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FINA 6220Professor Andrew Chen

- The Black-Scholes Option Pricing Model and Option

Greeks - Lecture Note 4

Outline

- The Black-Scholes Analysis
- Assumptions
- Call and put prices
- Using the Models in Practice
- Model Inputs
- Adjusting for Dividends
- Pricing American call options with Dividends
- The Greeks

AC 42

The Black-Scholes Model

- Assumptions
- 1. European options
- 2. Stock is log-normally distributed with mean

µt and standard deviation s(t)½ - 3. No dividends
- 4. Constant risk-free interest rate
- 5. No frictions in the market place
- Deriving the Black-Scholes Model
- Form a risk-less portfolio by long shares and

short one call option

AC 43

The Black-Scholes Model

- Consider the following portfolio Long shares

and short one call option. - V is the current Market value of the portfolio V

S C - Let C represent the amount that the value of the

call option will change if the price of the stock

moves by a tiny amount S - The change in the value of the portfolio, for a

tiny change in the value of the stock will be

VS C - To ensure the portfolio remains a risk-less

hedge, finding the value of sets V0

AC 44

The B-S Call and Put Formulas

- where
- N(x) is the cumulative normal distribution

function, i.e., the probability of observing a

value less than x when drawing randomly from a

standard normal distribution (zero mean and unit

variance).

(4.3)

(4.4)

(4.5)

AC 45

The B-S Call and Put Formulas

AC 46

B-S Call Option Formula

- Comments
- 5 Variables
- Interpretations
- European vs. American calls
- Delta of a call option
- Measures the change in value of he option for a

1 change in the stock price.

(4.6)

AC 47

B-S Put Option Formula

- Combining the B-S call option equation with the

put-call parity relation, leads easily to a

comparable B-S formula for a European put on a

non-dividend-paying stock. - Put Delta

(4.7)

(4.8)

AC 48

B-S Put Option Formula

AC 49

Black-Scholes Model in Practice

- Computational Issues
- Example S46, K45, r5, 30, and T6 months.
- From standard normal distribution tables, we get

N(d1) 0.62835

N(d2) 0.54595

AC 410

Black-Scholes Model in Practice

- Value of Call Option
- C (46)(0.62835) (43.889)(0.54595) 4.94
- Using DerivaGem with 99 time steps to value the

American call option produces an option price of

4.94. - Thus, the values of European and American call

options are the same when the underlying stock

does not pay dividends during the life of the

option.

AC 411

Black-Scholes Model in Practice

- Value of a European Put Option
- The Put price is
- P (43.889)(0.45405) (46)(0.37165) 2.83
- Using DerivaGem with 99 time steps to value the

American put option produces an option price of

2.91. - Thus, the Black-Scholes European put option

pricing model underprices the American put.

AC 412

Estimating Volatility from Past Prices

- Called the HSD (Historical Standard Deviation)
- Step 1 Take the natural logarithms of the prices
- Step 2 Compute the changes in logarithms. There

will be N changes - Step 3 Compute the mean of the changes
- Step 4 Compute the N deviations from the mean.

Square these deviations and sum them up, ie.,

compute Sum. - Step 5 The estimate of the daily variance is
- Annualize the volatility
- Step 6 The volatility is the square root of s2.

AC 413

Estimating Volatility from Past Prices

Closing

Change Squared Date

Price

ln(Price) in ln(P) Deviation 01

Feb 03 115.00 4.745 02 Feb 03 112.39

4.722 -0.023 0.000196 03 Feb

03 111.13 4.711 -0.011 0.000004 04 Feb

03 105.25 4.656 -0.055 0.002116 05 Feb

03 101.25 4.618 -0.038 0.000841 08 Feb

03 101.94 4.624 0.006 0.000225 09 Feb

03 95.94 4.564 -0.060 0.002601 10 Feb

03 98.56 4.591 0.027 0.001296 11 Feb

03 104.88 4.653 0.062 0.005041 12 Feb

03 99.06 4.596 -0.057 0.002304 16 Feb

03 99.06 4.596 0.000 0.000081 17 Feb

03 95.13 4.555 -0.041 0.001024 18 Feb

03 96.19 4.566 0.011 0.000400 19 Feb

03 97.13 4.576 0.010 0.000361 22 Feb

03 102.06 4.626 0.050 0.003481 23 Feb

03 102.94 4.634 0.008 0.000289 24 Feb

03 99.94 4.605 -0.029 0.000400 25 Feb

03 98.50 4.590 -0.015 0.000036 26 Feb

03 97.81 4.583 -0.007 0.000004 Sum -0.162

0.020700 Mean -0.009

AC 414

Estimating Volatility from Past Prices

- Example (continued)
- Daily Variance Sum/(18-1) 0.001218
- Annual Variance Daily Variance 252

0.306936 - Annual Standard Deviation 0.554
- The estimate of volatility for XYZ stock is

0.554.

