264 CHAPTER 7. THE BLACK-SCHOLES MODEL
(d) Draw the graph of Γ versusSandtfor a European call option
withK= 50,r= 0.10,σ= 0.25,T−t= 0.25.
(e) Comparing the graph of Γ versusS andtwith the graph ofVC
versusSandtin of Solution the Black Scholes Equation, explain
the shape and values of the Γ graph. This only requires an under-
standing of calculus, not financial concepts.- (a) Derive the expression for Θ for a European call option, as given
in the notes.
(b) Draw a graph of Θ versusS for K = 50,r = 0.10,σ = 0.25,
T−t= 0.25.
(c) Draw a graph of Θ versustfor an at-the-money stock, withK=
50,r= 0.10,σ= 0.25,T= 0.25. - (a) Derive the expression forρfor a European call option as given in
this section.
(b) Draw a graph of ρversus S for K = 50, r = 0.10, σ= 0.25,
T−t= 0.25. - (a) Derive the expression for Λ for a European call option as given in
this section.
(b) Draw a graph of Λ versus S forK = 50,r = 0.10, σ = 0.25,
T−t= 0.25.
Outside Readings and Links:
- Stock Option Greeks video on the meaning and interpretation of the
rates of change of stock options with respect to parameters.
7.7 Limitations of the Black-Scholes Model
Rating
Student: contains scenes of mild algebra or calculus that may require guid-
ance.