Questions and Answers of Introduces Quantitative Finance

Treating Pt in (8.27) as a random variable, show that if the hedge ratio δ is the negative delta ∆ of the option, then the variance of the change in portfolio Pt is a minimum with respect to the
What is the financial interpretation of d2?
Show that the delta of the European put option is −Φ(−d1).
From (8.29), we see that δ is the negative of the partial differential of the option with respect to the underlying asset, which is the delta ∆. In symbols,∆ := ∂O∂St.Using the result in Q4,
Using the result in Q3, show that StΦ(d1) − Ke−r(T−t)Φ(d2) = 0, whereΦ(x) := dΦ(x)dx = 1√2πe− −x2 2 .
Show that 12d2 1 − d2 2= ln St K+ r(T − t).
Although the European put option price can be obtained from the put-call parity, it is nonetheless a useful exercise to derive it directly from the heat equation. Starting from the analogue of(8.43).
Show that 1√2π ∞−∞exp −B2x2 2dx = 1 B.Hint: Perform a variable transformation y = Bx.
Instead of using the put-call parity, start from E0pτ) =n∑x=0n xpx(1 − p)n−x max K − uxdn−xS0, 0.(a) Go through the same steps in Section 7.6 to derive the Black-Scholes pricing
For the one-dimensional random walk model (7.1), a return to the origin S0 at time n occurs if Sn = 0. Note that a return to the origin can only happen if n is an even number.(a) Show that the
The butterfly spread is an option strategy that involves three different strike prices: K1 < K2 < K3. The exercise price K2 is at the money, and usually, K1 and K3 are chosen such that K2 = K1 + K3 2
A long strangle is an option strategy that involves the simultaneous buying of a slightly out-of-the-money put and a slightly out-of-the-money call of the same underlying asset and expiration
A long strip is another variation of straddle. It has one call and two puts.(a) Plot the payoff function at maturity.(b) What is the view of the option trader long in a strip?
A long strap is a variation of straddle. Instead of one call and one put, the strap has two calls and one put.(a) Plot the payoff function at maturity.(b) What is the view of the option trader long
A long straddle is an option strategy established by buying a call and a put on the same strike price written on the same underlying asset and with identical expiration date.(a) Plot the payoff
In Section 3.5, as an application of the three principles of Quantitative Finance, the fair price for the forward contract is obtained. The payoff function is ST − F0 for the buyer at maturity.(a)
The day count convention for SGD swap par rate is 365 days per year. Based on the bootstrapping approach adopted by the financial industry, and with reference to Table 5.2,(a) Verify that the
A popular trading strategy among currency traders is called the carry trade. It is a no-brainer. Borrow the currency that charges a low interest rate, exchange it into a currency that pays a higher
The definition of convexity requires a second derivative of the bond price with respect to the interest rate r. Show that the convexity of the zero-coupon bond is indeed T(T + 1), where T is the time
The total cash flow of the bond plus reinvestments should equal the time deposit of the same tenor and compounding scheme. Suppose p is the bond price in percent. Also denote the per period coupon
The prevailing interest rate is 10% per annum for the tenor of 20 years. Suppose a bond pays 7% per annum and the frequency of coupon payment is semiannual. This bond is traded at $74.26 per $100 par
Annuity is a stream of regular cash flows, either in the form of payment or receipt. Suppose you save an amount c every year, up to n years from time 0. The interest rate is y. Let x := 1 + y.(a)
The effective annual rate (EAR) is a useful concept to compare the yields of money market securities, which mature in less than a year. Suppose Bank U offers a structured product consisting of two
This is the First Illustration in one of the earliest books on financial mathematics by Moore [Moo29]: The Marshall National Bank pays 4% on time deposits. What is the annual interest payment on a
A futures contract is a standardized forward contract traded through a futures exchange rather than bilaterally over the counter. An index futures is a derivative product for investors trading and
The risk-free rate r0 is a key ingredient of the first principle of Quantitative Finance. Consider a frictionless world — absolutely no taxes, transaction costs and so on. In this idealized
