Homework II. November Universität Basel Wirtschaftswissenschaftliches Zentrum WWZ.

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1 Homework II November 2010 Universität Basel Wirtschaftswissenschaftliches Zentrum WWZ

2 Exercise 1 Explain the no-arbitrage and the risk-neutral valuation approaches to valuing a European option using a one-step binominal tree. No-Arbitrage Valuation: Erstellung eines Replikationsportfolios bestehend aus einer Cash- und Aktienposition. Der Preis des Replikationsportfolios muss dem Preis der Calloption entsprechen, weil ansonsten Arbitrage möglich ist. Risk-Neutral Valuation: Erfolgt durch Verwendung der Risikoneutralen Wahrscheinlichkeiten für die einzelnen Verzweigungen des Binomialbaums. Die Rendite des Underlyings entspricht hierbei dem risikolosen Zinssatz. Die Option wird anhand des Payoffs und dessen Diskontierung mit dem risikofreien Zinssatz bewertet. Folie 2

3 Exercise 2 In theory, using several assumptions, it is possible to price options by using dynamic replication. Illustrate why such a pricing is difficult or even impossible to use in practice. Dynamische Repl. verlangt eine stetige Anpassung des Replikationsportfolios. Dies führt zu verschiedenen Problemen: - stetiges Handeln ist in der Realität so gut wie unmöglich - Transaktionskosten - Zeitweise eingeschränkte Handelbarkeit von Basiswerten (Illiquidität) - Leerverkäufe nicht immer oder nur eingeschränkt möglich - Folie 3

4 Exercise 3 Binomial Option Pricing Asian Options or Path-Dependent Derivatives a) Use a three-time-step tree to price an American put option on the arithmetic average of the price of a non-dividend-paying stock when the stock price is $50, the strike is $55, the risk-free interest rate is 5% per annum (continuously compounded), the volatility is 25% per annum, and the time to maturity is three months. The arithmetic average is measured from today until the option matures. Hint: Use the CRR Approximation Excel Folie 4

5 Exercise 3 Binomial Option Pricing Asian Options or Path-Dependent Derivatives b) Explain why delta hedging is easier for such an option than for regular options. Der Payoff wird mit dem Zeitverlauf prognostizierbarer als bei regulären Optionen und eine hohe Volatilität des Basiswertes am Laufzeitende machen das Hedging nicht mehr so schwierig, da sich der Referenzkurs über den Durchschnitt von mehreren Perioden berechnet. Folie 5

6 Exercise 4 Assume you are given the following data on a non-dividend paying stock: - Today s stock price (S0) = 80 - Stocks volatility = 30% per annum - Risk free interest rate (continuously compounded) r = 10% p.a. - Time to maturity in years T-t = N(z1) = N(z2) = 0.52 a) Calculate the price of a European put and a European call option with a strike of $82. Excel b) Verify that the put-call parity is consistent with option prices in a). Excel Folie 6

7 Exercise 5 You heard and read about the Black-Scholes-Merton stock option pricing model. What are its four main assumptions about a) the statistical probability distribution of the price in T? Lognormal distribution of prices in T b) the risk neutral distribution of the expected return (with continuous compounding) in T? Normal distribution of returns in T c) the volatility of the stock price process? Constant volatility d) the risk-free interest rate? Constant risk-free interest rate for all maturities Folie 7

8 Exercise 6 On October 27, 2010, a European call option on the Microsoft stock is given by YahooFinance as follows: Option s conversion ratio: 1 Today s share price of Microsoft: For your calculation, assume that Microsoft will pay no dividend and the Black and Scholes model is correct - what would be the implied volatility the trader used to price this option? Assume further that the risk-free rate is 0.5% (continuously compounded). a) Excel Folie 8

9 Exercise 6 On October 27, 2010, a European call option on the Microsoft stock is given by YahooFinance as follows: Option s conversion ratio: 1 Today s share price of Microsoft: b) Is the call option out-of- the-money, at-the-money or in-the-money? Strike < Share price -> in-the-money Folie 9

10 Exercise 6 b) [ ] out-of- the-money, at-the-money or in-the-money? Give a short explanation for the difference between the three definitions. In-the-money S>K (Call) ( Put ) S<K die Option weist bei sofortiger Ausübung einen positiven Cash-Flow auf At-the-money S=K die Option weist bei sofortiger Ausübung einen Cash-Flow von Null auf Out-of-the-money S<K (Call) ( Put ) S>K die Option weist bei sofortiger Ausübung einen negativen Cash-Flow auf Folie 10

11 Exercise 7 Financial Engineering Structured Products BASF Discount Zertifikat a) Please explain how you could construct such a product (i.e. give an example of the building blocks of such a product)? - Zerobond & Short Put (Reverse Convertible) - Kauf Aktie oder LEPO & Short Call (Covered Call) b) How will the payoff diagram look at maturity? Plot the diagram and label it. Folie 11

12 Exercise 7 Financial Engineering Structured Products BASF Discount Zertifikat c) Today, you have to issue the product which is presented on the factsheet. For this, calculate the issue price and the option(s) of the product. What issue price would you recommend? Excel d) Is there a difference to the issue price of the RBS product which is EUR 45.58? If yes, what could be the reason(s) for it? - Gebühren - Andere Inputs, z.b. impl. Vola., Zinssatz, Laufzeit (Approximation) Folie 12

13 Exercise 7 Financial Engineering Structured Products BASF Discount Zertifikat d) To merchandise a discount certificate, regarding the attractiveness of the product, it is helpful to provide a certificate with a deep discount. In which market environment or for which types of equities could it be most interesting to issue such a product? - In Zeiten hoher Vola. (Call ist teurer -> Discount grösser) - Bei volatilien Aktien - Wenn eine grosse Nachfrage nach Produkten für auf seitwärts oder leicht steigende Aktienmärkte (Underlyings) herrscht. Folie 13