Fundamentals of financial mathematics

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Exam January 27, 2021 (1 PM)

• Question 1
  • Explain in detail the pricing of American options under a 3-step binomial tree model in general. Give general formulas (assume a general S_0, r, T, and payoff function). Assume q = 0 (no dividends). Draw the tree and show the details of price calculation at each step.
  • Illustrate it by the pricing of a digital put option paying 1 if the stock price at exercise is lower than the strike K and 0 otherwise under a setting with S_0 = 100; K=95; r=q=0; and u-1=1-d=0,1.
• Question 2: Give and prove the calendar spread inequality in a one price setting for the European Call (assuming no dividends).
• question 3
  • Discuss in a finite discrete market model the relationship between the existence and uniqueness of an equivalent Martingale measure with the arbitrage freeness and completness of the market model. Give briefly a definition/explanation of these concepts and state the main theorems.
  • Provide an example of an arbitrage free market model which is incomplete and relate it to your statements on the EMM.
• Question 4: Relate the following Black-Scholes PDE for the price O of an option under the Black-Scholes model dO/dt + (1/2) * sigma²S² * d²O/dS² + r * S * dO/dS - r * O = 0 to the statement "the total change in the value of a delta hedged portfolio is equal to 0 on average". ('d' was acutally a sigma symbol as well in the exam question)

Exam January 24, 2018 (9 AM)

The exam was closed-book. You got 3 hours to complete the question, for which it was important to write your answers down clearly. After that (or sooner if you are ready), there is a short oral part where the professor read your answers to see if he understand them and if you did not make any stupid mistakes. Before the exam, the professor gave some extra information about the questions: this is written between (). The points were distributed equally over the four questions.

• Q1. Consider a non-divende paying stock. Proof that if interest rates are zero (r = 0), that the put-call parity for European vanillas is also valid for American vanillas. (Give + proof the put-call parity. Proof the validity for American vanillas.)
• Q2. Discuss the pricing of an American put option in detail in a 3-step binomial tree model. (Do this step by step. Use formulas. Every detail is important.)
• Q3. What is a complete model? Give briefly a definition/explanation of the concept. How can you check whether a model is complete or not? Provide an example of a model which is not complete and relate this to your statements. (Why relevant? What does it actually mean?)
• Q4. Derive the Black-Scholes partial differential equations for the price O for options: $\frac{\partial O}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 O}{\partial S^2} + r S \frac{\partial O}{\partial S} - r O = 0$. (Proof that this is true, using a lemma. Your answer should be a mixture of correct mathematical formulas and sentences.)

Examen vrijdag 9 januari 2015 om 14.00 uur

1. Waarom hebben put en call gelijke prijs als K= S0*e^rT
2. Waar of niet waar?
   • Uitdrukking van EC(T,K)=... Met delta, gamma en nog een andere greek erin.
   • De prijs van een ODBC is altijd hoger dan IDBC als H<K<=S0
   • de integraal van Ws dWs = Ws/2
3. Wat is het verschil tussen real world en riskneutral world? Wat betekent dit voor het binomial tree model en het black scholes model?
4. Geef de black scholes SDE en leg uit wat deze voorstelt, geef de oplossing