58
5 Black-Scholes Differential Equation
possibilities of arbitrage. Trade commissions that supervise and authorize the use of
new financial instruments at exchanges check this to ensure that all trade is fair.
Exercises
1. Derive the Black-Scholes equation for the portfolio ˆ
= −p + S
∂ p
∂ S
.
2. Calculate the fair price of the option shown on the right-most plot in Fig. 5.4.
3. You have the bright idea to issue an option that pays twice the present stock value
S 0 , if the value of the stock S 1 after one year lies in the range K 1 < S 1 < K 2 .
You want to keep K 1 and K 2 general, in order to be able to offer variants of the
option to different customers. (a) Sketch the pay-off function. (b) Determine a
fair price for the options as a function of K 1 and K 2 .
4. Calculate a fair price for an option with the payoff function defined by
max(S
2
/K − K , 0).
5. Use your own words to explain the concept of a Martingale measure.
6. Explain why the valuation of options with the Martingale measure from (5.21)
yields the same price as the calculation using the Black-Scholes equation from
Sect. 5.2.
7. Calculate N
(z) = d N/dz and show that N (z) + N (−z) = 1.
8. Show that the forward contract f , given by (5.23), satisfies (5.6).
References
1. I. Steward, The Mathematical Equation That Caused the Banks to Crash, The Guardian (2012).
Available online at https://www.theguardian.com/science/2012/feb/12/black-scholes-equationcredit-crunch
2. D. Silverman, Solution of the Black Scholes Equation Using the Green’s Function of the Diffusion
Equation, unpublished note (UC Irvine, 1999)
3. P. Wilmott, S. Howison, J. Dewynne, The Mathematics of Financial Derivatives (Cambridge
University Press, Cambridge, 2005)
5 Black-Scholes Differential Equation
possibilities of arbitrage. Trade commissions that supervise and authorize the use of
new financial instruments at exchanges check this to ensure that all trade is fair.
Exercises
1. Derive the Black-Scholes equation for the portfolio ˆ
= −p + S
∂ p
∂ S
.
2. Calculate the fair price of the option shown on the right-most plot in Fig. 5.4.
3. You have the bright idea to issue an option that pays twice the present stock value
S 0 , if the value of the stock S 1 after one year lies in the range K 1 < S 1 < K 2 .
You want to keep K 1 and K 2 general, in order to be able to offer variants of the
option to different customers. (a) Sketch the pay-off function. (b) Determine a
fair price for the options as a function of K 1 and K 2 .
4. Calculate a fair price for an option with the payoff function defined by
max(S
2
/K − K , 0).
5. Use your own words to explain the concept of a Martingale measure.
6. Explain why the valuation of options with the Martingale measure from (5.21)
yields the same price as the calculation using the Black-Scholes equation from
Sect. 5.2.
7. Calculate N
(z) = d N/dz and show that N (z) + N (−z) = 1.
8. Show that the forward contract f , given by (5.23), satisfies (5.6).
References
1. I. Steward, The Mathematical Equation That Caused the Banks to Crash, The Guardian (2012).
Available online at https://www.theguardian.com/science/2012/feb/12/black-scholes-equationcredit-crunch
2. D. Silverman, Solution of the Black Scholes Equation Using the Green’s Function of the Diffusion
Equation, unpublished note (UC Irvine, 1999)
3. P. Wilmott, S. Howison, J. Dewynne, The Mathematics of Financial Derivatives (Cambridge
University Press, Cambridge, 2005)