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

Problem 1

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Consider an American call option. Does its value increase or decrease with time to expiration?…
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1. Consider an American call option. Does its value increase or decrease with time to expiration? Why?
2. 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.

1. G =
2. G =

Problem 3
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.
Problem 4

1. According to BSM partial differential equation (PDE), what is the relationship between theta, delta, and gamma of a portfolio ?? (Give the formula)
2. According to the BSM PDE, what is theta for a portfolio that is both delta-neutral and gamma-neutral? (Give the formula)
3. Specify the portfolio ? (that is, give its positions) that is used to derive the BSM PDE, for a general derivative with value f.
4. 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?
5. 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, GF = 2, ?G = 5, and GG = -2.

1. 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.
2. 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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