Interview training

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This is the homepage of the seminar given in summer term 2008. The page serves as a basis of communication between students and lecturer. It is updated regularly. You may check whether there are any changes of the page by viewing the date of the most recent update below.

Last update: 8.9.2008

1 Nature of the seminar
2 References
3 Unit 1
4 Unit 2
5 Unit 3
6 Unit 4
7 Unit 5
8 Unit 6
9 Unit 7
10 Unit 8
11 Unit 9
12 List of finance problems
13 List of problems in probability and statistics
14 List of logic problems
15 Examination

[edit] Nature of the seminar

We provide and discuss subjects which are part of the common body of knowledge in quantitative finance. In particular we discuss problems which typically are posed in job interviews.

[edit] References

Falcon Crack

Wilmott

Hull

[edit] Unit 1

The first unit was on 3.3.2008.

We talked about the goal of the seminar and agreed upon the exam.

There will be an written exam at the end of the seminar. The problems posed at the exam will be chosen from list of problem discussed during the seminar. The complete list is published on this homepage below.

We started posing (but not answering) some typical problems:

Financial: 1,2,3

Logical: 1-10

Statistical: 1,2,3

[edit] Unit 2

Unit 2 was on 10.3.2008.

The subject of this unit were facts on prices of financial derivatives that can be obtained without talking about special models.

We talked about forward prices, risk neutral models for stock prices, and the put-call parity. Moreover we discussed the qualitative behaviour of call and put prices and their dependence on the spot price and on maturity.

We discussed the answers of the financial problems 2-7.

[edit] Unit 3

Unit 3 was on 7.4.2008.

We gave an introduction to the Black-Scholes model for asset prices. We discussed the relation of the model parameters to the statistical properties of the asset prices. We dealt with some peculiarities of lognormality.

Related financial problems (which now can be answered): 9-13

[edit] Unit 4

Unit 3 was on 14.4.2008.

We discussed structure and details of the call price formula for the Black-Scholes model.

Related financial problems (which now can be answered): 14-18

[edit] Unit 5

Unit 5 was on 21.4.2008.

We discussed the logic problems 1-10.

[edit] Unit 6

Unit 6 was an 28.4.2008.

We started the discussion of continuous time models and of the Wiener process.

Our discussion was based on presentation Sheets 76-94 of the course Advanced Financial Mathematics.

[edit] Unit 7

Unit 7 was on 5.5.2008.

We continued the discussion of continuous time models and turned to aspects of the integral calculus.

Our discussion was based on presentation Sheets 95-118 of the course Advanced Financial Mathematics.

[edit] Unit 8

Unit 8 was on 19.5.2008.

We discussed the concepts of stochastic integration und the three basic principles of stochastic calculus.

Our discussion was based on presentation Sheets 119-141 of the course Advanced Financial Mathematics.

[edit] Unit 9

Unit 9 was on 26.5.2008.

We worked through typical examples and problems of stochastic integration.

Related finance problems: 26, 27, 28, 29, 30, 31, 32, 33, 34.

[edit] List of finance problems

1 What is the price of a call option at the money when the underlying has volatility zero ? (Spot price 100, riskless rate 5 percent, maturity 3 months)

2 What happens with the Gamma of an call or put option if maturity tends to zero ?

3 When is the call price equal to the put price (same underlying, strike and maturity, rate zero) ?

4 Give a rough sketch of the call price and the put price as a function of the spot price of the underlying ? Discuss the relation to the intrinsic value of the option.

5 Under what circumstances has an American call the same price as a European call ? Under what circumstances has an American put the same price as a European put ?

6 Give a rough sketch of the Deltas of a call price and a put price as a function of the spot price of the underlying. Do the same for the Gammas.

