Option pricing is determined by 5 key factors: the asset's cash price, strike price, volatility, time to expiration, and interest rates. The Black-Scholes model uses these factors to price European options, assuming the asset pays no dividends, markets are efficient, and other restricting assumptions. It models the asset's price as following geometric Brownian motion to derive a theoretical option price formula involving the standard normal distribution.
Want to understand how options work but don\'t have time to go through books? Read this presentation I prepared with couple of my classmates for a case study in Advanced Finance at AIM
This ppt is prepared to provide detailed information regarding Forwards and Futures contracts of Derivatives the topics covered under this are Meaning of Forwards contracts, Underlying Assets of Forwards contracts, FEATURES OF FORWARD CONTRACTS, Tailored made, Why Forwards contracts, FUTURES CONTRACT, What is A Futures Contract, Characteristics of Futures contracts, Mechanism of Trading in Futures Market, Margin requirement, Marking-to-market (M2M), SETTLING A FUTURE POSITION, OFFSETTING, CASH DELIVERY, by Sundar, Assistant Professor of commerce.
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Derivatives - Basics of Derivatives contract covered in this pptSundar B N
Derivatives - Basics of Derivatives including forward, futures, swap and options contracts which covers HISTORY OF DERIVATIVES, CHARACTERISTICS OF DERIVATIVES , FEATURES OF DERIVATIVES, FUNCTIONS OF DERIVATIVES MARKET, USES OF DERIVATIVES, DIFFERENCE BETWEEN SHARES AND DERIVATIVES SHARES DERIVATIVES, DEFINITION OF UNDERLYING ASSET, DERIVATIVES ADVANTAGES AND DISADVANTAGES, PARTICIPANTS/ TRADERS IN DERIVATIVES MARKET, SPECULATORS, ARBITRAGEURS, HEDGER
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Want to understand how options work but don\'t have time to go through books? Read this presentation I prepared with couple of my classmates for a case study in Advanced Finance at AIM
This ppt is prepared to provide detailed information regarding Forwards and Futures contracts of Derivatives the topics covered under this are Meaning of Forwards contracts, Underlying Assets of Forwards contracts, FEATURES OF FORWARD CONTRACTS, Tailored made, Why Forwards contracts, FUTURES CONTRACT, What is A Futures Contract, Characteristics of Futures contracts, Mechanism of Trading in Futures Market, Margin requirement, Marking-to-market (M2M), SETTLING A FUTURE POSITION, OFFSETTING, CASH DELIVERY, by Sundar, Assistant Professor of commerce.
Subscribe to Vision Academy for Video assistance
https://www.youtube.com/channel/UCjzpit_cXjdnzER_165mIiw
Derivatives - Basics of Derivatives contract covered in this pptSundar B N
Derivatives - Basics of Derivatives including forward, futures, swap and options contracts which covers HISTORY OF DERIVATIVES, CHARACTERISTICS OF DERIVATIVES , FEATURES OF DERIVATIVES, FUNCTIONS OF DERIVATIVES MARKET, USES OF DERIVATIVES, DIFFERENCE BETWEEN SHARES AND DERIVATIVES SHARES DERIVATIVES, DEFINITION OF UNDERLYING ASSET, DERIVATIVES ADVANTAGES AND DISADVANTAGES, PARTICIPANTS/ TRADERS IN DERIVATIVES MARKET, SPECULATORS, ARBITRAGEURS, HEDGER
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By www.ProfitableInvestingTips.com
What is Intrinsic Stock Value?
In the aftermath of the stock market crash of 1929 in the early days of the Great Depression Benjamin Graham introduced the concept of value investing. No longer would those buying and selling stocks need to act like they were at the casino. With the concepts of intrinsic value and margin of safety Graham taught investors a rational means of investing in stocks. With this in mind just what is intrinsic stock value? And how does this concept help with profitable stock investing?
What Is Intrinsic Stock Value?
The dictionary definition of intrinsic stock value is its fundamental value. It is obtained by adding up predicted future income of a stock and subtracting current price. It can also be seen as actual value of an equity versus its book value or market value. The concept of fundamental analysis of equities evolved from this concept. Using fundamental analysis the intrinsic value of a stock is the expected company cash flow discounted to current dollars. It is a discounted cash flow valuation. An inherent weakness in this concept is that too often the medium and long term prospects of a company and its stock price are not clear. So, what is intrinsic stock value of a company if the future is uncertain? The ability to see into the future to see how well a company will manage its assets, products, costs, R&D, and marketing is of utmost importance in calculating intrinsic stock value as a means of deciding whether or not to purchase a stock.
What is Intrinsic Stock Value as a Formula?
