Option Pricing Models: The Black-Scholes-Merton Model aka Black

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Option Pricing ModelsThe Black-Scholes-Merton
Model aka Black Scholes Option Pricing Model
(BSOPM)
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Important Concepts

• The Black-Scholes-Merton option pricing model
• The relationship of the models inputs to the
  option price
• How to adjust the model to accommodate dividends
  and put options
• The concepts of historical and implied volatility
• Hedging an option position

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The Black-Scholes-Merton Formula

• Brownian motion and the works of Einstein,
  Bachelier, Wiener, Itô
• Black, Scholes, Merton and the 1997 Nobel Prize
• Recall 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 Model

• Stock 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
  options life
• There are no taxes or transaction costs
• The stock pays no dividends
• The options are European

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Background

• Put and call prices are affected by
• Price of underlying asset
• Options exercise price
• Length of time until expiration of option
• Volatility of underlying asset
• Risk-free interest rate
• Cash flows such as dividends
• Premiums can be derived from the above factors

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Option Valuation

• The 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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The Concepts Underlying Black-Scholes

• The option price and the stock price depend on
  the same underlying source of uncertainty
• We can form a portfolio consisting of the stock
  and the option which eliminates this source of
  uncertainty
• The portfolio is instantaneously riskless and
  must instantaneously earn the risk-free rate

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Option Valuation Variables

• There are five variables in the Black-Scholes OPM
  (in order of importance)
• Price of underlying security
• Strike price
• Annual volatility (standard deviation)
• Time to expiration
• Risk-free interest rate

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Option Valuation Variables Underlying Price

• The 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 Price

• The strike (exercise) price is fixed for the life
  of the option, but every underlying security has
  several strikes for each expiration month
• For 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.

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Option Valuation Variables Volatility

• Volatility 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.

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Option Valuation Variables Time to Expiration

• The 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.

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Option Valuation Variables Risk-free Rate

• The risk-free rate of interest is the least
  important of the variables.
• It is used to discount the strike price
• The 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.

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Implied Volatility

• The implied volatility of an option is the
  volatility for which the Black-Scholes price
  equals the market price
• The is a one-to-one correspondence between prices
  and implied volatilities
• Traders and brokers often quote implied
  volatilities rather than dollar prices

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Nature of Volatility

• Volatility is usually much greater when the
  market is open (i.e. the asset is trading) than
  when it is closed
• For this reason time is usually measured in
  trading days not calendar days when options are
  valued

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A Nobel Formula

• The 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

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Screen Shot of the Excel template for the BSOPM
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OPM The Measurement of Portfolio Risk Exposure

• Because BS OPM isolates the effects of each
  variables effect on pricing, it is said that
  these isolated, independent effects measure the
  sensitivity of the options value to changes in
  the underlying variables.

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The Greeks

• Greeks are derivatives of the option price
  function
• Delta
• Gamma
• Theta
• Vega
• Rho
• The Greeks are also called hedge parameters as
  they are often used in hedging operations by big
  financial institutions

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Delta

• Delta (D) describes how sensitive the option
  value is to changes in the underlying stock
  price.
• Change in option price Delta
• Change in stock price

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Delta Application

• Suppose that the delta of a call is .8944.
• What does this mean????

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Delta Neutral

• In other words, we want the delta to be zero.
• Example
• Current stock price is 100
• Call price (per opm) is 11.84
• Delta .8944
• We must buy .8944 shares of stock for each option
  sole to produce a delta neutral portfolio

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Delta Neutrality

• Exists when small changes in the price of the
  stock does not affect the value of the portfolio.
• However, this 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.

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Gamma

• Gamma (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

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Gamma Application

• Can be either positive or negative
• The 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.

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Theta

• Theta (Q) of a derivative is the rate of change
  of the 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

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Theta Application

• If 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.

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Vega

• Vega (n) is the rate of change of the value of a
  derivatives portfolio with respect to volatility
• For 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

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Vega Application

• Proves to us that the more volatile the
  underlying stock, the more volatile the option
  price.
• Vega is always a positive number.

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Rho

• Rho 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

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Rho Application

• Rho for calls is always positive
• Rho for puts is always negative
• A Rho of 25 means that a 1 increase in the
  interest rate would
• Increase the value of a call by .25
• Decrease the value of a put by .25