7 What happens with the Delta of an call or put option if maturity tends to zero ?

8 Given the path of the underlying describe the path of a call price and of a put price as maturity tends to zero.

9 Explain the idea of risk neutral pricing.

10 Suppose that $X$ is normally distributed with mean 0 and variance $σ 2$ . What is the expectation of $e X$ ?

11 Describe the relation between the mean of the asset prices and mean of the log-returns.

12 Given the mean of the log-returns, what is the effect of increasing volatility on the average asset prices ?

13 When is it possible to replace ordinary returns by log-returns and why ?

14 Give a rough estimate of the price of a call at the money (interest rate zero). Be prepared to answer numerical questions.

15 How can the variance of the terminal stockprice be used for estimating the price of a call at the money. Be prepared to answer numerical questions.

16 Given a the price of a call with a given maturity how can you estimate the price of the same call with doubled maturity ?

17 If a pizza of 12 inches diameter serves 6 persons which diameter serves 8 persons ?

18 What is more valuable if it is 10 percent out of the money: a call or a put ? Explain carefully !

19 Explain the notion of a Wiener process ! State its basic properties !

20 Explain the intuitive background of the notion of an integral !

21 Explain the notion of quadratic variation ! Discuss the special case of the Wiener process !

22 Explain the term semimartingale with respect to its role in stochastic integration ! Give typical examples of semimartingales !

23 Let $(X t)$ be a continuous semimartingale. Expand $X_t^2$ by the integration-by-parts formula.

24 (Continuation) Discuss the special cases of the Wiener process and of smooth function.

25 State Ito's formula for the Wiener process and for general continuous semimartingales.

26 Evaluate $\int_0^t W_s dW_s$ !

27 Expand $W_t^2$ as Ito-process !

28 Expand $W_t^3$ as Ito-process !

29 Expand $e^{W_t}$ as Ito-process !

30 Let $(X t)$ be a continuous semimartingale. Expand $e^{X_t}$ !

31 Explain the notion of a stochastic exponential (continuous case) !

32 Discuss the solution of $d X t = X t d W t$ !

33 Discuss the solution of $d X t = X t d Z t$ when $(Z t)$ is a continuous semimartingale ! Explain the special case when $(Z t)$ is a smooth function !

34 Derive the Black-Scholes model from its defining stochastic differential equation !

[edit] List of problems in probability and statistics

1 A die is thrown up to three times. You may say stop and earn the last face value in dollars. Find the optimal strategy.

2 A die is thrown three times How much do you earn on average if you get the maximal face value in dollars ?

3 Discuss the exchange paradox.

[edit] List of logic problems

1 Add the numbers 1 to 100.

2 How long is the shortest way from a corner of a cube to the (diagonnally) opposite corner ?

3 A clock with analogic display shows 15:15. Calculate the the angle between the hour and minute hands ?

4 What are the decimal expansions of 13/16 and 9/16 ?

5 Two bikers start at a distance of 25 miles moving towards each other with speeds 20 mph and 30 mph, respectively. A fly starts at the same time flying forwards and backwards between the helmets of both bikers until they all collide. How many miles will the fly have travelled before it dies ?

6 On a 20x20 chessboard dollars are placed on the following way: The first row is filled with 1,2,3,... dollars. The second row is filled with 2,3,4,... dollars, the third row with 3,4,5,... dollars, an so on. How many dollars are placed on the board ?

7. How to evaluate $\int e^{-x^2} dx$ ?

8. Find the age of three children: The product is 36. I tell you the sum, but that is not enough. When I tell you that the eldest is left-handed, then you find the answer. How ?

9. You want to place dominoes (2x1) on a chessboard (8x8). Is it possible to do it in way such that two opposite corners remain empty ?

10. Is it possible that the average of two subsequent prime numbers is prime ?

[edit] Examination

In order to get credits for the course students have to take an exam. For the exam we propose Tueday, 30.9.2008, 18:00, at the Department of Statistics and Mathematics, UZA 2, 5th floor.

Please, let me know whether you will take the exam.

There will be posed 10 problems, taken from the lists above. The problems should be answered within 60 minutes. The exam is closed books.

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