Exam 14: The Black-Scholes Model

The current price of a stock is $100. What is the Black-Scholes model price of a six-month put option at strike $98, given an interest rate of 2% and a dividend rate of 1%? The volatility is 45%.

A stock is currently trading at $S _ { 0 } = 21.30$ . It is not expected to pay dividends over the next year. You price a one-month put option on the stock with a strike of $K = 22.50$ using the Black-Scholes model and find the following numbers: =-0.666 =-0.738 N =0.253 N =0.230 Given this information, the delta of the put is

The current price of a stock is $100. What is the Black-Scholes model price of a six-month call option at strike $101, given an interest rate of 2% and a dividend rate of 1%? The volatility is 25%.

The implied volatility of an option

Consider a call option on a stock that pays dividends at the rate $q > 0$ . Which of the following statements is most valid for the Black-Scholes model?

Let $E ^ { * } ( . )$ denote risk-neutral expectations in the Black-Scholes setting. Then, the Black-Scholes formula may calculated by taking the following expectation:

The dollar-euro exchange rate is $1.30/€. The dollar interest rate is 2% and the euro interest rate is 3%. What is the price of a six-month call option to buy euros at a strike price of $1.25/€? The volatility of the $/€ exchange rate is 40%.

A stock is currently trading at $S _ { 0 } = 26.15$ . It is not expected to pay dividends over the next year. You price a one-month put option on the stock with a strike of $K = 25$ using the Black-Scholes model and find the following numbers: =0.717 =0.645 N =0.763 N =0.740 Given this information, the delta of the put is

The current price of a stock is $100. Consider the Black-Scholes model price of a six-month call option at strike $101, given an interest rate of 2% and a dividend rate of 1%? The volatility is 25%. What is the real-world (physical) probability of the option ending up in the money if the growth rate of the stock is expected to be 5% per year?

A stock is currently trading at $S _ { 0 } = 21.30$ . It is not expected to pay dividends over the next year. You price a one-month put option on the stock with a strike of $K = 22.50$ using the Black-Scholes model and find the following numbers: =-0.666 =-0.738 N =0.253 N =0.230 Given this information, the probability of the put finishing out-of-the-money is

A put option can be replicated by holding a position in stock and bonds, i.e., $P = B + \Delta S$ where $\Delta$ is the delta of the put option. Comparing the replication formula to the Black-Scholes formula, and assuming no dividends, what can you say about the delta of the option?

A volatility swap is an option on the realized standard deviation of a stock's return over a defined period of time. A volatility swap may be replicated using

The Black-Scholes model differs from the binomial in that

If the Black-Scholes call delta (assume a non-dividend-paying stock) is equal to 1/2, then which of the following statements is most valid?

Which of the following quantities associated with equity option pricing is model dependent?

The S&P 500 index is trading at a level of 1200. The rate of interest is 2%. The average rate of dividends for stocks in the index is 3%. Index volatility is 20%. What is the Black-Scholes price of a one-year at-the-money put option on the index?

Which of the following is not an assumption underlying the Black-Scholes model?

The implied volatility skew observed in stock indices cannot be attributed to which of the following reasons?

The current price of a stock is $100. Consider the Black-Scholes model price of a six-month call option at strike $101, given an interest rate of 2% and a dividend rate of 1%? The volatility is 25%. What is the risk-neutral probability of the option ending up in the money?

Most major stock indices, like the S&P 500, exhibit an implied volatility skew. This means if we consider three put options, $P \left( K _ { 1 } \right) , P \left( K _ { 2 } \right) , P \left( K _ { 3 } \right)$ at strikes $K _ { 1 } < K _ { 2 } < K _ { 3 } \leq S$ (where $S$ is the current index level) and the options have implied volatilities $\sigma _ { 1 } , \sigma _ { 2 } , \sigma _ { 3 }$ , respectively, then the most likely pattern is