# 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

- Construct a package containing
- 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

- Construct a package containing
**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

- 11.80%
- 25.00%
- 41.40%
- Cannot be determined

Solution

**A.**

(11.80% )