AC 415

Implied Volatility or Implied Standard Deviation

- where C is the observed price of call option,

c(s) is the model price of the option, and solve

for s. - Why implied volatilities are different
- Non-Simultaneity of prices
- Bid-Ask Prices
- Model Mis-specification
- Using implied volatility in practice
- Volatility smile

AC 416

Adjusting B/S Model for dividends

- To adjust the B/S model for dividends, subtract

the present value of the dividends during the

life of the option from the current stock price. - Example S 46, K 45, r 5, s 30, T 6

months, and two dividends of 0.35 with ex-dates

at the end of the 1st and 4th months. - PV(D) 0.35e-0.05(1/12) 0.35e-0.05(4/12)

0.6928 - S S PV(D) 46 0.6928 45.3072
- Using S for the current stock price, the

DerivaGem software gives C 4.52 - and P 3.10. Notice that without dividends, C

4.94 and P 2.83

AC 417

Black-Scholes Option Values

- Sample Black-Scholes Option Values (S100)

AC 418

Black-Scholes Option Values

- Sample Black-Scholes Option Values (S100)

AC 419

Black-Scholes Option Values

- Sample Black-Scholes Option Values (S100)

AC 420

The Option Greeks

- Delta (d)
- Essential in managing risk
- Hedge portfolio
- Short(long) option is hedged by buying(shorting)

delta shares of underlying stock. - Delta Neutral
- A position that is made riskless (for small price

changes)

AC 421

The Option Greeks (example)

- Suppose we wanted to hedge a long position in the

3 month 105 strike price call option when the

volatility was 0.15 and the interest rate was 5

percent. - The options delta is 0.328, so we could short

0.328 shares of the underlying asset for each

option. - For time spread of 105 strike calls, we could

sell (0.328/0.157) 2.089 1-month for each

3-month call long. - For money spread for 3-month calls, we could sell

(0.328/0.581) 0.565 100-strike call for each

105-strike call long.

AC 422

The Option Greeks (example)

- Example (continued)
- To illustrate the bear-money-spread above,

suppose the stock falls by 1 and we have a delta

neutral position that is long 100 of the

105-strike calls and short 57 of the 100-strike

calls. - The value of the long position should drop by

about - 100 x (-1) x 0.33 -33
- The short position should make a profit of about
- (-57) x (-1) x 0.58 33.06
- So the hedged position should have very low risk.

AC 423

Delta Neutral Strategies

- Value of Options Position or Portfolio
- Delta Neutral Position implies that
- That is,
- Therefore,

V N1C1 N2C2

AC 424

Portfolio Property of Option Deltas

- The delta of a portfolio of options is the sum of

the deltas of the individual option positions. - This allows one to summarize the price

sensitivity of even a very complex portfolio of

options based on a given underlying stock in a

single number, the delta. (Of course, the delta

of a portfolio of options is not limited to be

less than 1.0.) - Example Suppose a portfolio of options on a

given underlying stock has a delta of 5000. A 1

increase (decrease) in the underlying stock will

produce a 5,000 increase (decrease) in the

option position.

AC 425

Lambda ()

- Delta gives the dollar change in the option value

caused by a one dollar change in the price of the

underlying stock. - Lambda of an option is the percentage change in

the option value due to a one percentage point

increase in the underlying asset.

(4.10A)

Lambda (call) C (S/C)

Lambda (put) P (S/P)

(4.10B)

AC 426

Lambda () (S 100)

AC 427

Lambda () (S 100)

AC 428

Option Standard Deviation (s)

- Since s is the standard deviation of the

underlying stock, then the standard deviation

(SD) of a call is equal to the lambda of the call

times the standard deviation of the stock. - The standard deviation of a put is equal to the

absolute value of lambda of the put time the

standard deviation of the stock.

(4.12A)

(4.12B)

AC 429

Option Systematic Risk (beta)

- The systematic risk of a call is equal to the

lambda of the call times the systematic risk of

stock, and the systematic risk of a put is equal

to the lambda of the put times the systematic

risk of the stock.

(4.13A)

(4.13B)

AC 430

Example

- Compute the beta and SD of the 3 month 100 strike

price call option when the stock price volatility

is 30 and the stock beta is 0.95. (See 4-27) - Compute the beta and SD of the 1 month 100 strike

price put option when the stock price volatility

is 30 and the stock beta is 0.95. (See 4-28)

AC 431

Theta (T)

- Theta refers to the rate of time decay for an

option. - Wasting assets
- Theta measures the rate at which the value

decays. - A common way of expressing decay is simply as the

dollar loss in the option value over the next day

if the underlying stock remains at the same

price.

AC 432

Theta (T) (S 100)

AC 433

Theta (T) (S 100)

AC 434

Vega (v)

- Vega measures volatility sensitivity
- Since volatility is such an important determinant

of option value, many options are quite sensitive

to a change in volatility. - Some people adopt a frequently used alternative

term for the dollar change in option value caused

by a one percentage point increase in volatility,

the Greek letter kappa, written . - Example

AC 435

Vega (v) (S 100)

AC 436

Vega (v) (S 100)

AC 437

Rho

- The final parameter in the option formula is the

riskless interest rate. - The change in the option value for a one

percentage point increase in the interest rate is

known as rho. - The time value for a call option comes partly

from the interest that can be earned by investing

the strike price from the present to the

expiration date. - The higher the interest rate, the greater the

calls time value, other things equal hence rho

is positive for a call. - The opposite is true for a put, since the put

holder loses interest while waiting until option

maturity to receive the strike price.

AC 438

Rho (S 100)

AC 439

Rho (S 100)

AC 440

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