Derivatives Final UPDATED Exam
Questions and CORRECT Answers
The delta of a call option of 0.7. What does this mean? - CORRECT ANSWER - A delta
of 0.7 means that when the price of the stock increases by a small amount, the price of the option
increases by 70% of this amount. Similarly, when the price of the stock decreases by a small
amount, the price of the option decreases by 70% of this amount.
The theta of a call option -0.1. What does this mean? - CORRECT ANSWER - A theta of -
0.1 means that if Delta(t) units of time pass with no change in either the stock price or its
volatility, the value of the option declines by 0.1Delta(t). A trader who feels that neither the
stock price nor its implied volatility will change should write an option with as high a negative
theta as possible. Relatively short-life-at-the-money options have the most negative thetas.
"The procedure for creating an option position synthetically is the reverse of the procedure for
hedging the option position". Explain this statement. - CORRECT ANSWER - To hedge
an option position, it is necessary to create the opposite option position synthetically. For
example, to hedge a long position in a put, it is necessary to create a short position in a put
synthetically. It follows that the procedure for creating an option position synthetically is the
reverse of the procedure for hedging the option position.
What is the difference between the volatility smiles that are typically observed for equities and
currencies? - CORRECT ANSWER - The volatility smile for equities is mostly downward
sloping. The volatility smile for currencies is a symmetrical smile.
What volatility smile is likely to be caused by jumps in the underlying asset price? Is the pattern
likely to be more pronounced for a 2-year option than for a 3-month option? - CORRECT
ANSWER - Jumps tend to make both tails of the stock price distribution heavier than
those of the lognormal distribution. This creates a volatility smile similar to that in Figure 20.1.
The volatility smile is likely to be more pronounced for the three-month option
A stock price is currently $20. Tomorrow, news is expected to be announced that will either
increase the price by $5 or decrease the price by $5. What are the problems in using Black-
Scholes-Merton to value 1-month options on the stock? - CORRECT ANSWER - The
probability distribution of the stock price in one month is not lognormal. Possibly, it consists of
, two lognormal distributions superimposed upon each other and is bimodal. Black-Scholes is
clearly inappropriate, because it assumes that the stock price at any future time is lognormal.
What volatility smile is likely to be observed for 6-month options when the volatility is uncertain
and positively correlated with the stock price? - CORRECT ANSWER - When the asset
price is positively correlated with volatility, the volatility tends to increase as the asset price
increases, producing less heavy left tails and heavier right tails. Implied volatility then increases
with the strike price.
A European call option on a certain stock has a strike price of $30, a time to maturity of 1 year,
and an implied volatility of 30%. A European put option on the same stock has a strike price of
$30, a time to maturity of 1 year, and an implied volatility of 33%. What is the arbitrage
opportunity open to a trade? Does the arbitrage work only when the lognormal assumption
underlying Black-Scholes-Merton holds? Explain carefully the reasons for your answer -
CORRECT ANSWER - Put-call parity implies tha European put and call options have the
same implied volatility. If a call option has an implied volatility of 30% and a put option has an
implied volatility of 33%, the call is priced too low relative to the put. The correct trading
strategy is to buy the call, sell the put and short the stock. This does not depend on the lognormal
assumption underlying Black-Scholes-Merton. Put-call parity is true for any set of assumptions.
A company uses an EWMA model for forecasting volatility. It decides to change the parameter
Lambda from 0.95 to 0.85. Explain the likely impact on the forecasts. - CORRECT
ANSWER - Reducing Lamba from 0.95 to 0.85 means that more weight is put on recent
observations of u^2, and less weight is given to older observations. Volatilities calculated with
lamba = 0.85 will react more quickly to new information and will "bounce around" much more
than volatilites calculated with lamba = 0.95.
Explain the difference between a regular credit default swap and a binary credit default swap -
CORRECT ANSWER - Both provide insurance against a particular company defaulting
during a period of time. In a credit default swap, the payoff is the notional principal amount
multiplied by one minus the recovery rate. In a binary swap, the payoff is the notional principal
Explain the difference between a cash CDO and a synthetic CDO - CORRECT
ANSWER - A cash CDO is created by buying bonds and tranching out the risks. A
synthetic CDO is created from a portfolio of short CDSs (i.e., CDS that are selling protection)
Questions and CORRECT Answers
The delta of a call option of 0.7. What does this mean? - CORRECT ANSWER - A delta
of 0.7 means that when the price of the stock increases by a small amount, the price of the option
increases by 70% of this amount. Similarly, when the price of the stock decreases by a small
amount, the price of the option decreases by 70% of this amount.
