Teachers, Fall 2017

Lecturer and examiner: Fredrik Viklund, Office 3527 (Dept of Mathematics), email: fredrik.viklund@math.kth.se

TA (problem sessions): Alexander Aurell, email: aaurell@kth.se

Course literature

Björk: Arbitrage Theory in Continuous Time, 3rd Edition, ISBN 978-0199574742

News

Suggested solutions for the exam: SF2975_exam_171020_SOLUTIONS.pdf
PS5: 25.1, 25.2, 26.1, 26.2, 27.2, 27.3
L16: Pricing formula in LIBOR model, swaps, swaptions, swap rate models.
More on the LIBOR market model, including the non-rigorous argument for Black's formula alluded to in Björk: http://www.fabiomercurio.it/booksample.pdf
L15: The change of numeraire technique (i.e., identify factors in tricky expectation as suitable RN derivative, compute under weighted measure instead via Girsanov) applied to price bond options and an extension of the BS formula for random interest rates. Example: Hull-White. Market models: how to price interest rate caps/caplets, the LIBOR market model.
No lecture on Oct 10.
Hints for Project II now available below.
L14: Recap of ``inversion of yield curve'' program. Forward rate models: Heath-Jarrow-Morton framework, HJM drift condition. Worked the example of constant volatility, leading to the calibrated Ho-Lee model. Change of numeraire revisited, T-forward measures using T-bond as numeraire.
PS4: 18.3, 22.1, 22.5, 23.2, 23.3, 24.2. Next problem session a subset of 25.1-3, 26.1-2, 27.1-2 and 29.1-3 will be solved.
L13: Interest rate models, the Term Structure Equation. Martingale modeling, affine term structures. A class of ATSs, with several examples. The Hull-White model, on calibration of the HW model.
Project II has been updated, see below for the correct assignment.
L12: Fixed income assets, contd. Forward rates, short rate. Bonds, coupon bearing bonds, yield, yield-curve. Modeling of bond market. Given dynamics for p(t,T), f(t,T), r(t), expression of f in terms of p-dynamics. The argument involves a slightly tricky differentiation wrt T.
Guest lecture: Friday Oct 6, Robert Thorén, Head of Risk Solutions, Algorithmica Research.
L11: finished discussion of path dependent derivatives. Reflection principle and related formulas, example of down-out contract. Fixed income assets: zero-coupon bonds. Bonds and interest rates.
Project II has been posted, see below.
PS3: 16.2, 16.6, 17.2, 17.3 (for an interpretation of sigma_d = sigma_X, see this pdf), 18.2. Next problem session a subset of 18.3, 22.1, 22.5, 23.2, 23.3, 24.1, 24.5, 25.1 and 25.3 will be solved.
L10: Finished discussion of currency derivatives and foreign equity markets. Path dependent derivatives. Stopping times: first passage time and first exit time. Reflection principle: set-up.
I skipped a few steps in the derivation of the PDE for the continuous yield dividend paying stock.I did not put back V^h(t) = F(t,S_t) which was the reason for bringing back V^h(t) into the differential. This then gives the -rF term in the PDE.
L9: Finished discussion of dividend paying assets; BS-PDEs with jump condition; continuous dividends also with BS-PDEs (but no jumps). Problem: try to derive the PDE when dD=\delta S dt by passing to the limit in the discrete dividend model. Currency derivatives, start of discussion.
Thanks for very good comments on the mini-evaluation. We will try to address at least some of the things pointed out!
L8: Change of numeraire. Derivation of the formula for RN-derivative relating EMMs corresponding to different numeraires. I sketched an example of how change of numeraire can be used: the derivative was max S_1(T), S_2(T) in the 1+2 dimensional BS model. Models with dividends: self-financing portfolio, ex dividend convention, jump of stock price process.
PS2: 9.4, 13.3, 15.1, 15.4.
L7: Incomplete markets. Pricing formula for T-claims. The market price of risk determines the EMM. How to choose EMM/market price of risk.
L6: Pricing and hedging in the multidimensional BS model. The martingale approach to multidimensional models: arbitrage, completeness and hedging.
L5: Martingale approach. First and second fundamental theorems. Martingale representation theorem. Details worked out for BS model. BS model in N dimensions, N >1.
Note! There was a typo in Project 1, see below for updated file. Thanks to the group of students spotting this!
L4: The Greeks. Martingale approach to arbitrage pricing. Change of measure, Girsanov.
On Friday Sept 8, at 2pm, there is a MSc presentation in Room F11 that you may find interesting and useful (for Project 2): Simon Carmelid presents his thesis ``Calibrating the Hull-White Model using adjoint algorithmic differentiation''.
Sept 7. Re: project groups. In case you are unable to find a group to join, get in touch with we and I will try to connect you with other students in the same situation. Note: before doing this, please do try yourself to find or create a group.
--> The lecture room for Friday September 8 is E3 <--
PS1: Ex 5.12, Ex 7.1, Ex 7.2, Ex 8.3. There were some errors in the solution of 8.3 presented on the board, you can find a detailed and correct solution here.
Lecture 3: Derivation of the BS PDE, Part 2. Risk-neutral pricing formula. Explicit computation of the price of a European call in the BS model.
Lecture 2: Self-financing portfolios, arbitrage. The Black-Scholes model, derivation of the BS equation, Part 1: we determined half of the hedging portfolio.
Lecture 1: Review of basic probability theory: stochastic processes, Brownian motion, martingales. Ito integrals, Ito's formula, SDEs, the infinitesimal operator connecting SDEs to PDEs. The Feynman-Kac formula in a simple setting (will return to this).
First lecture on Aug 28. Welcome!

