MATH 425, spring 2016
Course information
- Instructor: Dr G.Berkolaiko, office: Blocker 625c,
email: berko AT math.tamu.edu
- No office phone: budget cuts
- Lectures: TR 9:35-10:50, BLOC 149
- Office hours: TW 11-12 (subject to availability);
other times by appointment.
- Exams: A midterm on Thu, March 10 and the final on
Thu, May 5, 12:30-2:30.
- 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:
- Other resources:
- Projects
(for honors contracts).
- Matlab demonstrations:
- Past exams.
- Notes from the pre-midterm review.
Homework
Exercise with stars next to them are compulsory for honors students.
- Submit Thu, Jan 28: Updated for 2016
- (*) 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:
- short 95C@5.50, 100C@2.70, 105C@1.15
- long 95P@1.05, 100P@2.50, 105P@5.50
- short 95P@1.05, 100P@2.50, 105P@5.50
(draw 3 figures with 3 curves on each).
- 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).
- 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).
- Submit Thur, Feb 11:
- 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.
- 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.
- (*) 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,
\[F(t) = S(t) e^{r(T-t)}.\]
- 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
Solutions: HW2_sol.pdf and
a Matlab script HW2.m.
- Submit Tue, Mar 1: (NB: no office hours Monday, Feb 29)
- 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).
- 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.
- From the book: 6.1, 6.4, 6.5, 6.10.
- (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.
- Submit Thur, Mar 10:
- (Prepare for Quiz on Thur, Mar 2) Derive the formula for
the price of call in the 1-level tree model when \(r>0\),
\[ C = (S_0 - S_d e^{-r\tau}) \Delta, \qquad
\mbox{where } \Delta = \frac{S_u - E}{S_u - S_d}. \]
- 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)?
- 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?
- (*) 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 \(E\).
- 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.
- 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.
- From the book: 16.4, 16.6
Solutions: HW4_sol.pdf and
a Matlab script HW4.m.
- Submit Thu, Apr 14:
- 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
- (P*) Look up and program the sample paths of a
Brownian Bridge (\(B_0=0\), \(B_1=0\)),
generalized Brownian Bridge (\(B_0=a\), \(B_1=b\)).
- 8.2, 8.3, 8.8*, 8.10, 8.11 (use the result of 8.10, not
8.7), P8.1
- Derive the expressions for \(\rho\) and vega; 10.4,
10.6
- 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.
- 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.
- What's the implied volatility? Corresponding delta?
- Simulate delta-hedging with daily close prices and above
volatility (download daily close prices from Yahoo Finance for
the above period of August 2014).
- Calculate the volatility realized by these historical
prices. Re-do delta-hedging using the historical volatility to
calculate deltas.
- Does hedging work better or worse with realized volatility?
- (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:
- Use "Close" column from Yahoo data (not the "Adj. Close").
- It makes sense to measure time in trading days (since there
is no market and therefore no volatility on weekends). Assume
there are 252 trading days a year.
- For the last (optional) part, you should use one-step
updating scheme to recalculate volatility,
$$ \sigma_{next}^2 = w \sigma_{prev}^2 + (1-w) \frac{LogRet^2}{\delta t},$$
where \(LogRet\) is the last day's log-return and \(w\) is a
number smaller but close to 1 (something between 0.8 and
0.95).
This file was last modified on Wednesday, 10-Jan-2024 15:58:24 CST.