Black and scholes

Question 1

What is the Black-Scholes option valuation model

Answer 1

An options-pricing formula as a function of the value of the underlying asset and times

Question 2

What is involved with the Black-Scholes model

Answer 2

Requires specification of the 2 parameters
- The risk free rate
- Asset volatility

Question 3

Assumptions of the BS model

Answer 3

-No transaction costs or taxes
-All securities are perfectly divisible
-No riskless arbitrage opportunities
-Security trading is continuous
-The stock price follows a Markov continuous-time stochastic process
-Riskfree rate and volatility are constant
-No dividends on the underlying asset

Question 4

What is a Markov continuous-time stochastic process ?

Answer 4

where only the present value of a variable is relevant for predicting the future. The past history of the variable and the way that the present has emerged from the past are irrelevant

Question 4

Notations of the Black-Scholes model

Answer 4

C0 = Call option value
P0 = Put option value
S0 = Current stock price
X = Exercise price
r = annualized risk-free interest rate
T = Time to expiration of option in years
Sigma = the annualised standard deviation of returns on the underlying asset, the “volatility” measure

Question 4

The N(x) function in BS

Answer 4

Probability that a normally distributed variable with mean of zero and a standard deviation of 1 is less than x

Question 5

Black-Scholes Formula Call Price

Answer 5

Question 6

Probability of BS call being exercised

Answer 6

-The N(d) can be (loosely) viewed as risk-adjusted probabilities that the call option will expire in the money
-Ln(S0/X) is approximately the percentage amount by which the option is currently in or out of the money
-For instance, if S0 = 160 and X = 100 the option is 60% in the money and ln(160/100) = 0.47
-If S0 = 90 and X = 100 the option is 10% out of the money and ln(90/100) = - 0.105

Question 7

In Black-Scholes, what would happen if Nd1 = 1 and Nd2 = 1 ??

Answer 7

C0 = S0×N(d1) – Xe^-rT×N(d2) = S0 – Xe-rT
- The option would be approx equal to its intrinsic value

Question 7

Black-Scholes Put price Formula

Answer 7

-You first have to rearrange Put-Call Parity
-Then sub into Black-Scholes formula for a call

Question 7

So what is the probability of BS call being Exercised ??

Answer 7

IF the N(d) terms are close to 1.00 there is a very high probability that the option will be exercised

Question 7

The Framework of the BS model with dividends

Answer 7

Stock price is the sum of 2 components
- A riskless component that corresponds to
the known dividend during the life of the
option
- Risky component

Question 7

When is the BS model with dividends correct ??

Answer 7

the BS formula is correct if S0 is equal to the risky component of the stock price and σ is the volatility of the process followed by the risky component

Question 7

Answer this Black-Scholes Example

Answer 7

Question 8

Answer to the Black-Scholes example

Answer 8

Question 8

Properties of Black-Scholes formula

Answer 8

-As S0 becomes very large C0 tends to S0 – Xe-rT and P0 tends to zero
-As S0 becomes very small C0 tends to zero and P0 tends to Xe-rT – S0
-As the volatility, σ, approaches zero, the call price is always max(S0 – Xe-rT, 0):
If S0>Xe-rT,d1 and d2 tend to +∞, so that
N(d1) and N(d2) tend to 1 and C0 = S0 – Xe-
rT
If S0<Xe-rT,d1 and d2 tend to -∞, so that
N(d1) and N(d2) tend to zero and C0 = 0
-Similarly, the put price is always max(Xe-rT – S0 , 0) as σ tends to zero

Question 9

What happens to the Black-Scholes model when dividends are taken into account

Answer 9

-Only dividends with ex-dividend dates should be included
-The “dividend” should be the expected reduction in the stock price
-We assume the amount and timing of the dividends during the life of an option with certainty
-For short-life options, it is usually possible to estimate dividends during the life of the option with reasonable accuracy
-For options lasting several years, there is likely to be uncertainty about dividend growth rates making option pricing more difficult

Question 10

What is the Riskless component in the price of a stock ??

Answer 10

The riskless component, at any time, is the present value of all the dividends during the life of the option discounted from the ex-dividend dates to the present at the risk-free rate.

Question 11

What can the BS formula be used for operationally?

Answer 11

this means that the BS formula can be used provided that the stock is reduced by the present value of all dividends during the life of the option, the discounting being done from the ex-dividend dates at the risk-free date.

Question 12

Nature of Volatility in BS

Answer 12

-Volatility is much greater when the market is open than when it is closed
-For this reason time is usually measured in “trading days” not calendar days when volatility parameters are being estimated and used
-It is assumed that there are 252 trading days in one year for most assets

Question 13

Trading days example

Answer 13

-Suppose it is April 1 and an option lasts to April 30 so that the number of days remaining is 30 calendar days or 22 trading days
-The time to maturity would be assumed to be 22/252 = 0.0873 years

Question 14

What is the implied volatility of an option with BS

Answer 14

-The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price (quoted price)
-If the actual stock standard deviation exceeds the implied volatility, the option is considered a good buy
-A higher implied volatility than the actual volatility indicates that the fair price of the option is lower than the observed one

Question 15

Investor fear gauge

Answer 15

-During periods of turmoil, implied volatility -can spike quickly
-Due to its correlation with financial crisis, implied volatility is often called “investor fear gauge”
-Whereas historical volatility is backward looking, implied volatility is forward looking.
-Observers use implied volatility to infer market assessments of the possibility of turmoil in the near future
-While the B/S formula is derived assuming that stock volatility is constant, the time series of implied volatility derived from the model is far from constant!

Question 16

B/S Dividends and volatility

Answer 16

-Important that dividends are included in the model
-Volatility has a significant impact on option values, and it is more difficult to be estimated than dividends. If we know the value of all the other variables, we can determine the “implied volatility

Question 17

What are some other reasons the B/S model deviates from market price ??

Answer 17

-Biases in the Black-Scholes model
Market prices may be incorrect. E.g. market is -inefficient or prices are temporarily out of equilibrium

Question 18

When is there an arbitrage opportunity in the B/S model

Answer 18

When the B/S gives the correct value but the market makes a pricing error

Question 19

What would your arbitrage strategy be if the put was mis priced on the market

Answer 19

Buy the undervalued asset (Put) and sell the equivalent asset

Question 20

What is the equivalent asset ??

Answer 20

A portfolio comprised of:
C = long call
-S = short stock
+PV(D) = lend PV of dividend
+Xe-rT = lend PV of exercise price

Question 21

B/S Arbitrage, options payoffs at expiration

Answer 21

If X > ST
Put is exercised to receive = X - ST
Call is not exercised

If  ST > X Put is not exercised Call is exercised resulting in a liability =  (ST - X) = X-ST

Therefore, in either case payoff will be    X-ST

Question 22

Overall Position at expiration

Answer 22

Share Value + ST
FV of dividend received + FV(D)
Payoff from Options + X - ST
Repay Loans -FV(D) - X
________________
Net position 0