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1. Is there an arbitrage in the following data? If this is the case, how much can be made risk-free? S=I20. K.= 120, r=6%. T=0.50, D0. c=10, ce6 2. A non-dividend paying stock has a 3-month forward price of 4100 and a put option price p=11.5. The three-month interest rate is I% (cont. compounding). Assuming no arbitrage is possible, what is the highest possible strike price K? 3. A company located in Canada and another company located in Australia discuss the opportunity of entering into a swap agreement directly without using the services of a financial institution:
Fixed Floating Company in Canada 2.2% 6-month LIBOR + 0.1% Company in Australia 5.8% 6-month LIBOR + 2.1%
The company in Canada needs to borrow a floating loan, while the company in Australia seeks loan financing at a fixed rate (semi-annually compounded). Principal amount and maturity for both loans are the same. a) The Australian company targets borrowing at a fixed rate of 4.9″%. Describe how the swap contract can be designed? What is the effective interest rate on the loan of the Canadian company? In addition, both companies discuss the option of entering into a second swap agreement: borrowing in foreign currency. The company located in Canada – in AUD, and the company located in Australia – in CAD, respectively:
CAD AUD Company in Canada 2.9% 3.5% Company in Australia 4.8% 4.0%
b) What is the total gain for both companies from entering into the second swap? Describe how the swap contract will be designed so both companies share the total gain equally without any exposure to exchange rate risk for the company located in Australia. c) Is the gain for each company under b) economically significant? d) Plot a graphical overview of the two swap contracts. 4. The following data is available: a dividend-paying stock with a stock price S=32 and strike price I@38. The risk-free interest rate is r=3.5%, the time to maturity is 12 months, and volatility is 10%. It is expected that a dividend of $2.5 will be paid after six and twelve months. a) Using DerivaGem*, an Excel tool provided with the textbook, find the price of a European call and European put on the stock. b) Using DerivaGem*, find the price of the European put on the stock if not paying dividends. When is the price of the put option equal to its intrinsic value? c) Using DerivaGem•, analyze the effect on the put and call option prices in a) with variations in dividends, 2) volatility, or 3) the time to maturity. Graph the results and discuss the reasons for the observed effects. •see Hull, p.838 — Chapter: DerivaGem; Select Option type: Black-Scholes — European

 

 

 

 

Sample Solution

As listed above, there exists a long list of methods and algorithms that can be used for facial recognition. Four of them are used frequently and are most known in the literature, i.e. Eigenface Method, Correlation Method, Fisherface Method and the Linear Subspaces Method. But how do these facial recognition work? Because of word limitations, only one of those four facial recognition techniques, i.e The Eigenface Method, will be discussed. Hopefully this will give an general idea of how facial recognition works and can be used.

One of the major difficulties of facial recognition, is that you have to cope with the fact that a person’s appearance may change, such that the two images that are being compared differentiate too much from each other. Also environmental changes in pictures, like lightning, have to be taken into account, in order to have successful facial recognition. Thus from a picture of a face, as well as from a live face, some yet more abstract visual representation must be established which can mediate recognition despite the fact that in real life the same face will hardly ever form an identical image on successive occasions. Our ability to do this shows that we can derive structural codes for faces, which capture those aspects of the structure of a face essential to distinguish it from other faces[6].

One of the four most famous facial recognition methods is the Eigenface Method. This method focuses on the aspects of the face stimulus that are important for identification. This is done by decoding face images into significant local and global ‘features’[24]. Such features may or may not be directly related to our intuitive notion of face features such as the eyes, nose, lips and hair. Scientists Matthew Turk and Alex Pentland [24] developed a computer system for the eigenface approach which works as following: “In the language of information theory, we want to extract the relevant information in a face image, encode it as efficiently as possible, and compare one face encoding with a database of

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