Prime among
them are the foreign currency and stock index options, which though traded on the CBOE,
follow the European style (Robert W. Kolb, 1995).
Options as an investment tool, offers many advantages over more traditional stocks or
bonds. Cost efficiency is a major advantage of using stock option as in investment tool. The
investor only pays for the options and not the underlying asset, which could cost a great deal
more. Since there is less money involved, risk is greatly reduced. As an investment tool it
provides variety to one’s portfolio. However, there are some notable disadvantages too. Firstly,
the trading volume is much lower than stocks or bonds, lowering the liquidity of options.
Options usually entail higher commissions, although these days options and stocks tend to incur
similar charges. Unlike stocks or bonds, options are complicated and not readily available to
entry level investors (Eric Banks, 2003).
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Unlike stocks or bonds, options are priced differently based on the underlying assets. The
story of options prices really begins with the French mathematician Louis Bachelier, who
derived a closed formula for the pricing of standard CALLs and PUTs in his 1900 PhD thesis
dissertation. His primary assumption was that stock prices follow an arithmetic Brownian
motion, where one can imagine rapid movements about the origin. In contrast to what is used
today, returns are normal as opposed to lognormal.
Louis Bachelier showed that for a non-dividend paying stock and zero interest rate, the
price of a European style option should be as follows:
c(S, T) = SN (S – K) – KN (S – K) – σ √T n(S – K)
(σ √T)
Where S is the stock spot price (price at which a stock can be bought or sold at a specified time
and place), K is the STRIKE price, σ the volatility of the stock price (the instantaneous standard
deviation of the stock price), T the time to the options maturity, N the cumulative normal density
function and n the probability density function.
However, the Bachelier formula above ignores any discounting and assumes that stock
prices can be negative and as such not suitable for stock prices. Sprenkle, who in 1961, first
started to adapt the approach of Bachelier to non-negative prices by assuming lognormal returns.
Investors risk averseness was another factor he assumed and modified the above formula to the
following:
C(S,T) = eρT SN(d1) – (1 – A) KN (d2)
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Where d1 = 1 / σ√T [ ln (S/K) + (ρ + σ2 / 2)T ] , d2 = d1 - σ√T and ρ is the average rate of growth
of the stock price and A is the degree of risk aversion. However, there were parameters to
estimate such as degree of risk aversion and average growth of stock price and as such this
pricing formula did not receive much attention. Later, Boness (1964) improved the formula by
accounting for the time value of money through discounting of the terminal stock price, using
expected rate of return of the stock. The formula was then changed to
C(S,T) = SN(d1) – Ke-ρT N(d2)
With the component definitions remaining the same (Boness A.J.., 1973).
Although the above approaches provided formulae close to the now used Black Scholes
formula, it was the groundbreaking work of Black, Scholes and Merton that the option price was
explicitly connected to hedging strategy. In 1973, Black Scholes was to realize that the expected
return of the option price should be the risk free rate and that by holding a certain amount of
stock, referred to as delta, the option position could be completely hedged (Black, F. and
Scholes, M., 1973).
Compared to the previous works, the Black Scholes formula had distinct advantages over
the previously formulae. It had an explicit hedging strategy for the replication of the CALL,
which only depends on the volatility of the stock price and other observable quantities like risk
free rate, time to maturity of the option, its strike and spot price. It also had a universal price in
that despite the investor risk aversion the price of the option should be the same for all investors.
And above all the formula was easy to use with volatility being the only not easy to calculate
parameter (Black, F., 1989).
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The famous formula is as follows:
C(S,T) = SN(d1) – Ke-rT N(d2)
Where d1 = 1 / σ√T [ ln (S/K) + (r + σ2 / 2)T ] and d2 = d1 - σ√T. Like many academic
breakthroughs, it took a while for the formula to be acknowledged and Black, Scholes and
Merton were awarded the Nobel Prize in 1997.
