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 Formula
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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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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
• We might expect the implied volatilities of
  options on the same underlying asset to have the
  same implied volatility

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Implied Volatilities
IN FACT THIS IS NOT THE CASE!

• The term structure of volatility describes the
  way at-the-money implied volatility varies with
  time to expiration.
• - There is evidence that implied volatilities are
  higher for longer time-to-expiration index
  options
• The volatility smile is the way in which implies
  volatility varies with strike price for options
  of a fixed expiration
• Here there is evidence that deep out-of-the money
  puts and deep in-the-money calls have higher
  implied values than at-the-money puts and calls.
• The smile became more pronounced after the
  crash of 87 and some attribute it to
  crashophobia
• The smile is an important challenge to option
  pricing theories

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Executive Options

• Importance
• FASB may soon require firms to calculate and
  recognize as a cost the value of employee stock
  options when granted
• Even if FASB backs off, 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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Executive Options

• Why not use B-S using an historically estimated
  expected time to exercise?
• Exercise experience will depend in large part on
  stock returns during the sample period
• Because option value is a nonlinear function of
  the term of the option, use of he expected term
  will introduce additional error
• Since the time until exercise will generally
  vary inversely with the stock price (and
  therefore the option payoff), replacing stated
  term with expected term will overstate the value
  of the option

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Executive Options

• Example Employee has two sources of wealth
• Call option
• Risk-free investments

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Executive Options

• At maturity the exercise rule is trivial
• At node B, EV 24.20 gt 20
• Here the exercise rule depends on investors
  risk tolerance
• It would not take much risk aversion to prefer
  20 for sure to the gamble of 44 with p 0.55 and
  0 with p .45

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Executive Options

• 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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Executive Options

• If we assume risk aversion is sufficient to
  dictate early exercise at node B, the value of
  the firms obligation and the employees
  compensation is reduced by 10

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Executive Options

• Note the difficulty in using historical rates of
  exercise to forecast future rates Exercise will
  never occur at node C!!
• Rates of exercise depend on past price movements
  (which do not predict future price movements!)
• In this example, expected term is 0.55 x 1
  period 0.5 x 2 periods

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Executive Options

• 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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Executive Options

• 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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Conclusions

• 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