Introduction to Mathematical Finance - 2020
Thea, 3. August, 11:00-11:30
They let me in 15 minutes late. 1) Definition of arbitrage, arbitrage opportunity of 1st and 2nd kind 2) Explain the binomial model an its conditions to be arbitrage free 3) What if u=r, what would be an arbitrage opportunity and its associated wealth process. 3) Taking a payoff H, what would be a fair price? -> sellers and buyers price, what are you using when, you are saying that the extended market (with price \pi_b < \pi < \pi_s) is arbitrage free? -> FTAP 4) State the FTAP 1st, which are the easy implications? Prove one of the hard ones. Then he offered me to prove other directions as well. 5) Prove: a local martingale bounded from below is a super-martingale. 6) Prove: a local martingale bounded from below is a true martingale. 7) What is the trinomial model? When m = r and u>r>d, it is easy to see that the market is arbitrage free. What is the explanation? He wanted to here: q_1*u + q_2*r + q_3*d= r and r lies strictly between u and d and this is just a convex combination, so it is easy to see that there actually exists one.
Selim, 5. August, 10:30-11:00
There was a delay of 5 minutes. 1) Write down the problem of utility maximisation of terminal wealth (including consumption) 2) Write down the expression of terminal wealth (using the strategy theta and x) 3) Write are the results used for this? (MOP and DPP) 4) Explain how we proved the MOP? So, I defined the set A_k, gamma_k and J_k 5) What is the property of J? (generalised supermartingale). Prove it. What prerequistes do we need for the MOP and state it. (Here it is important to talk about a supermartingale, and no more a generealised supermartingale. Furthermore, it is important to state the property for the sets in F_k, which gives that the set of ... is closed under taking maxima, and which gives us the possibility to approximate J_k by an increasing sequence). Prove it then. 6) Now consider the trinomial model, what do we need for it to be arbitrage-free? (like 7) for Thea) 7) Can you now maximise the utility of terminal wealth of this model, given that the utility is log? How would you do? Here, it was a bit strange, I didn't really know how to do it, and we talked then about the duality problem, I stated it and finally, he asked me what is the link between the solution of the primal and dual problem, which is f_x*=I(h_y*). I was really surprised by this kind of exam since I was sure that I will not spend the whole time on the two last chapters. However this is exactly what happened so have a careful look at it before taking the exam.
Viola, 5. August, 14:00 - 14:30
At first the professor asked me if I wanted to start with expected utility from terminal wealth. I said I preferred something from chapter 1 or 2 and that we could talk about expected utility later. He said okay and we started with chapter 2.
1. Question: Definition of arbitrage
2. Question: The characterisation theorem of arbitrage (i.e. the 6 equivalences; Proposition 2.2.1). I stated the theorem and explained the proof and in detail I explained the direction “There doesn’t exist arbitrage of the first kind with a 0-admissible strategy” implies “For every self-financing strategy \(\psi\) with \(V_{0}(\psi) = 0 \: P-a.s. \) and \(V_{T} \geq 0 \: P-a.s\) then \(V_{T} = 0 \: P-a.s.\) ”
3. Question: We then switched to attainability of contigent claims and I stated the theorem of the 3 equivalences of attainability (Theorem 2.5.2) and said that the interesting direction would be 3–>1 and I said we need the essential supremum for that. At which point I also mentioned that we use the Optional Decomposition Theorem
4. Question: He then wanted me to state the Optional Decomposition Theorem at which point he said “I guess now we have to do the hard direction of the proof” I told him I haven’t even looked at the proof as I found it super long and really didn’t think it would come up in the exam..
5. Question: We then switched back to the attainability statement of question 3. He wanted the definition of the essential supremum and the idea of the proof. In particular he wanted to know how we would proof 1 -> 3. I didn't really know what he wanted with that question and I explained the proof of theorem 2.5.2 where we showed 1 -> 2 -> 3 and I gave an overview of each implication. Then the time was over.
Overall the Prof. Czichowsky was quite nice and allowed to change to the topics I felt more comfortable with.