Despite its widespread use, the Black Scholes formula has many limitations associated
with the assumptions it makes. These assumptions can be considered as non-real life assumptions
about the market and how the market behaves.
The first assumption that volatility is a constant over time has been challenged and
empirically proved to be incorrect. While volatility – a measure of how much a stock can move
in the near term – is relatively constant in very short term, it never is constant in the medium to
long term. Large price changes are generally followed by large price changes and vice versa.
The Black Scholes formula also assumes that the price of the underlying stock moves up
or down with equal probability. This is not true as the price of a given stock is determined by
many economic factors that cannot be assigned the same probability. And it is important to note
that the price of stock at time t+1 is independent from the price of stock at t or t+2. More
importantly, there may not be a single source of uncertainty or risk driving two assets, even if
one is derived from the other (Chen Hua, Roderick Wong, 2004).
Third assumption is reasonable and assumes that the returns of log normally distributed
underlying stock prices are normally distributed. Although reasonable in the real world, it does
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not fit observed data accurately. Asset returns, unlike log normal, have finite variance and heavy
tails. As the time scale over which the returns are calculate increases, the distribution of the asset
prices look more like normal distribution with heavy tails. This is despite the fact that the
autocorrelation of asset prices is often insignificant (Rama, Cont, Peter et. al, 2003).
The Black Scholes formula uses the risk-free rate to represent the interest rate and
assumes it to be constant. However, in real life there is no such thing as risk-free and even the
US Government treasury bills 30-day rate, which is supposed to be credible enough, can change
in times of volatility.
In the real world, most firms pay dividends to their stockholders. The basic BlackScholes model assumed that the underlying stock does not pay any dividends during the option’s
life. This assumption was corrected for dividends and a workaround was added. The workaround
subtracts the discounted value of a future dividend from the underlying stock price.
The Black-Scholes model assumes that there is no fee or commission for buying or
selling options and stocks and there are no barriers to trading. We all know that there are nontrivial fees for buying and selling any derivative.
The Black-Scholes model is limited to the European-style options, which can be
exercised only on expiration date of the option. The American-style options can be exercised at
any time during leading up to the expiration date, making the American-style option more
valuable due to this flexibility.
The assumption that the financial markets are perfectly liquid and it is possible to buy or
sell any amount of stock or options at any time is not plausible. Investors are limited by the
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amount of money they can invest, policies of the company they work and the wish of sellers to
sell. In addition, volume of trade also depends on market volatility at a given time. Moreover, it
is not possible to sell a fraction of an option. Options, generally comes in bundles of one hundred
shares.
The above assumptions, which have limitations, relate to the fundamental aspects of the
financial market. In order to make the model more accurate and fitting, it is necessary to develop
models that take into consideration assumptions not addressed or fully addressed by the BlackScholes model. Many models, which are variants of the Black-Scholes model, have been
proposed over time, with each trying to mimic the market characteristics fully. Each and every
aspect of the financial market cannot be mimicked and cannot be considered in a single model. It
is not possible to mathematically model every factor affecting the price of a financial security.
This is a limitation and most advanced models only attempt to capture most of the aspects. Also,
the Black-Scholes model’s basis is Brownian motion, which itself is affected by scaling, timechange properties and other factors (Teneng, D, 2011).
The original Black-Scholes option pricing model assumes that stock returns follow
Brownian motion. Stock returns, however, differ from this assumption in at least three important
ways. These assumptions were briefly described above. Firstly, stock prices jump leading to nonnormal returns. Secondly, return volatilities vary over time. This variation is primarily due to
conjecture and does not follow a pattern. Finally, returns and the volatilities are correlated. The
correlation is generally negative for stocks.
In order to price assets, in the case of options underlying assets, and manage risks it is
important to consider the stochastic process that governs the asset dynamics. Although Brownian
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motion is a good model to follow, it has its shortcomings. The Time-Change Lévy process
addresses these by using a continuous time stochastic process with stationary independent
increments. This model also captures the stochastic volatility by using stochastic time change to
the process itself (Peter Carr, Liuren Wu, 2002).
