MATH 425, spring 2015

Course information

• Instructor: Dr G.Berkolaiko, office: Blocker 625c, email: berko AT math.tamu.edu
• No office phone: budget cuts
• Lectures: MWF 9:10-10, BLOC 164
• Office hours: MWF 10:10-11:00 (subject to availability); other times by appointment.
• Course description (first day handout)
• Homework and exam grades on eCampus.
• FAQ

Extras

• Get Matlab (free for Texas A&M Students) or Octave (open-source analogue of Matlab)
• Matlab programs from the book's author.
• Some websites with relevant quotes:
  • SPX quote on Chicago Board Options Exchange (enter SPX and press Submit)
  • SPY quote on Yahoo Finance
  • E-mini S&P 500 Futures on CME
• Other resources:
  • Options glossary.
  • Investopedia, for example Futures Fundamentals.
  • An illustration of CLT: sampling distributions.
• Matlab demonstrations:
  • ch01_add.m PNL diagrams for different options and their combinations, uses the function plotOptionPortfolioPNL.
  • ch04_add.m testing uniform and normal random number generators, CLT.
  • ch05_add.m are close-to-close price movements log-normal?
  • ch07_add.m simulating Brownian motion and more.
  • ch09_add.m hedging an options portfolio using BSM formulae (uses functions BSMcall.m and BSMfutcall.m).
  • ch16_add.m using the tree model to evaluate options
  • ch16_add_hedge.m hedging within the tree model (uses function BSMcall.m from earlier).
  • ch13_add.m experimenting with bisection and Newton-Raphston methods; using Newton's to compute the implied volatility: function BSMiVolCall.m; computing the implied volatility for a range of real-life option prices ch14_add.m.

Homework

Exercise with stars next to them are compulsory for honors students.

1. Submit Wed, Jan 28:
   • (*) Plot prices of SPX call options around the money (20 data points) for 3 different expirations on the same plot.
   • In class we compared PNL diagrams for 3 long calls (95C@5.50, 100C@2.70, 105C@1.15). Draw PNL diagrams for:
     1. short 95C@5.50, 100C@2.70, 105C@1.15
     2. long 95P@1.55, 100P@3.70, 105P@7.10
     3. short 95P@1.55, 100P@3.70, 105P@7.10
     (draw 3 figures with 3 curves on each).
   • From the book: exercises 1.1, 1.3, P1.2 (from pages 7, 8 and 9 correspondingly).
2. Submit Wed, Feb 11:
   • Argue that it is never profitable to exercise an American call early. Therefore, its value is identical to that of a European call.
   • Prove that the call price \(C_t(E,T)\) is a decreasing function of the strike price \(E\).
   • (*) Prove that the call price \(C_t(E,T)\) is a convex function of the strike price \(E\), i.e. $$C(E) \leq \alpha_1 C(E_1) + \alpha_2 C(E_2),$$ where \(E = \alpha_1 E_1 + \alpha_2 E_2\) and \(\alpha_1 + \alpha_2=1\). (Hint: consider first \(\alpha_1 = \frac12\) and price a butterfly).
   • From the book: 2.3, 2.4. 2.5
   • Find out why prices for E-mini S&P 500 Futures on CME do not follow the formula we derived for futures, \[F(t) = S(0) e^{r(T-t)}.\]
   • Find the arithmetical errors in this example of the dichotomy model. Why does the correct answer dealer's profit makes perfect sense (and therefore the erroneous answer makes no sense)?
   • You assumed the dichotomy model, priced and sold a call with strike \(E\), \(S_d < E < S_u\), followed the hedging procedure but the market ended up at \(S_1\), \[ S_d \le S_1 \le S_u. \] Did you lose or make money?
   • (*) Assume the dichotomy model with \(r=0\) and \(S_d < S_0 < S_u\) and plot the call price \[ C = (S_0 - S_d) \frac{S_u - E}{S_u - S_d} \] for a range of strike prices \(E\) that starts below \(S_d\) and ends above \(S_u\). Observe that our call price violates the bound we derived earlier \[ C \geq \max(S_0-E, 0). \] This suggests the above formula is not valid for \(E < S_d\) or \(E > S_u\). Derive the appropriate formulas for the call prices in these ranges of \(E\).
   • Show that \[ f(x,t) = \frac1{\sqrt{2\pi t}} e^{-\frac{x^2}{2t}} \] satisfies the heat equation \(f_t = \frac12 f_{xx}\).
   • From the book: 3.4, 3.6, 3.7 (may use the differentiation trick), 3.8; P3.2
3. Submit Wed, Feb 25:
   1. From the book: 4.1, 4.3, 4.2, 4.4, P4.1 (Hint: in 4.2 you need to derive the denisty of a transformed random variable, see, e.g. Wiki).
   2. Redo the Matlab analysis ch05_add.m we did for chap 5 with prices from the stock (or index) of your choice. Submit the figure; make sure to include in the figure the stock symbol and the dates for which the data was gathered.
   3. From the book: 6.1, 6.2 (calculate (6.5) and (6.6) in the circumstances specified; ignore 3rd and 4th moments), 6.5*, 6.10, and P6.2.
4. Submit Wed, Mar 11:
   1. Calculate \(\mathbb{E} \left[ \left(W_{t+\delta t} - W_t \right)^2\right]\), \(\mathrm{var} \left[\left(W_{t+\delta t} - W_t \right)^2\right]\); 7.2
   2. 8.2, 8.3, 8.8*, 8.10, 8.11 (use the result of 8.10, not 8.7), P8.1
   3. Derive the expressions for \(\rho\) and vega; 10.4, 10.6
   4. Check (by calculating derivatives) that $$ V(t,S) = e^{-r(T-t)} N(d_2) $$ satisfies the Black-Scholes-Merton PDE. Show (without calculating derivatives) that $$ V(t, S) = S N(d_1) $$ also satisfies Black-Scholes-Merton PDE (see Chap. 17 for help).
5. Submit Wed, Apr 8:
   1. Calculate value of a Euro call with a knockout barrier using binomial tree with 3 levels: S0=100, u=1.1, d=0.9, r=0.02, dt=1/12, E=85, barrier=82.
   2. (*) For the American Call computation via the tree model, prove that at every node the immediate excercise profit is less or equal than the computed option value.
6. Submit Fri, Apr 24:
   1. For the American Put with the parameters S0=100, E=120, r=0.05, expiration in 3 months, use the tree model with dt = 1/12, u=1.1, d=0.9. Calculate the option price and deltas. Generate two random paths through the tree and describe hedging procedure and results.
   2. Read Sec 16.2, do exercise 16.6
   3. Read Sec 16.3, do exercise 16.2
   4. 12.2
   5. Show that the change of variables \(\ln F - \tau/2 = z\) converts the equation \[ \frac{\partial Q}{\partial \tau} = \frac12 F^2 \frac{\partial^2 Q}{\partial F^2}\] into the heat equation \[ \frac{\partial H}{\partial \tau} = \frac12 \frac{\partial^2 H}{\partial z^2}. \]
7. Submit Fri, May 1: Matlab mini-project: (10 homework-equivalent points)
   On Aug 7, 2014 the stock AAPL closed at 94.48. At this time, the call with strike 94 and expiration 8/29 was traded at 2.425 (midprice). Assume the rate is 0.0015.
   1. What's the implied volatility? Corresponding delta?
   2. Simulate delta-hedging with daily close prices and above volatility (download daily close prices from Yahoo Finance).
   3. Calculate the volatility realized by these historical prices.
   4. Does hedging work better or worse with realized volatility?

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This file was last modified on Wednesday, 10-Jan-2024 15:58:23 CST.