1. Introduction
The Black-Scholes option pricing model (BSOPM) has been one of the most important developments in finance in the last 50 years
Has provided a good understanding of what options should sell for
Has made options more attractive to individual and institutional investors
Continuous time and multiple periods
Future security prices are not limited to only two values
There are theoretically an infinite number of future states of the world
The pricing logic remains: A risk less investment should earn the riskless rate of interest
Derivation from:
Physics
Mathematical short cuts
Arbitrage arguments
Fischer Black and Myron Scholes utilized the physics heat transfer equation to develop the BSOPM

2. The Model

3. The Model Variable definitions: S = current stock price
K = option strike price
e = base of natural logarithms
R = riskless interest rate
T = time until option expiration
 = standard deviation (sigma) of returns on the underlying security
ln = natural logarithm
N(d1) and N(d2) = cumulative standard normal distribution functions

4. Determinants of the Option Premium
The lower the striking price for a given stock, the more the option should be worthBecause a call option lets you buy at a predetermined striking price
The longer the time until expiration, the more the option is worth
The option premium increases for more distant expirations for puts and calls
The higher the stock price, the more a given call option is worth
A call option holder benefits from a rise in the stock price
The greater the price volatility, the more the option is worth
The volatility estimate sigma cannot be directly observed and must be estimated
Volatility plays a major role in determining time value
A company that pays a large dividend will have a smaller option premium than a company with a lower dividend, everything else being equal
Listed options do not adjust for cash dividends and the stock price falls on the ex-dividend date
The higher the risk-free interest rate, the higher the option premium, everything else being equal A higher “discount rate” means that the call premium must rise for the put/call parity equation to hold

5. Assumptions of BSOPM Model
European exercise style: A European option can only be exercised on the expiration date. Thus, American options are more valuable than European options. In reality, few options are exercised early due to time value
Markets are efficient: BSOPM assumes informational efficiency—i.e., People cannot predict the direction of the market or of an individual stock and the Put/call parity implies that you and everyone else will agree on the option premium, regardless of whether you are bullish or bearish
No transaction costs: no commissions and bid-ask spreads (Not true and it causes slightly different actual option prices for different market participants
Interest rates remain constant: There is no real “riskfree” interest rate (often the 30-day T-bill rate is used)
Prices are lognormally distributed: The logarithms of the underlying security prices are normally distributed—this is a reasonable assumption for most assets on which options are available
The stock pays no dividends during the option’s lifesee next slide

6. Relaxing the assumption of no Dividends During the Option’s Life
If you apply the BSOPM to two securities, one with no dividends and the other with a dividend yield, the model will predict the same call premium
Robert Merton developed a simple extension to the BSOPM to account for the payment of dividends

7. Intuition in BSOPM Cash Outflow Cash Inflow
The valuation equation has two parts: One gives a “pseudo-probability” weighted expected stock price (an inflow), the other gives the time-value of money adjusted expected payment at exercise (an outflow) The value of a call option is the difference between the expected benefit from acquiring the stock outright and paying the exercise price on expiration day
Cash Outflow
Cash Inflow

8. Black-Scholes Prices from Historical Data
To calculate the theoretical value of a call option using the BSOPM, we need:
The stock price
The option striking price
The time until expiration
The riskless interest rate
The volatility of the stock
Valuing a Microsoft Call Example
We would like to value a MSFT OCT 70 call in the year Microsoft closed at $70.75 on August 23 (58 days before option expiration). Microsoft pays no dividends.
We need the interest rate and the stock volatility to value the call. Consulting the “Money Rate” section of the Wall Street Journal, we find a T-bill rate with about 58 days to maturity to be 6.10%.
To determine the volatility of returns, we need to take the historical returns and determine their volatility. Assume we find the annual standard deviation of MSFT returns to be

9. Calculating Black-Scholes Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using the BSOPM:

10. Calculating Black-Scholes Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
Using normal probability tables, we find:

11. Calculating Black-Scholes Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
The value of the MSFT OCT 70 call is:

12. Calculating Black-Scholes Prices from Historical Data
Valuing a Microsoft Call Example (cont’d)
The call actually sold for $4.88.
The only thing that could be wrong in our calculation is the volatility estimate. This is because we need the volatility estimate over the option’s life, which we cannot observe.