Considering the European style option pricing, which is modeled on Brownian motion
implies constant drift and volatility. In addition, stock prices are considered to be independent of
stock price history. We know that investors look at the stock price history before investing. In
order to better model real-life conditions, drift and volatility were tied to stock price history
using delayed responses and finite different approximation for an infinite dimensional BlackScholes equation (Mou-Hsiung Chang, Tao Pang, Moustapha Pemy, 2009).
The Black-Scholes model assumes that in a perfectly competitive, liquid (ability to buy or
sell any amount of stock) market there exist no opportunities to earn a risk-free profit. This can
be considered as no opportunities for arbitrage. Although all of the assumptions gave been
tweaked in some way to make the model more meaningful, some assumptions are easier to tweak
than others. The assumption of stock prices following Brownian motion is easier to tweak than
the assumption of perfectly liquid markets (Samuel W. Malone, 2002).
It has also been shown that the Binomial option pricing model converges to the BlackScholes model for European style options and some American style options. The Binomial
option pricing model is essentially a tree construction with branches that indicates the possible
value of the underlying stock and the corresponding option value. This tree is constructed on the
assumption that the underlying stock can go up or down by a factor that is related to the length of
time and the volatility of the stock (Simon Benninga and Zvi Wiener, 1997).
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An improved version of the Binomial option pricing model is the Trinomial option
pricing model. Unlike, the Binomial model which considers underlying stock price going up or
down, the Trinomial model includes a third option where the stock price remains unchanged.
Trinomial trees also afford the advantage of preserving the Binomial trees within them.
Binomial model suffers from speed and for increased accuracy, where more nodes have
to be created then speed does become an issue. For some types of options, inaccuracies creep in
until a large number of time intervals are used. Trinomial model is faster than Binomial and is
better equipped to handle exotic options – more complex than vanilla or commonly traded
options (Paul Clifford, Oleg Zaboronski, 2008).
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Materials

1 calculator for data analysis.

Windows Laptop running Windows 7, software to record, develop, analyze, print data
charts, and graphs.

Microsoft Visual C++ 2010 software package, version 10.0.30319.1 RTMRel.

BlueJ Java software package, version 3.0.4.
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Methods of Procedure
The following C++ functions are used to simulate the Black-Scholes and
Binomial methods. The list of functions and their short descriptions are provided below.
The actual C++ code is also included following the definition.
1. AssetPrices() is a function that returns the price of an asset or stock over a
user defined time period. This function prices per user defined time tick in an
array that holds the results.
2. AssetBinomial() is a function that uses the Binomial method for an American
Option call.
3. AssetBlackScholes() is a function that uses the Black Scholes formula for a
European call. This is used for comparison purposes.
4. AssetForward() uses the forward method to evaluate an American Option call.
This is used for comparison purposes.
5. In addition, there are utility calls that timestamp the output, print the output to
an output file and compare and return the greater of two numbers. There are
also real life data files that contain all of the market data for options for
specific periods.
//**********************************************************************
double *AssetPrices ( double AssetPriceT0, double AssetGrowth, double AssetVolatility,
double OptExpiryDate, int NumSteps, int *AssetSeed )
// Name:
AssetPrices
// Purpose: This function simulates the behavior of an asset over a user defined time
//
period. The time tick defined by the user is used to populate an array of
//
asset prices.
// Parameters:
//
Input, double AssetPriceT0, the asset price at time 0.
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//
Input, double AssetGrowth, the expected growth rate.
//
Input, double AssetVolatility, the volatility of the asset.
//
Input, double OptExpiryDate, the expiry date.
//
Input, integer NumSteps, the number of steps to take between
//
AssetPriceT0 and OptExpiryDate.
//
Input/output, int AssetSeed, a starting point for the random number
//
generator.
//
//
Output, double Asset_Path[N+1], the option values from time
//
AssetPriceT0 to OptExpiryDate in equal steps.
//
{
// Local variables
double dSteps;
int i;
double p;
double *r;
double *s;
// calculate the number of steps asked for in equal intervals
dSteps = OptExpiryDate / ( double ) ( NumSteps );
r = r8vec_normal_01_new ( NumSteps, AssetSeed );
// Allocate memory to hold the results, which is the asset price for every time interval
// defined.
s = new double[NumSteps+1];
// Initialize the array
s[0] = s0;
p = s0;
// For the number of steps user has asked for, apply the formula to find out the price
// for each time tick. Store it in the array created earlier.
for ( i = 1; i <= NumSteps; i++ )
{
p = p * exp ( ( AssetGrowth - AssetVolatility * AssetVolatlity ) * AssetPriceT0 +
AssetVolatility * sqrt ( AssetPriceT0 ) * r[i-1] );
s[i] = p;
}
// Delete the seed
delete [] r;
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// Return the head of the array so that it can be used for further processing.
return s;
}
//**********************************************************************
double AssetBinomial ( double AssetPriceT0, double AssetExercise, double
AssetInterestRate, double AssetVolatility, double OptExpiryDate,
int NumSteps )
// Name:
AssetBinomial
// Purpose: Uses the binomial method for a European call
// Parameters:
// Input, double AssetPriceT0, the asset price at time 0.
// Input, double AssetExerciseP, the exercise price.
// Input, double AssetInterestRate, the interest rate.
// Input, double AssetVolatility, the volatility of the asset.
// Input, double OptExpirtyDate, the expiry date.
// Input, int NumSteps, the number of steps to take between 0 and T1.
// Output, double AssetBionomial the option value at time 0.
{
double a;
double b;
double c;
double d;
double dt;
int i;
int n;
double p;
double u;
double *w;
//
// Determine the number of steps in time intervals
//
dt = AssetPriceT0 / ( double ) ( NumSteps );
a = 0.5 * ( exp ( - AssetInterestRate * dt ) + exp ( ( AssetInterestRate + AssetVolatility *
AssetVolatility ) * dt ) );
d = a - sqrt ( a * a - 1.0 );
u = a + sqrt ( a * a - 1.0 );
p = ( exp ( AssetInterestrate * dt ) - d ) / ( u - d );
// Create an array to hold the results
w = new double[NumSteps+1];
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for ( i = 0; i <= NumSteps; i++ )
{
w[i] = r8_max ( AssetPriceT0 * pow ( d, NumSteps - i ) * pow ( u, i ) AssetExercisePrice, 0.0 );
}
//
// Trace backwards to get the option value at time 0.
//
for ( n = m - 1; 0 <= n; n-- )
{
for ( i = 0; i <= n; i++ )
{
w[i] = ( 1.0 - p ) * w[i] + p * w[i+1];
}
}
for ( i = 0; i < m + 1; i++ )
{
w[i] = exp ( - AssetInterestRate * t1 ) * w[i];
}
c = w[0];
delete [] w;
return c;
}
/***********************************************************************
double AssetBlackScholes ( double AssetPriceT0, double AssetTimeKPrice, double
AssetExercisePrice, double AssetInterestRate, double AssetVolatility, double
AssetExpiryDate )
// Purpose: This function evaluates the Black-Scholes formula for an American call.
// Parameters:
// Input, double AssetPriceT0, the asset price at time zero.
// Input, double AssetTimeKPrice, the time at which the asset price is known.
// Input, double AssetExercise, the exercise price of the asset or the option.
// Input, double InterestRate, the risk free interest rate.
// Input, double AssetVolatility, the volatility of the stock, asset or option.
// Input, double AssetExpiryDate, the expiry date of the option by which it should be
// exercised or it will expire.
