- Consider an American call option. Does its value increase or decrease with time to expiration? Why?
- Consider a European call option on a stock that does not pay dividends. Assume a Black-Schole-Merton (BSM) framework. Is theta for an at-the-money call positive or negative? Why?
Suppose S follows geometric Brownian motion – with dynamics:
Use Ito’s lemma to determine the dynamics of the processes G below. In each case, express the coefficients of dt and dz in terms of G rather than S.
- G =
- G =
Assume a BSM framework with underlying process St. Again,
Consider a derivative with payoff function:
?(ST) = (ST)1/2 – K
If the initial value of the underlying is so, what is the initial value of the derivative? Show all work.
- According to BSM partial differential equation (PDE), what is the relationship between theta, delta, and gamma of a portfolio ?? (Give the formula)
- According to the BSM PDE, what is theta for a portfolio that is both delta-neutral and gamma-neutral? (Give the formula)
- Specify the portfolio ? (that is, give its positions) that is used to derive the BSM PDE, for a general derivative with value f.
- Consider a European call option on a stock that does not pay dividends. Recall that the delta of such a call equals N[d1], and gamma equals . For the portfolio ? in part (c) above, but with f = c for the call option, what is the formula for gamma?
- For delta hedging, what are the risks in the situation where the gamma of a portfolio is highly negative and the delta is zero?
Suppose that you have a portfolio ? with ?? = 2 and G? = 3. You want to make this portfolio both delta-neutral and gamma-neutral. There are two derivatives in the marketplace, F and G, with ?F = -1, GF = 2, ?G = 5, and GG = -2.
- Determine the appropriate hedge using the two derivatives (and not the underlying). Indicate the amount of each derivative, and specify whether the positions are long or short.
- Now instead, consider taking a position in the underlying. Determine an appropriate hedge that uses the underlying along with only one derivative (you choose).