Mr. Graham presented investors with a formula for calculating intrinsic stock value in 1962 and modified it in 1974. The 1974 version considers the following:
• Earnings per share, EPS, for the preceding twelve months
• A constant of 8.5 representing an expected price to earnings ratio, P/E ratio, for a company that is not growing
• An estimate of long term growth, five years = g
• A constant of 4.4 which was the average yield of high grade corporate bonds in the early 1960 decade
• The current yield of AAA corporate bonds = Y
• Where V = intrinsic value
The formula is as follows:
V = (EPS x (8.5 + 2g) x 4.4)/Y
The way the investors were encouraged to use intrinsic value was to derive what is referred to as a Relative Graham Value, RGV. This is to divide the calculated intrinsic value of the stock by its current price. If the result, the RGV, is less than one the stock is overvalued and a bad investment and if the ratio is above one it is undervalued and may be a good investment.
What is Intrinsic Stock Value as an Investing Tool?
There are a couple of difficulties in using the simple calculation above to determine the forward looking earnings of a stock and therefore its intrinsic value. First of all the formula does not account for inflation. Thus one could use the formula and end up with a stock valued higher in dollars but in dollars that are inflated.
Option Pricing ModelsThe Black-Scholes-Merton Model a.docxhopeaustin33688
Option Pricing Models:
The Black-Scholes-Merton Model aka Black – Scholes Option Pricing Model (BSOPM)
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Important ConceptsThe Black-Scholes-Merton option pricing modelThe relationship of the model’s inputs to the option priceHow to adjust the model to accommodate dividends and put optionsThe concepts of historical and implied volatilityHedging an option position
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The Black-Scholes-Merton FormulaBrownian motion and the works of Einstein, Bachelier, Wiener, ItôBlack, Scholes, Merton and the 1997 Nobel PrizeRecall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
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Some Assumptions of the ModelStock prices behave randomly and evolve according to a lognormal distribution. The risk-free rate and volatility of the log return on the stock are constant throughout the option’s lifeThere are no taxes or transaction costsThe stock pays no dividendsThe options are European
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BackgroundPut and call prices are affected byPrice of underlying assetOption’s exercise priceLength of time until expiration of optionVolatility of underlying assetRisk-free interest rateCash flows such as dividendsPremiums can be derived from the above factors
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Option ValuationThe value of an option is the present value of its intrinsic value at expiration. Unfortunately, there is no way to know this intrinsic value in advance. Black & Scholes developed a formula to price call options This most famous option pricing model is the often referred to as “Black-Scholes OPM”.
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Note: There are many other OPMs in existence. These are mostly variations on the Black-Scholes model, and the Black-Scholes model is the most used.
The Concepts Underlying Black-ScholesThe option price and the stock price depend on the same underlying source of uncertaintyWe can form a portfolio consisting of the stock and the option which eliminates this source of uncertaintyThe portfolio is instantaneously riskless and must instantaneously earn the risk-free rate
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Option Valuation VariablesThere are five variables in the Black-Scholes OPM (in order of importance):Price of underlying securityStrike priceAnnual volatility (standard deviation)Time to expirationRisk-free interest rate
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Option Valuation Variables: Underlying PriceThe current price of the underlying security is the most important variable.For a call option, the higher the price of the underlying security, the higher the value of the call.For a put option, the lower the price of the underlying security, the higher the value of the put.
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Option Valuation Variables: Strike PriceThe strike (exercise) price is fixed for the life of the option, but every underlying security has several strikes for each expiration monthFor a call, the higher the strike price, the lower the value of the call.For a put, the higher t.
Black-Scholes Model
Introduction
Key terms
Black Scholes Formula
Black Scholes Calculators
Wiener Process
Stock Pricing Model
Ito’s Lemma
Derivation of Black-Sholes Equation
Solution of Black-Scholes Equation
Maple solution of Black Scholes Equation
Figures
Option Pricing with Transaction costs and Stochastic Volatility
Introduction
Key terms
Stochastic Volatility Model
Quanto Option Pricing Model
Key Terms
Pricing Quantos in Excel
Black-Scholes Equation of Quanto options
Solution of Quanto options Black-Scholes Equation
A Comparison of Option Pricing ModelsEkrem Kilic 11.0.docxevonnehoggarth79783
A Comparison of Option Pricing Models
Ekrem Kilic �
11.01.2005
Abstract
Modeling a nonlinear pay o¤ generating instrument is a challenging work. The mod-
els that are commonly used for pricing derivative might divided into two main classes;
analytical and iterative models. This paper compares the Black-Scholes and binomial
tree models.
Keywords: Derivatives, Option Pricing, Black-Scholes,Binomial Tree
JEL classi�cation:
1. Introduction
Modeling a nonlinear pay o¤ generating instrument is a challenging work to
handle. If we consider a European option on a stock, what we are trying to do is
estimating a conditional expected future value. In other words we need to �nd out
the following question: what would be the expected future value of a stock given
that the price is higher than the option�s strike price? If we �nd that value we can
easily get the expected value of the option. For the case of the American options
the model need to be more complex. For this case, we need to check the path that
we reached some future value of the stock, because the buyer of the option might
exercise the option at any time until the maturity date.