//
// Output: the value of the call option.
//
{
double c;
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double d1;
double d2;
double n1;
double n2;
double tau;
// Get the time interval
tau = AssetTimeKPrice - AssetExpiryDate;
// If it is a valid time, i.e., it is in the future then..
if ( 0.0 < tau )
{
d1 = ( log ( AssetPriceT0 / AssetExcersiePrice ) + ( AssetInterestRate + 0.5 *
AssetVolatility * AssetVolatility ) * tau ) / ( AssetVolatility * sqrt ( tau ) );
d2 = d1 - AssetVolatility * sqrt ( tau );
n1 = 0.5 * ( 1.0 + erf ( d1 / sqrt ( 2.0 ) ) );
n2 = 0.5 * ( 1.0 + erf ( d2 / sqrt ( 2.0 ) ) );
c = AssetPriceT0 * n1 - AssetExercisePrice * exp ( - AssetInterestRate * tau ) * n2;
}
else
{
// If it is now, then
c = r8_max ( AssetPriceT0 - AssetExercisePrice, 0.0 );
}
return c;
}
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Results
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Conclusions
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Reference List
Zvi Bodie, Alex Kane, Alan J. Marcus (1999). Investments (Fourth Edition). Irwin McGraw-Hill
Philip McBride, Johnson (1999). Derivatives: A manager’s guide to the world’s most powerful
financial instruments, McGraw-Hill
Eric Banks (2003). Exchange Traded Derivatives, John Wiley & Sons Ltd.
Robert W. Kolb (1995), Understanding Options, A Wiley Finance Edition.
Chen Hua, Roderick Wong, Differential equations and asymptotic theory in Mathematical
Physics, World Scientific Publishing Pte. Ltd.
Boness A.J.., 1973, Elements of a Theory of Stock Option Value. Journal of Political Economy,
72 (April) 163-75
Black, F., Scholes, M., (1973), The Pricing of Options and Corporate Liabilities, Journal of
Political Economy, Vol. 81, pp. 637-654.
Black, F., (1989), How We Came Up with the Option Formula, The Journal of Portfolio
Management, Vol. 15, pp. 4-8.Austad, S.N. (1997). Why we age. John Wiley & Sons, Inc.
Nonparametric Cures, ANNALS OF ECONOMICS AND FINANCE 4, 73–101 (2003).
Teneng, D (2011), Limitations of the Black Scholes model, International Research Journal of
Finance and Economics, ISSN 1450-2887 Issue 68 (2011).
Rama, Cont, Peter, Tankov, 2003. “Financial Modelling with Jump processes” Chapman
Hall/CRC Press, 2003.
Peter Carr, Liuren Wu, 2002. “Time change Levy processes and option pricing” [Online],
August 2002, accessed 20. 11. 2010,
http://www.math.nyu.edu/research/carrp/papers/pdf/jfetchgepaper.pdf
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Mou-Hsiung Chang, Tao Pang, Moustapha Pemy, 2009 “An Approximation Scheme for BlackScholes Equations with Delays”, AMS 2000 subject classi_cations: primary 90A09, 60H30;
secondary 60G44, 90A16
Samuel W. Malone, 2002, “Alternative price processes for Black Scholes: Empirical evidence
and theory”, NSF VIGRE grant number DMS-9983320.
Simon Benninga and Zvi Wiener, 1997, “Binomial Option Pricing, the Black-Scholes Option
Pricing Formula, and Exotic Options”, Mathematica in Education and Research Vol. 6 No. 4
1997.
Paul Clifford, Oleg Zaboronski, 2008, “Option Pricing Model using Trinomial tree”, [online]
accessed 11.23.2011, University of Warwick, United Kingdom.
http://www2.warwick.ac.uk/fac/sci/maths/people/staff/oleg_zaboronski/fm/trinomial_tree_2010_
kevin.pdf
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