A Note on Continuous Compounding

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A Note on Continuous Compounding
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Continuous Compounding

• If your money earns at an APR (annual percentage
  rate) of 6 per year compounded semi-annually,
  what is the effective annual rate of return?
• FV (1.03)2 1.0609
• reff 6.09
• The general formula for the effective annual
  rate is
• As the compounding frequency increases,
• Where e is the number 2.71828

K
K
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Continuous Compounding
For Example
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Continuous Compounding

• We can also calculate the continuously
  compounded rate of return on a stock for some
  period as follows
• If ST is the end-of-period price and S0 is the
  starting price, then the return is ln(ST / S0)
  where ln is the natural logarithm
• For example, if the price one year ago was 100
  and the current price is 110, the continuously
  compounded return id ln( 110 / 100), or 0.0953,
  and e.0953 1 10

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The Black-Scholes FormulaGroup Project II
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Models of Stock Price Movements

• Predictive Models
• - The perfect model would indicate the exact
  future price
• Probabilistic Models
• - Stock price models used in option pricing are
  probabilistic
• Binomial Model
• - Each period the stock price moves up or down by
  a constant percentage amount
• Black-Scholes Option Pricing Formula
• - The evolution of the stock price over time is
  governed by a Geometric Brownian Motion

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Geometric Brownian Motion

• The return on a stock price between now and some
  short time in the future (?t) is normally
  distributed
• The returns between any two time intervals are
  independent
• The returns between any two time intervals are
  identically distributed
• The mean of the distribution is µ times the
  amount of time (µ?t)
• The standard deviation of returns is
• µ instantaneous rate of return
• ? instantaneous standard deviation

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Geometric Brownian Motion

• A numerical example Assume there are two stocks
  with an identical expected return of 12. Stock
  S1 has annualized instantaneous volatility of 10
  while S2 has annualized instantaneous volatility
  of 25. What is the probability that the stock
  price will change by more than a certain amount
  for each stock?

 

• Higher volatility implies a stock is riskier in
  the sense that the probability is higher that
  future price changes will be greater

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The Distribution of Stock Prices

• We have already noted that the above assumptions
  imply that the return on the stock is distributed
  normally. This implies that the distribution of
  stock prices will be log-normal

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The Distribution of Stock Prices
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Do Stock Prices Follow a Geometric Brownian
Motion?

• The geometric Brownian motion model predicts that
  large price movements will be far less likely
  than is fact the case. The most extreme example
  of this is that of the crash of Oct 19, 1987. If
  we assume annualized volatility of 20, the
  probability of a price move of the magnitude
  experienced is approximately 10-160 (a virtual
  impossibility)
• There is also evidence that returns do not scale
  the way they should (returns should be
  proportional to elapsed time and the standard
  deviation of returns should be proportional to
  the square root of elapsed time) There is
  evidence that monthly and quarterly volatilities
  are too high to be consistent with annual
  volatilities under the assumptions of the model.
  
• Finally, there is evidence that volatilities
  change through time. This may be related to 1)
  and 2) above.

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The Black-Scholes Formula

• In 1973 Black and Scholes published a closed
  form solution to the problem of pricing European
  call options. The formula is as follows
• Where
• And

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The Black-Scholes Formula

• Notation is consistent with that developed
  previously. New notation includes the following
• N(d) The probability that a random draw from a
  standard normal distribution will be less than d
• e 2.71828, the base of the natural log
  function (ln)
• r The annualized, continuously compounded rate
  of return on a risk-free asset with the same
  maturity as the expiration of the option
• ? Standard deviation of the annualized
  continuously compounded rate of return on the
  stock

 
 
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The Black-Scholes Formula

• The Black-Scholes formula can be easily used
  with a hand calculator. See the text for an
  example. By contrast, the binomial model requires
  a computer
• Note that of the 5 inputs necessary (S, X, r, ?,
  and T), only the standard deviation of the return
  on the stock must be computed
• Online services (such as Bloomberg) report
  standard deviations, and both the Black-Scholes
  and binomial model option prices

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The Relation Between the Black-Scholes Model and
the Binomial Model

• It can be shown that if we choose the parameters
  governing the up and down movements in the stock
  appropriately, as below for example
• Then the larger n (the number of periods), the
  closer the binomial call option price to the
  Black-Scholes call option price
• In the limit, as n??, and ?T ? 0, the two are
  equal
• In this case the stock price process will be the
  geometric Brownian motion discussed above, the
  assumed stochastic process governing stock price
  movements in the Black-Scholes model.