To solve the problem that summarized above, �rst we need to model the move-
ment of the stock during the pricing period. The common model for the change
of the stock prices is Geometric Brownian Motion. Secondly, the future outcomes
of the model might have the same risk. Risk Neutrality assumption provides that.
By constructing a portfolio of derivative and share makes possible to have same
�E-mail address: [email protected]
A Comparison of Option Pricing Models 2
outcome with canceling out the source of the uncertainty.
The models that are commonly used for pricing derivative might divided into
two main classes. The �rst classes is the models that provide analytical formulae to
get the risk neutral price under some reasonable assumptions. The Black-Scholes
formula is in this group. The formulae that we have to price the derivatives
are quite limited. The reason is that we are trying to solve a partial di¤erential
equation at the end of the day. But mathematician could manage to solve just
someof thepartialdi¤erential equations; therefore, weareboundedto some limited
solutions.
The second classes models provide numerical procedures to price the option.
Binomial trees that �rst suggested by Cox, Ross and Rubenstein, is in this group,
because we need to follow an iterative procedure called �backwards induction�to
get option price. Monte Carlo simulations are another type of models that belongs
to this class. Also �nite di¤erencing methods are a type of numerical class.
In this paper, �rst I will introduce Black-Scholes and Binomial Tree models for
option pricing. Second I will introduce the volatility estimation methods I used
and calculate some option prices to compare models. Finally I will conclude.
2. Option Pricing Models
2.1. Black-Scholes Model
Black-Scholes formula s.
1. Option Pricing
• There are 5 determinants of Option pricing or
premiums:
1. Cash Price of Asset (S )t
2. Strike Price (K)
3. Volatility of the underlying asset’s price (σ)
4. Time to expiration (T)
5. Interest Rates (r)
These factors affect the premium / price of both
American & European options in several ways.
2. Cash Price of Asset
• Keeping all other factors constant, if cash
price of underlying asset goes up value of the
call option increases but value of the put
option diminishes.
3. Effect of Strike Price
• If all other factors remain constant but the
strike price of option increases, intrinsic value
of the call option will decrease and hence its
value will also begin to decrease.
• On the other hand, with all other factors
remaining constant, increase in strike price
will increase the intrinsic value of the put
option and it will therefore become dearer.
4. Effect of Volatility
• Volatility in the price of the underlying asset
affects both call & put options in the same
way. As higher volatility escalates the chances
of an option going in-the-money at any point
in time during the life of the contract. It
increases the risk to the option seller and
consequently makes the option, both call &
put, more expensive.
5. Effect of time to Expiration
• The effect of time to expiration on both call &
put options is similar to that of volatility on
option premiums.
• The longer the maturity of the option the
greater is the uncertainty and hence the
prices of both call & put options are higher,
keeping all other factors constant.
• Therefore, longer maturity options are always
more expensive than shorter maturity
options.
6. Effect of Interest Rates
• Higher interest rates has the same effect as
lowering the strike price, and therefore as
seen, higher interest rate will result in an
increase in the value of a call option and a
decrease in the value of a put option.
7. Black-Scholes Model for Options Pricing
• Black-Scholes model for pricing European options
published by Fischer Black & Myron Scholes in 1973 is by
far the most popular model to price an option.
• Black-Scholes model start by specifying a simple & well
known equation that the manner in which stock prices
fluctuates.
• This equation is called Geometric Brownian Motion
implies that stock returns will have a lognormal
distribution i.e., the logarithm of the stock’s return will
follow the normal (bell shaped) distribution.
8. Cont…
• Black & Scholes then propose that the price of
an option is determined by the only two
variables that are allowed to change – time
and the underlying stock price.
• While other factors such as volatility, exercise
price and risk free rate do affect the price of
the option they are not allowed to change.
9. Main assumptions are as follows:
1. The stock pays no dividend during the
option’s life.
2. European exercise terms are used.
3. Markets are efficient.
4. No commissions are charged.
5. Interest rates remain constant & known.
6. Returns are log-normally distributed.
10. The Model: (-rt)
C = SN(d1 ) – Ke N(d 2 )
Where,
C = Theoretical Call Premium
S = Current Stock Price
t = time until option Expiration
K = Option’s strike price
r = Risk-free interest rate
N = Cumulative Standard Normal Distribution
E = Exponential Term (2.7183)
d1 = 1n(S/K) + {r + s 2 /2)}t
s√t
d 2 = d1 - s√t
s = Standard Deviation of Stock returns
1n = Natural logarithm