The theta of a call option -0.1. What does this mean? - CORRECT ANSWER - A theta of -
0.1 means that if Delta(t) units of time pass with no change in either the stock price or its
volatility, the value of the option declines by 0.1Delta(t). A trader who feels that neither the
stock price nor its implied volatility will change should write an option with as high a negative
theta as possible. Relatively short-life-at-the-money options have the most negative thetas.
"The procedure for creating an option position synthetically is the reverse of the procedure for
hedging the option position". Explain this statement. - CORRECT ANSWER - To hedge
an option position, it is necessary to create the opposite option position synthetically. For
example, to hedge a long position in a put, it is necessary to create a short position in a put
synthetically. It follows that the procedure for creating an option position synthetically is the
reverse of the procedure for hedging the option position.
What is the difference between the volatility smiles that are typically observed for equities and
currencies? - CORRECT ANSWER - The volatility smile for equities is mostly downward
sloping. The volatility smile for currencies is a symmetrical smile.
What volatility smile is likely to be caused by jumps in the underlying asset price? Is the pattern
likely to be more pronounced for a 2-year option than for a 3-month option? - CORRECT
ANSWER - Jumps tend to make both tails of the stock price distribution heavier than
those of the lognormal distribution. This creates a volatility smile similar to that in Figure 20.1.
The volatility smile is likely to be more pronounced for the three-month option
A stock price is currently $20. Tomorrow, news is expected to be announced that will either
increase the price by $5 or decrease the price by $5. What are the problems in using Black-
Scholes-Merton to value 1-month options on the stock? - CORRECT ANSWER - The
probability distribution of the stock price in one month is not lognormal. Possibly, it consists of
, two lognormal distributions superimposed upon each other and is bimodal. Black-Scholes is
clearly inappropriate, because it assumes that the stock price at any future time is lognormal.
What volatility smile is likely to be observed for 6-month options when the volatility is uncertain
and positively correlated with the stock price? - CORRECT ANSWER - When the asset
price is positively correlated with volatility, the volatility tends to increase as the asset price
increases, producing less heavy left tails and heavier right tails. Implied volatility then increases
with the strike price.
A European call option on a certain stock has a strike price of $30, a time to maturity of 1 year,
and an implied volatility of 30%. A European put option on the same stock has a strike price of
$30, a time to maturity of 1 year, and an implied volatility of 33%. What is the arbitrage
opportunity open to a trade? Does the arbitrage work only when the lognormal assumption
underlying Black-Scholes-Merton holds? Explain carefully the reasons for your answer -
CORRECT ANSWER - Put-call parity implies tha European put and call options have the
same implied volatility. If a call option has an implied volatility of 30% and a put option has an
implied volatility of 33%, the call is priced too low relative to the put. The correct trading
strategy is to buy the call, sell the put and short the stock. This does not depend on the lognormal
assumption underlying Black-Scholes-Merton. Put-call parity is true for any set of assumptions.
A company uses an EWMA model for forecasting volatility. It decides to change the parameter
Lambda from 0.95 to 0.85. Explain the likely impact on the forecasts. - CORRECT
ANSWER - Reducing Lamba from 0.95 to 0.85 means that more weight is put on recent
observations of u^2, and less weight is given to older observations. Volatilities calculated with
lamba = 0.85 will react more quickly to new information and will "bounce around" much more
than volatilites calculated with lamba = 0.95.
Explain the difference between a regular credit default swap and a binary credit default swap -
CORRECT ANSWER - Both provide insurance against a particular company defaulting
during a period of time. In a credit default swap, the payoff is the notional principal amount
multiplied by one minus the recovery rate. In a binary swap, the payoff is the notional principal
Explain the difference between a cash CDO and a synthetic CDO - CORRECT
ANSWER - A cash CDO is created by buying bonds and tranching out the risks. A
synthetic CDO is created from a portfolio of short CDSs (i.e., CDS that are selling protection)
