Option Pricing Models: The Black-Scholes-Merton Model aka Black – Scholes Option Pricing Model (BSOPM) * 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 * 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. * 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 * 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 * 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”. * 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 * 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 * 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. * 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.

1. Option Pricing Models:
The Black-Scholes-Merton Model aka Black – Scholes Option
Pricing Model (BSOPM)
*
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
*
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

2. to continuous time.
*
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
*
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
*
Option ValuationThe value of an option is the present value of
its intrinsic value at expiration. Unfortunately, there is no way

3. 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”.
*
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
*
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
*

4. 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.
*
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 the strike price, the higher the value of the put.
*
Option Valuation Variables: VolatilityVolatility is measured as
the annualized standard deviation of the returns on the
underlying security.All options increase in value as volatility
increases.This is due to the fact that options with higher
volatility have a greater chance of expiring in-the-money.
*

5. Option Valuation Variables: Time to ExpirationThe time to
expiration is measured as the fraction of a year. As with
volatility, longer times to expiration increase the value of all
options.This is because there is a greater chance that the option
will expire in-the-money with a longer time to expiration.
*
Option Valuation Variables: Risk-free RateThe risk-free rate of
interest is the least important of the variables.It is used to
discount the strike priceThe risk-free rate, when it increases,
effectively decreases the strike price. Therefore, when interest
rates rise, call options increase in value and put options
decrease in value.
*
Implied VolatilityThe implied volatility of an option is the
volatility for which the Black-Scholes price equals the market
priceThe is a one-to-one correspondence between prices and
implied volatilitiesTraders and brokers often quote implied
volatilities rather than dollar prices
*

6. Nature of VolatilityVolatility is usually much greater when the
market is open (i.e. the asset is trading) than when it is
closedFor this reason time is usually measured in “trading days”
not calendar days when options are valued
*
A Nobel FormulaThe Black-Scholes-Merton model gives the
correct formula for a European call under these
assumptions.The model is derived with complex mathematics
but is easily understandable. The formula is
*
Screen Shot of the Excel template for the BSOPM
*
OPM & The Measurement of Portfolio Risk ExposureBecause
BS OPM isolates the effects of each variable’s effect on
pricing, it is said that these isolated, independent effects

7. measure the sensitivity of the options value to changes in the
underlying variables.
*
The Greeks
Greeks are derivatives of the option price function :
Delta Gamma Theta VegaRho
The Greeks are also called hedge parameters as they are
often used in hedging operations by big financial institutions
*
DeltaDelta (D) describes how sensitive the option value is to
changes in the underlying stock price. Change in option price
= Delta
Change in stock price

8. A
B
Slope = D
Stock price
Option
price
*
Delta ApplicationSuppose that the delta of a call is .8944.What
does this mean????
*
Delta NeutralIn other words, we want the delta to be
zero.Example:Current stock price is $100Call price (per opm) is
$11.84Delta = .8944We must buy .8944 shares of stock for each
option sole to produce a delta –neutral portfolio
*
Delta NeutralityExists when small changes in the price of the
stock does not affect the value of the portfolio.However, this

9. “neutrality” is dynamic, as the value of delta itself changes as
the stock price changes. This idea of neutrality can be extended
to the other sensitivity measures.
*
GammaGamma (G) is the rate of change of delta (D) with
respect to the price of the underlying asset. For example, a
Gamma change of 0.150 indicates the delta will increase by
0.150 if the underlying price increases or decreases by 1.0.
Change in Delta = Gamma
Change in stock price
*
Gamma ApplicationCan be either positive or negativeThe only
Greek that does not measure the sensitivity of an option to one
of the underlying assets. – it measures changes to its Greek
brother – Delta, as a result of changes to the stock price.
*
ThetaTheta (Q) of a derivative is the rate of change of the

10. value with respect to the passage of time.Or sensitivity of
option value to change in time Change in Option Price =
THETA
Change in time to Expiration
*
Theta ApplicationIf time is measured in years and value in
dollars, then a theta value of –10 means that as time to option
expiration declines by .1 years, option value falls by $1.AKA
Time decay: A term used to describe how the theoretical value
of an option "erodes" or reduces with the passage of time.
*
VegaVega (n) is the rate of change of the value of a derivatives
portfolio with respect to volatilityFor example: a Vega of .090
indicates an absolute change in the option's theoretical value
will increase by .090 if the volatility percentage is increased by
1.0 or decreased by .090 if the volatility percentage is decreased
by 1.0. Change in Option Price = Vega
Change in volatility
*

11. Vega ApplicationProves to us that the more volatile the
underlying stock, the more volatile the option price.Vega is
always a positive number.
*
RhoRho is the rate of change of the value of a derivative with
respect to the interest rate
For example:a Rho of .060 indicates the option's
theoretical value will increase by .060 if the interest rate is
decreased by 1.0.Change in option price = RHO
Change in interest rate
*
Rho ApplicationRho for calls is always positiveRho for puts is
always negativeA Rho of 25 means that a 1% increase in the
interest rate would:Increase the value of a call by $.25Decrease
the value of a put by $.25
*
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12. where
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