# Consider an American call option. Does its value increase or decrease with time to expiration?…

- 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?

Problem 2

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 =

Problem 3

Assume a BSM framework with underlying process S_{t}. Again,

Consider a derivative with payoff function:

?(S_{T}) = (S_{T})^{1/2} – K

If the initial value of the underlying is s_{o}, what is the initial value of the derivative? Show all work.

Problem 4

- 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[d
_{1}], 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?

Problem 5

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, G_{F} = 2, ?_{G} = 5, and G_{G} = -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).

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