Financial Mathematics Assignment three solution
- ([2;p55]) Draw the expiry payoff diagrams for each of the following portfolios:
(a) Short one share, long two calls with exercise price E (this combination is called a
straddle);
(b) Long one call and one put, both with exercise price E (this is also a straddle: why?);
(c) Long one call and two puts, all with exercise price E (a strip);
(d) Long one put and two calls, all with exercise price E (a strap);
(e) Long one call with exercise price E1 and one put with exercise E2. Compare the
three cases E1 > E2 (known as a strangle), E1 = E2, E1 < E2.
(f) As (e) but also short one call and one put with exercise price E (when E1 < E < E2,
this is called a butterfly spread). - Derive the price formula of an European put based on the Black-Scholes model.
- Show that the payoff function of a portfolio c−p is S −E. From this and the Black-Scholes
formula, show the formula of the put-call parity. - 4. ([6;p56]) Find the most general solution of the Black-Scholes equation that has the special
form
(a) V = V (S)
(b) V = A(t)B(S) - 4. ([6;p56]) Find the most general solution of the Black-Scholes equation that has the special
form
(a) V = V (S)
(b) V = A(t)B(S) - 4. ([6;p56]) Find the most general solution of the Black-Scholes equation that has the special
form
(a) V = V (S)
(b) V = A(t)B(S) - What is the put-call parity relation for options on an asset that pays a constant
continuous dividend yield? - Derive the put-call parity result for the forward/futures price in the form
C − P = (F − E)e
−r(T −t)
What is the corresponding version when the asset pays a constant continuous dividend
yield? - What is the forward price for an asset that pays a single dividend dyS(td) at
time td?