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Hedge Ratios and the Black-Scholes Formula

• An options hedge ratio is the change in the
  price of an option for a 1 change in the stock
  price.
• A call option therefore has a positive hedge
  ratio, and a put option has a negative hedge
  ratio. A hedge ratio is commonly called the
  options delta (? )
• In the single period binomial model, the hedge
  ratio was easily calculated as
• Black-Scholes hedge ratios are also easy to
  compute. The hedge ratio for a call is N(d1),
  while the hedge ratio for a put is N(d1) 1

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Hedge Ratios and the Black-Scholes Formula

• It is important to note that the hedge ratios
  change as the price of the stock changes
• For a call option, an option deeply in the money
  will be exercised at expiration with high
  probability. Therefore, each dollar change in the
  value of the stock will change the value of the
  option by close to one dollar.
• If an option is far out of the money near
  expiration, exercise will be unlikely, so each
  dollar change in the value of the stock will have
  little impact on the value of the option

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Hedge Ratios and the Black-Scholes Formula

• Note that an options delta also will change
  with time, since time to expiration is an
  important determinant of the probability that an
  option will expire in the money. (As time
  approaches expiration, the value of the option
  approaches its intrinsic value)

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Hedge Ratios and the Black-Scholes Formula

• The hedge ratio is an important tool in
  portfolio management because it shows the
  sensitivity of the value of a portfolio to
  changes in the value of n underlying security
• Delta Hedging If we know the hedge ratio of a
  call, it tells us the number of calls that must
  be sold to hedge the stock position
• Assume for example that the option price is 10,
  the stock price is 100, and ? .6
• This means that if the stock price changes by a
  small amount, then the option price changes by
  about 60 of that amount

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Hedge Ratios and the Black-Scholes Formula

• If an investor had sold 10 options contracts
  (options to buy 1000 shares), then the investors
  position could be hedged by buying 600 shares of
  stock (.6 x 1000) because a 1 increase in the
  value of the stock will offset the change in the
  value of the call portfolio
• Remember that a portfolio will only remain delta
  hedged for a short period of time because of the
  impact of both time and the price of the stock on
  the hedge ratio

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Hedge Ratios and the Black-Scholes Formula

• Dynamic Hedging Schemes refer to the frequent
  rebalancing that is necessary in order to
  maintain a particular hedge

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Summary and Review of Determinants of Option
Values

• Signs of option function partial derivatives

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Implied Volatilities

• The Black-Scholes model yields a price as an
  output after inputting the variables listed above
• Recall that the only input that must be
  calculated is the volatility estimate
• An alternative is to use an option market price
  to yield the implied volatility of the
  underlying asset

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Employee Stock Options

• Importance
• Accounting for employee stock options is an
  important issue
• Investors, analysts, and employees need to be
  able to value these options
• Why Black-Scholes doesnt work
• Employee stock options are not transferable
• The only way to liquidate a position is sell it,
  forfeiting the time value
• Early exercise is based on portfolio
  diversification motives

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Employee Stock Options (cont.)

• The propensity to exercise early will depend on
• The employees level of risk aversion
• The extent to which the employees human capital
  is firm specific
• The fraction of total wealth the option(s)
  represent

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Employee Stock Options (cont.)

• The following figure shows the influence of the
  assumed degree of employee risk aversion and
  non-option wealth on the value of an option
• Assuming a gamma of about 4 is reasonable for
  this type of exercise

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Employee Stock Options (cont.)

• Although the values of tradable options rise
  with the volatility of the underlying, the
  value of restricted employee options can actually
  fall.
• - When ? is high, a risk averse investor is more
  likely to exercise early, possibly dominating the
  effect of ?

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Employee Stock Options (cont.)

• For reasonable parameters, early exercise
  reduces the time the option is alive and option
  value by more than 50
• Employee stock options will need to be valued in
  a manner similar to that used for mortgage backed
  securities
• Instead of valuing each mortgage in a pool
  separately, take a statistical approach in which
  the characteristics of the mortgage pool
  determine the value in different economic
  environments
• Statistical analysis employing monte carlo
  simulations is useful here