Math 459/559 Mathematical Finance 2 Spring 2010: B. Hassard (replacing J. Reineck)

Course Materials:

syllabus/course page (J. Reineck)

Homework Assignments

CBOT Corn Futures and Options Market CBOT Soybean Futures and Options Market

Nonlinear risk worksheet Hedging to minimize R(Cmax(h),h)-R(Cmin(h),h), R(C,h) = revenue from corn + revenue from futures

Simulation solutions of a Stochastic Differential Equation giving rise to a Wiener process (parameters mu, sigma)

Simulation solutions of a Stochastic Differential Equation giving rise to a lognormal random walk (parameters mu, sigma)

Procter and Gamble call spread investment example

Procter and Gamble option prices Nov 20 2009

Computations for Luenberger exercise 10.3 Maple worksheet
PDF version of the above (view without needing Maple)

Explanation why hedge in 20100304 midterm question 5 was not necessarily a minimum variance hedge

Implied volatility computation:

• Google "yahoo finance PG" to find the current P&G stock price
• click on "Options" to find current call options prices
• choose an expiration a month or two in the future
• choose a strike price that has some nonzero volume of trading, write down the call option price
• click on "Key Statistics", find the dividend rate
• Google "US Treasury yield curve rates" and take the 3mo rate as the "risk-free"
• plug all of these into the following worksheet, and use "Tools, Goal Seek" (Excel 2007: Data, Data Tools, What if Analysis, Goal Seek) to compute the standard deviation sigma that makes the Black-Scholes Call price formula give the actual call option price.

https://www.worldscibooks.com/etextbook/p556/p556_chap04.pdf
Pages 111-115 give derivation of Black-Scholes PDE starting with processes for the asset S and for the the derivative f (result of Ito's lemma) as in Luenberger section 13.1 p. 354. The derivation uses the original delta-hedge portfolio argument rather than a replicating portfolio argument, and is extended to allow an asset S which pays a continuously compounded dividend yield.
See class notes 2010-03-18 for the (Euro) Call solution of the Black-Scholes PDE extended to account for continuously compounded dividend payments. This is the solution is encoded in the Black-Scholes_call_option_implied_volatility_goal_seeker worksheet above.
HW Higham p. 138 P14.2, but choose an NYSE traded stock whose symbol starts with the same letter as your name (first or last), and use the worksheet above to determine implied volatility

Lemma relating density function g(y) for Y = ln(X/X_0) to density function f(x) for X.
Application giving density function for S(T) in terms of density function for ln(S(T)/S(t))

Monte Carlo Call Option Pricing
S = 20, K=18, r = 0.05, T = 1
M = 25 price computations using averages of N=1000 (red) N=10000 (blue) N=100000 (black) simulations
Monte Carlo Call Option Pricing Maple worksheet
PDF version of the above (view without needing Maple)

S = 20, K=18, r = 0.05, sigma=0.25, T = 1
Price computation using average of N=10000 simulations,
with mu = 0.04, mu = r = 0.05, mu = 0.06 illustrating that
obtain Black-Scholes value when mu = r, not if mu < r or mu > r
Monte Carlo Call Option Pricing Excel spreadsheet mu=r
PDF version of the above (view without needing Excel)

Online option calculators

Try
S = 20, K = 18, dividend = 0, r = 0.05, sigma=0.25, time T = 1 yr
(same values as in MCC.pdf, MC_mu.pdf) above, and compute Euro and Amer Call and Put Options.
For comparison, the Black-Sholes Euro Call value is $3.628 (see MCC.pdf).

Smirnov's Option Calculator (binary tree for all, also B-S for Euro)
Chicago Board Options Exchange Online Calculator (methods unknown)
Dr. Robert Lums's Option Calculator (methods unknown)

Solutions to Higham 9.3, 9.5, 12.2, 12.3 2010-04-20--14.17.36.pdf

Course web page: http://www.math.buffalo.edu/~hassard/459-559/