13. Implied Volatility
Instead of solving for the call premium, assume the market-determined call premium is correct
Then solve for the volatility that makes the equation hold
This value is called the implied volatility

14. Calculating Implied Volatility
Sigma cannot be conveniently isolated in the BSOPM
We must solve for sigma using trial and error
Valuing a Microsoft Call Example
The implied volatility for the MSFT OCT 70 call is 35.75%, which is much lower than the 57% value calculated from the monthly returns over the last two years.

15. An Implied Volatility Heuristic
For an exactly at-the-money call, the correct value of implied volatility is:

16. Historical Versus Implied Volatility
The volatility from a past series of prices is historical volatility
Implied volatility gives an estimate of what the market thinks about likely volatility in the future
Strong and Dickinson (1994) find
Clear evidence of a relation between the standard deviation of returns over the past month and the current level of implied volatility
That the current level of implied volatility contains both an ex post component based on actual past volatility and an ex ante component based on the market’s forecast of future variance

17. Pricing in Volatility Units
You cannot directly compare the dollar cost of two different options because
Options have different degrees of “moneyness”
A more distant expiration means more time value
The levels of the stock prices are different
Volatility smiles are in contradiction to the BSOPM, which assumes constant volatility across all strike prices
When you plot implied volatility against striking prices, the resulting graph often looks like a smile

18. Volatility Smiles (cont’d)

19. Using Black-Scholes to Solve for the Put Premium
Can combine the BSOPM with put/call parity; alternatively,

20. Problems Using the Black-Scholes Model
Does not work well with options that are deep-in-the-money or substantially out-of-the-money
Produces biased values for very low or very high volatility stocks
Increases as the time until expiration increases
May yield unreasonable values when an option has only a few days of life remaining

21. The Greeks
There are several partial derivatives of the BSOPM, each with respect to a different variable:
Delta
Gamma
Theta
Etc.

22. Delta Delta is an important by-product of the Black-Scholes model
There are three common uses of delta
hedge Ratio
Option sensitivity
 likelyhood to become in the money
Delta is the change in option premium expected from a small change in the stock price

23. Measure of Option Sensitivity
For a call option:
For a put option:

24. Measure of Option Sensitivity
Delta indicates the number of shares of stock required to mimic the returns of the option
E.g., a call delta of 0.80 means it will act like 0.80 shares of stock
If the stock price rises by $1.00, the call option will advance by about 80 cents
For a European option, the absolute values of the put and call deltas will sum to one
In the BSOPM, the call delta is exactly equal to N(d1)
Though, adjustments for dividend yield exists
N(d1) x Exp(-DT)
The delta of an at-the-money option declines linearly over time and approaches 0.50 at expiration
The delta of an out-of-the-money option approaches zero as time passes
The delta of an in-the-money option approaches 1.0 as time passes

25. DeltaHedge Ratio Delta is the hedge ratio
Assume a short option position has a delta of If someone owns 100 shares of the stock, writing four calls results in a theoretically perfect hedge

26. DeltaLikelihood of Becoming In-the-Money
Delta is a crude measure of the likelihood that a particular option will be in the money at option expiration
E.g., a delta of 0.45 indicates approximately a 45 percent chance that the stock price will be above the option striking price at expiration

27. Theta
Theta is a measure of the sensitivity of a call option to the time remaining until expiration:
Theta is greater than zero because more time until expiration means more option value
Because time until expiration can only get shorter, option traders usually think of theta as a negative number
The passage of time hurts the option holder
The passage of time benefits the option writer

28. Calculating Theta For calls and puts, theta is: Calculating Theta
The equations determine theta per year. A theta of –5.58, for example, means the option will lose $5.58 in value over the course of a year ($0.02 per day).

29. Gamma
Gamma is the second derivative of the option premium with respect to the stock price
Gamma is the first derivative of delta with respect to the stock price
Gamma is also called curvature
As calls become further in-the-money, they act increasingly like the stock itself. For out-of-the-money options, option prices are much less sensitive to changes in the underlying stock
An option’s delta changes as the stock price changes
Gamma is a measure of how often option portfolios need to be adjusted as stock prices change and time passes. For example, options with gammas near zero have deltas that are not particularly sensitive to changes in the stock price.
Also, for a given striking price and expiration, the call gamma equals the put gamma

30. Calculating Gamma
Calculating Gamma
For calls and puts, gamma is:

31. Sign Relationships Long call + - Long put Short call Short put Delta
Delta
Theta
Gamma
Long call + -
Long put
Short call
Short put
The sign of gamma is always opposite to the sign of theta