Tentative plan

NB: this plan, including lecture dates, is very likely to change. Please check the schedule every week.

Week 35 2017

Mon 28 aug	10:00 - 12:00	LE E31	Björk Ch. 4-6
Tue 29 aug	13:00 - 15:00	LE E31	Björk Ch. 7-8
Wed 30 aug	15:00 - 17:00	LE D34	Björk Ch. 7-8
Fri 1 sep	10:00 - 12:00	LE M32	Problem session
Week 36 2017
Fri 8 sep	10:00 - 12:00	LE E3	Björk Ch. 9+10-12. (Only a small portion of this material!)
Week 37 2017
Mon 11 sep	10:00 - 12:00	LE Q21	Björk Ch. 12-13
Tue 12 sep	13:00 - 15:00	LE L52 (Aurell)	Björk Ch. 13-14
Wed 13 sep	15:00 - 17:00	LE D34	Björk Ch. 15
Fri 15 sep	10:00 - 12:00	LE L52	Problem session
Week 38 2017
Mon 18 sep	10:00 - 12:00	LE M33	Björk Ch. 16
Tue 19 sep	13:00 - 15:00	LE M33	Björk Ch. 17 + 26
Fri 22 sep	10:00 - 12:00	LE M33	Björk Ch. 18
Week 39 2017
Mon 25 sep	08:00 - 10:00	LE D34	Problem session
Tue 26 sep	13:00 - 15:00	LE Q33	Björk Ch. 18+22
Wed 27 sep	15:00 - 17:00	LE K51	Björk Ch. 22-23
Fri 29 sep	10:00 - 12:00	LE D34	Björk Ch. 24-25
Week 40 2017
Mon 2 oct	10:00 - 12:00	LE L52	Problem session
Tue 3 oct	13:00 - 15:00	LE L52	Björk Ch. 25-26
Wed 4 oct	15:00 - 17:00	LE D34	Björk Ch 26-27
Fri 6 oct	10:00 - 12:00	LE L51	Björk Ch. 27, 29 GUEST LECTURE
Week 41 2017
Mon 9 oct	10:00 - 12:00	LE Q33	Problem session
Tue 10 oct	13:00 - 15:00	LE E35	No lecture
Wed 11 oct	15:00 - 17:00	LE K51	Review
Fri 13 oct	10:00 - 12:00	LE M33	Review
Week 42 2017
Fri 20 oct	14:00 - 19:00	EX E34, D35, D41, ...	Tentamen
Week 51 2017
Wed 20 dec	08:00 - 13:00	OM Q21, Q24	Omtenta

Projects (more information to come!)

General instructions: Projects2017_general.pdf
Project 1
- Reading for Project 1: VolatilitySurface.pdf
- Assignment, Project 1: Projects2017_project_1_edited.pdf
- Datafiles, Project 1: New_option_prices_1.csv, New_option_prices_2.csv
- NEW: The spot price of the underlying asset has been requested. For the data set New_option_prices_1.csv the spot price is approximately 1593, for New_option_prices_2.csv the spot price is approximately 1624.

Project 2
- Reading for Project 2: calibration-hull-white.pdf
- On Jamshidian's trick: https://en.m.wikipedia.org/wiki/Jamshidian%27s_trick
- Assignment: Projects2017_project_2-1.pdf
- Quantlab workspace (right click and choose "Save link as...", then upload to your catalogue "appdata" through home.ug.kth.se): get_zero_rate.qlw
- Guide to the workspace: workspace_guide.pdf
- General information about Quantlab is available here.
- NEW: Hints for problem 2 For the pricing of a caplet, first follow the arguments in Chapter 26.8 in Björk to get an expression of the caplet price in terms of the price of a put option on the a zero-coupon bond. Then use the log-normality of the zero-coupon bond (see article) to calculate the put price. You can find some inspiration for this in the Wikipedia page on the Hull-White model https://en.wikipedia.org/wiki/Hull%E2%80%93White_model#Derivative_pricing
  
  For the pricing of a swaption, start from equation (27.46) in Björk. Following the notation on the Wikipedia page for Jamshidian's trick, let the stochastic short rate in the Hull-White model play the part of the random variable $W$ and let $p_{n} (t)$ play the role of $K$ . You do not need to derive your corresponding $f_{i} (w)$ explicitly (cf. the article), but define them implicitly using the short rate. After this step, Jamshidian's trick should get you to a linear combination of put options prices for zero-coupon bonds. For some inspiration on this, see the following short text http://wwwf.imperial.ac.uk/~mdavis/IRM17/HW_SWAPTION_FORMULA.PDF
  Remeber to justify all steps in your derivation, if you use any results - make sure their conditions are satisfied.

Old exams

t151023.pdf (exam), s151023.pdf (solution)
t161021.pdf (exam), s161021.pdf (solution)

Formal information

Intended learning outcomes

To pass the course, the student should be able to do the following:

explain the properties and determine the price of the most common financial derivatives, such as call and put options, bonds and forwards and futures.
define martingale measures and use them for pricing financial derivatives.
explain and analyse the following models (the assumptions behind them and the limitations these imply) and be able to use them for pricing
- Black-Scholes model and its extensions, for example to several currencies and to assets paying dividends.
- Short rate models (especially those with an affine term structure)
- Forward rate models
- LIBOR market models
- Swap rate models
use the change of numeraire technique to price financial derivatives

To receive the highest grade, the student should in addition be able to do the following:

Combine all the concepts and methods mentioned above in order to solve more complex problems.

Course main content

The martingale approach to arbitrage pricing of financial derivatives. Black-Scholes model and extensions thereof. Short rate models. Forward rate models. LIBOR market models. Pricing using the change of numeraire technique.