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Problem 1 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: dS= µSdt+sSdz Use It HYPERLINK "http://en.wikipedia.org/wiki/Kiyoshi_It%C5%8D" \o "Kiyoshi Ito" o’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 = S1/2 G = Ser(T-t) Problem 3 Assume a BSM framework with underlying process St. Again, dS= µSdt+sSdz 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 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 N'[d1]s0 * sT. 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, GF = 2, ?G = 5, and GG = -2. Determine the appropriate hedge using the two derivatives (and not the...

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Robert · Accepted Answer

Problem 1 
a) Consider an American call option. Does its value increase or decrease with time to expiration? 
Why? 
The various factors on which the price of stock options depends are the current stock price, the 
strike price, the time to expiration, the volatility of the stock price, the risk free interest rate and 
the dividends expected during life of an option. Greater the stock price greater will be the value 
of the call option and vice versa for put option. Greater will be the strike price, greater will be 
the value of the put option and vice versa for the put option. The time to expiration has positive 
effects on both call and put i.e. greater the value of it, larger will be the value of both call and 
put option. Both put and call American options become more valuable ( or at least do not 
decrease in value)as the time to expiration increases 
b) 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? 
As per BSM, the theta is usually negative because as time passes and when everything else will 
remain the same, the value of option decreases which in turn decreases the value of theta. In 
case of at the money option it is negative. It is very large and decreases as the maturity comes 
nearer because of same reason as explained above. 
Problem 2 
Suppose S follows geometric Brownian motion – with dynamics:
Use Itō’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. 
a) G =      
Using first derivative we will get,

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