32. Vega
Vega is the first partial derivative of the OPM with respect to the volatility of the underlying asset:
All long options have positive vegas. The higher the volatility, the higher the value of the option.
E.g., an option with a vega of 0.30 will gain 0.30% in value for each percentage point increase in the anticipated volatility of the underlying asset.
Vega is also called kappa, omega, tau, zeta, and sigma prime
Two derivatives measure how vega changes:
Vomma measures how sensitive vega is to changes in implied volatility and Vanna measures how sensitive vega is to changes in the price of the underlying asset

33. Calculating Vega
Calculating Vega

34. Rho
Rho is the first partial derivative of the OPM with respect to the riskfree interest rate:
Rho is the least important of the derivatives. Unless an option has an exceptionally long life, changes in interest rates affect the premium only modestly

35. Position Derivatives
The position delta is the sum of the deltas for a particular security
Position gamma
Position theta
Position derivatives change continuously
E.g., a bullish portfolio can suddenly become bearish if stock prices change sufficiently
The need to monitor position derivatives is especially important when many different option positions are in the same portfolio

36. Delta Neutrality
Delta neutrality means the combined deltas of the options involved in a strategy net out to zero
Important to institutional traders who establish large positions using straddles, strangles, and ratio spreads

37. Calculating Delta Hedge Ratios
A Strangle Example
A stock currently trades at $44. The annual volatility of the stock is estimated to be 15%. T-bills yield 6%.
An options trader decides to write six-month strangles using $40 puts and $50 calls. The two options will have different deltas, so the trader will not write an equal number of puts and calls.
How many puts and calls should the trader use?

38. Calculating Delta Hedge Ratios
A Strangle Example (cont’d)
Delta for a call is N(d1):

39. Calculating Delta Hedge Ratios (cont’d)
A Strangle Example (cont’d)
For a put, delta is N(d1) – 1.

40. Calculating Delta Hedge Ratios (cont’d)
A Strangle Example (cont’d)
The ratio of the two deltas is -.11/.19 = This means that delta neutrality is achieved by writing .58 calls for each put.
One approximate delta neutral combination is to write 26 puts and 15 calls.

41. Why Delta Neutrality Matters
Strategies calling for delta neutrality are strategies in which you are neutral about the future prospects for the market
You do not want to have either a bullish or a bearish position
The sophisticated option trader will revise option positions continually if it is necessary to maintain a delta neutral position
A gamma near zero means that the option position is robust to changes in market factors

42. Two Markets: Directional and Speed
Directional market
Speed market
Combining directional and speed markets

43. Directional Market
Whether we are bullish or bearish indicates a directional market
Delta measures exposure in a directional market
Bullish investors want a positive position delta
Bearish speculators want a negative position delta

44. Speed Market
The speed market refers to how quickly we expect the anticipated market move to occur
Not a concern to the stock investor but to the option speculator
In fast markets you want positive gammas
In slow markets you want negative gammas

45. Combining Directional and Speed Markets
Directional Market
Down
Neutral Up
Speed Market
Slow
Write calls
Write straddles
Write puts
Write calls; buy puts
Spreads
Buy calls; write puts
Fast
Buy puts
Buy straddles
Buy calls

46. Introduction to Dynamic Hedging
A position delta will change as
Interest rates change
Stock prices change
Volatility expectations change
Portfolio components change
Portfolios need periodic tune-ups

47. Minimizing the Cost of Data Adjustments
It is common practice to adjust a portfolio’s delta by using both puts and calls to minimize the cash requirements associated with the adjustment

48. Position Risk
Position risk is an important, but often overlooked, aspect of the riskiness of portfolio management with options
Option derivatives are not particularly useful for major movements in the price of the underlying asset
Position Risk Example
Assume an options speculator holds an aggregate portfolio with a position delta of –155. The portfolio is slightly bearish.
Depending on the exact portfolio composition, position risk in this case means that the speculator does not want the market to move drastically in either direction, since delta is only a first derivative.

49. Position Risk (cont’d)
Position Risk Example (cont’d)
Profit
Stock Price

50. Position Risk Example (cont’d)
Because of the negative position delta, the curve moves into profitable territory if the stock price declines. If the stock price declines too far, however, the curve will turn down, indicating that large losses are possible.
On the upside, losses occur if the stock price advances a modest amount, but if it really turns up then the position delta turns positive and profits accrue to the position.