MATH 425, spring 2016

Course information

Extras

Homework

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

  1. Submit Thu, Jan 28: Updated for 2016
    1. (*) Plot prices of SPX call options around the money (20 data points) for 3 different expirations on the same plot.
    2. 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@1.05, 100P@2.50, 105P@5.50
      3. short 95P@1.05, 100P@2.50, 105P@5.50
      (draw 3 figures with 3 curves on each).
    3. Consider two put options with the same expiration but different strikes, \(E_1\) and \(E_2\) (with \(E_1 < E_2\)). Formulate the conditions on their prices \(P_1\) and \(P_2\) so that the PNL graphs of the two puts (both long) intersect. Justify your answer by drawing all possible types of the graphs' arrangement (hint: there are three cases).
    4. From the book: exercises 1.1, 1.3, P1.2 (from pages 7, 8 and 9 correspondingly).
    Solutions: HW1_sol.pdf and Matlab file HW1.m (uses the function plotOptionPortfolioPNL).
  2. Submit Thur, Feb 11:
    1. Argue that it is never profitable to exercise an American call early (assuming the stock pays no dividends). Therefore, its value is identical to that of a European call.
    2. Prove that the call price \(C_t(E,T)\) satisfies $$ -1 \leq \frac{C(E_2)-C(E_1)}{E_2-E_1} \leq 0. \tag{1}$$ (You may assume that \(r=0\).) Prove that, if \(C\) is differentiable with respect to \(E\), then this is equivalent to $$ -1\leq \frac{\partial C}{\partial E}\leq0. \tag{2}$$ Hints: For the first part you may (a) see problem (iii) of HW1 and argue similarly, or use P-C parity or (b) use the Portfolio Lemma. For the second part, let \(E_2=E_1+\Delta E_1\) and take the limit \(\Delta E_1 \to 0\) to get from (1) to (2). To go from (2) to (1), use Mean Value Theorem.
    3. (*) 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).
    4. From the book: 2.3, 2.4. 2.5
    5. Find out why prices for E-mini S&P 500 Futures on CME do not follow the formula we derived for futures, \[F(t) = S(t) e^{r(T-t)}.\]
    6. Show that \[ f(x,t) = \frac1{\sqrt{2\pi t}} e^{-\frac{x^2}{2t}} \] satisfies the heat equation \(f_t = \frac12 f_{xx}\).
    7. From the book: 3.4, 3.6, 3.7 (may use the differentiation trick), 3.8; P3.2
    Solutions: HW2_sol.pdf and a Matlab script HW2.m.
  3. Submit Tue, Mar 1: (NB: no office hours Monday, Feb 29)
    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 whose name or ticker (code) matches the first few letters of your surname. Submit the figure; make sure to include in the figure the stock symbol and the dates for which the data was gathered. (*) Find out what happened on the day of the biggest amplitude change in the stock price.
    3. From the book: 6.1, 6.4, 6.5, 6.10.
    4. (P*) Generate a random sequence \(\{X_n\}\) of standard normal variables. Plot the partial sums of the series $$ W_t = \frac{X_0}{\sqrt{2\pi}}t + \frac{2}{\sqrt{\pi}}\sum_{n=1}^M \frac{X_n}{n} \sin(nt), $$ over \(t\in[0,\pi]\) with \(M=10,\ 100,\ 1000\). Make sure all three curves lie on the same figure and the same sequence \(\{X_n\}\) is used in all curves.
    Solutions: HW3_sol.pdf and a Matlab script PaleyWiener.m.
  4. Submit Thur, Mar 10:
    1. (Prepare for Quiz on Thur, Mar 2) Derive the formula for the price of call in the 1-level tree model when \(r>0\), \[ C = (S_0 - S_d e^{-r\tau}) \Delta, \qquad \mbox{where } \Delta = \frac{S_u - E}{S_u - S_d}. \]
    2. Find the arithmetical errors in this example of the 1-level tree model. Why does the correct answer for the dealer's profit makes perfect sense (and therefore the erroneous answer makes no sense)?
    3. You assumed the 1-level tree 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?
    4. (*) Assume the 1-level tree 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 \(E\).
    5. For the European Put with the parameters S0=100, E=110, r=0, expiration in 3 months, use the 3-level tree model (dt = 1/12) with 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.
    6. 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.
    7. From the book: 16.4, 16.6
    Solutions: HW4_sol.pdf and a Matlab script HW4.m.
  5. Submit Thu, Apr 14:
    1. Calculate \(\mathrm{var} \left[\left(W_{t+\delta t} - W_t \right)^2\right]\) (Hint: Use the definition of the Wiener process and moments of the Gaussian, exercise 3.7); 7.2
    2. (P*) Look up and program the sample paths of a Brownian Bridge (\(B_0=0\), \(B_1=0\)), generalized Brownian Bridge (\(B_0=a\), \(B_1=b\)).
    3. 8.2, 8.3, 8.8*, 8.10, 8.11 (use the result of 8.10, not 8.7), P8.1
    4. Derive the expressions for \(\rho\) and vega; 10.4, 10.6
    5. 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).
    Solutions: (up to a minor renumbering) HW5_sol.pdf and a Matlab script HW5.m.
  6. Submit Thur, Apr 28: 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 for the above period of August 2014).
    3. Calculate the volatility realized by these historical prices. Re-do delta-hedging using the historical volatility to calculate deltas.
    4. Does hedging work better or worse with realized volatility?
    5. (optional) Of course, one cannot know realized (historical) volatility in advance. But one can adjust hedging by combining the initial guess (implied vol) with the data obtained over the past days. Implement this and compare with previous results.
    Some remarks:

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