1. Consider a binomial model. The current value of a stock is $70 and the strike price for a put option is $67. The *up* factor is equal to 1.10 and the *down* factor is equal to 0.80. The risk-free rate is 5%. What is the value of the put option?

2. Use the Black-Scholes formula to find the value of a call option on the following stock: *T* = 6 months; *σ* = 50%; *X* = 50; *S*_{0} = 50; *r*_{f }= 3%.

3. A call option with *X* = 50 on a stock currently priced at 55 is selling for 10. Using a volatility estimate of 30%, you find that *N(d*_{1}*)* = 0.6 and *N(d*_{2}*)* = 0.5. The risk-free rate is 0%. Is the implied volatility more or less than 30%? Explain.

4. A delta-neutral portfolio is hedged against: