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Limitations of Black Scholes Model

 

  • Limitations:
    • The model does not allow for early exercise
    • Not suitable for valuing American Options that can be exercised any time during their life
    • The stepwise binomial method is superior for valuing American Options, particularly American Puts and American Calls on stocks that pay dividends
    • Not suitable for valuing warrants as warrants are long term options and it is quite likely that the underlying stock will pay dividends during the life of the warrant
    • Also, when exercised warrants increase the total number of shares which adds another level of complication in valuing warrants using Black and Scholes formula

Summary

  • Complications arise in valuing options because its impossible to quantify risks associated with options
  • Options can be valued using the binomial method
  • Replicating options
  • Risk neutral method
  • European options on non dividend paying stocks can be valued using the Black Scholes method
  • Option Delta is defined as:
  • Replicating a call option
    • Construct a package containing
      • Buy delta stocks and
      • Borrow a sum of money which is equal to the difference between the pay-offs from the option and pay offs from the delta shares
  • This package has the same pay-off as a call option
  • The value of the package is the value of the call option
  • Replicating a put option
    • Construct a package containing
      • Sell delta stocks and
      • Deposit a sum of money which is equal to the difference between the pay-offs from the option and pay offs from the delta shares
    • This package has the same pay-off as of a put option
    • The value of the package is the value of the put option
  • Risk Neutral Method
  • Determine the probability of upside and downside changes in stock price
  • Assume investors are risk neutral
  • Discount the future expected pay-off at the riskfree rate to derive the option value
  • Black Scholes Model
  • Assumes log normal distribution of stock prices
  • Provides a model for valuing European options on non dividend paying stocks:
  • Where,
 
  •  Log is the natural log with base e
    • N (d) = cumulative normal probability density function
    • X = exercise price option;
    • T = number of periods to exercise date
    • P =present price of stock
    • σ = standard deviation per period of (continuously compounded) rate of return on stock
  • Value of Put =
  • Black Scholes model can be used to derive Implied Volatility
  • Reflects market opinion in the likely volatility of a stock

Example

Company X owns a property with a book value of €80,000. There is a buyer willing to pay €200,000 for the property. However, Company X must also provide the buyer with a put option to sell the property back to Company X for €200,000 at the end of 2years. Moreover, Company X agrees to pay the buyer €40,000 for a call option to repurchase the property for €200,000 at the end of 2 years. In effect, with this transaction Company X “borrows” money from the buyer. What is the annually compounded interest rate per year on this implied loan

  1. 11.80%
  2. 25.00%
  3. 41.40%
  4. Cannot be determined
     
Solution

A.
(11.80% )
 





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