Chapter 15: The Black Scholes option pricing formula

Question 1

State the assumptions underlying the Black Scholes option pricing pricing formula

Answer 1

1. The price of the underlying share follows a geometric brownian motion
2. There are no risk-free arbitage opportunities
3. The risk free rate of interest is constant, the same for all maturities and the same for borrowing and lending - can be relaxed
4. Unlimites short-selling is allowed
5. There are no taxes or transaction costs
6. The underlying asset can be traded continously and in infinitesimally small numbers of units

Question 2

What do the underlying assumptions of the Black Scholes option pricing formula infer

Answer 2

That the market is complete - all derivatives have payoffs which can be replicated

Question 3

How realistic is the following assumption underlying the Black Scholes model
1. The price of the underlying share follows a geometric brownian motion
2. There are no risk-free arbitage opportunities
3. The risk free rate of interest is constant, the same for all maturities and the same for borrowing and lending
4. Unlimites short-selling is allowed
5. There are no taxes or transaction costs
6. The underlying asset can be traded continously and in infinitesimally small numbers of units

Answer 3

Constant volatility
* Model assumes constant volatility, which may not be constant all the time - depends on when the estimate is taken and its frequency.

Constant drift
* Long-term drift may not be constant over time - impacted by interest rates.

Independent increments
* Evidence of mean reverting behaviour in markets, which is inconsitent with the assumption of independent increments
* Evidence of short-term momentum in real markets, which is inconsistent with the indepenent increments assumption

Normal distribution
* The distribution of log(St/Su) has a taller peak than that implied by the normal ditribution, this is because there are more days of little to no movement in the shareprice
* The distribution of log(St/Su) has fatter tails than that implied by the normal ditribution, this is because there are more extreme movements in the share price.
* The sample prices of securities are no continous and appear to jump occassionally

Question 4

How realistic is the following assumption underlying the Black Scholes model
1. The price of the underlying share follows a geometric brownian motion
2. There are no risk-free arbitage opportunities
3. The risk free rate of interest is constant, the same for all maturities and the same for borrowing and lending
4. Unlimites short-selling is allowed
5. There are no taxes or transaction costs
6. The underlying asset can be traded continously and in infinitesimally small numbers of units

Answer 4

A consequence of jumps in the share price is that it is not possible to rebalance the replicating portfolio at each moment as to eliminate movements in the value of the portfolio. So the portfolio is not entirely risk free

Hedging strategies can be constructed to reduce the level of risk.

Question 5

How realistic is the following assumption underlying the Black Scholes model
1. The price of the underlying share follows a geometric brownian motion
2. There are no risk-free arbitage opportunities
3. The risk free rate of interest is constant, the same for all maturities and the same for borrowing and lending
4. Unlimites short-selling is allowed
5. There are no taxes or transaction costs
6. The underlying asset can be traded continously and in infinitesimally small numbers of units

Answer 5

The risk free rate does vary and in an unpredictable way. And different rates can apply for lending and borrowing

Question 6

How realistic is the following assumption underlying the Black Scholes model
1. The price of the underlying share follows a geometric brownian motion
2. There are no risk-free arbitage opportunities
3. The risk free rate of interest is constant, the same for all maturities and the same for borrowing and lending
4. Unlimites short-selling is allowed
5. There are no taxes or transaction costs
6. The underlying asset can be traded continously and in infinitesimally small numbers of units

Answer 6

Unlimites short selling may not be allowed, except in penal interest rates.
Can hold mixtures of derivatives which decrease the need for short-selling

Question 7

How realistic is the following assumption underlying the Black Scholes model
1. The price of the underlying share follows a geometric brownian motion
2. There are no risk-free arbitage opportunities
3. The risk free rate of interest is constant, the same for all maturities and the same for borrowing and lending
4. Unlimites short-selling is allowed
5. There are no taxes or transaction costs
6. The underlying asset can be traded continously and in infinitesimally small numbers of units

Answer 7

• Shares deallings attract transaction costs. There is also income and capital gains tax

Question 8

How realistic is the following assumption underlying the Black Scholes model
1. The price of the underlying share follows a geometric brownian motion
2. There are no risk-free arbitage opportunities
3. The risk free rate of interest is constant, the same for all maturities and the same for borrowing and lending
4. Unlimites short-selling is allowed
5. There are no taxes or transaction costs
6. The underlying asset can be traded continously and in infinitesimally small numbers of units

Answer 8

Shares can normally be dealt with in integer amounts of 1 unit and not cotinously