1. Option Pricing Models: The Black-Scholes-Merton Model aka Black – Scholes Option Pricing Model (BSOPM)

2. Important Concepts The Black-Scholes-Merton option pricing model
The relationship of the model’s 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

3. 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.

4. 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 option’s life
There are no taxes or transaction costs
The stock pays no dividends
The options are European

5. Background Put and call prices are affected by
Price of underlying asset
Option’s 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

6. 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”.
Note: There are many other OPMs in existence. These are mostly variations on the Black-Scholes model, and the Black-Scholes model is the most used.

7. 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

8. 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

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. 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

15. 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

16. 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

17. Screen Shot of the Excel template for the BSOPM

18. OPM & The Measurement of Portfolio Risk Exposure
Because BS OPM isolates the effects of each variable’s effect on pricing, it is said that these isolated, independent effects measure the sensitivity of the options value to changes in the underlying variables.

19. 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

20. 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
Option
price A B
Slope = D
Stock price

21. Delta Application Suppose that the delta of a call is .8944.
What does this mean????

22. 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 shares of stock for each option sole to produce a delta –neutral portfolio

23. 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.

24. 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 indicates the delta will increase by if the underlying price increases or decreases by 1.0.
Change in Delta = Gamma
Change in stock price

25. 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.

26. 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

27. 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.

28. 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

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

30. 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

31. 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