The adjective “uncertain” from which “uncertainty” is derived has many different meanings in different contexts. In Quantitative Finance, uncertainty refers to the state of limited knowledge
Consider the following FX quotes:Bid Ask EUR/USD 1.3501 1.3502 USD/SGD 1.2711 1.2734 EUR/SGD 1.7077 1.7100 A trader has spotted a triangular arbitrage opportunity starting with SGD. The trader who is
A Treasury bond has one year left before maturity. The annual coupon rate is c and the coupon is paid semi-annually. An investor has invested an amount of A for this bond at the price of P.(a) Solve
An investor buys a Treasury bond at 103% of the par value and holds it to maturity, which is two years later. The coupon rate is 6%.(a) What is the return on this investment for holding the bond to
Show that, for the three averages of Pythagoras, namely, arithmetic meana, geometric mean g, and harmonic mean h, they satisfy the following inequality:h ≤ g ≤ a.That is, T 1x1 + 1 x2 + ··· +
In addition to the arithmetic and geometric means, another average is called the harmonic mean. These three means were proposed and studied by Pythagoras. The harmonic mean h of T positive real
Incorporated in 1974, Temasek is a sovereign wealth fund based in Singapore. Temasek manages a portfolio (asset under management (AUM)) valued at S$223 billion as of March 31, 2014. The 1-year total
Given a sample rt, t = 1, 2, . . . , T, of simple returns or total returns over one period, say, a year. The arithmetic mean a is defined as a := 1 TT∑t=1 rt.The geometric mean g is an average
Total return is simple return with the inclusion of dividends received. In the typical long first and short later setting, let Pt be the selling price and Pt−1 be the buying price one period ago.
The simple return (2.1) can be rewritten as r := simple return = selling price buying price − 1.The ratio R defined by R := selling price buying price is known as the simple gross return. The
An investor from the eurozone bought 500 shares of Microsoft a year ago for $38.13 per share. The exchange rate was $1.3591 per euro then. Today, Microsoft is traded at $47.74 and the exchange rate
Suppose you are a forex trader domiciled in the U.S. and you hold the view that the euro is going to weaken against the Greenback. The exchange rate GBP/USD is nicknamed cable. You decide to short
Suppose one in 100 actively managed mutual funds in the database is truly based on skill and not luck. Moreover, suppose a statistical test can diagnose the difference between skill and luck. If a
A stock mutual fund’s performance is usually measured relative to a stock index such as the S&P 500 index. For a start, consider simply the difference between the return on the fund and the return
Over the same period from 1977 to 1990 when Peter Lynch was in charge of the Magellan Fund, the total return of S&P 500 index had only three negative annual returns. In other words, the whole stock
The legendary fund manager Peter Lynch was in charge of an actively managed mutual fund called the Magellan Fund between 1977 and 1990. Over the decade of 1980, the asset under management (AUM)
How many combinations are there in picking k unordered outcomes from n possibilities? The answer to this question is known as the binomial coefficient, which is the number of combinations in choosing
Professor Nouriel Roubini, nicknamed Dr. Doom, is another prominent economist who had forecasted correctly the 2008 financial crisis. On September 7, 2006, to the laughter and scepticism from an
An organisation has recently purchased a company car for one of its senior managers at a cost of £25 000 and is trying to determine the most suitable time for replacement. It is estimated that the
A large organisation is considering replacing part of its vehicle fleet with the latest equipment. Although such a decision is expensive, it is felt that the investment will be worthwhile in terms of
A local authority is under severe financial pressure and is considering selling some of the land it owns to a building company. The building company is offering to pay£150 000 for the land now or a
A hospital has decided to replace an important piece of medical equipment at a cost of £30 000. The expected life of the equipment is five years and it will have a resale value of £4000 at this
A production company is considering purchasing a new piece of equipment for the production process for one of its products. The product itself is due to be withdrawn from the market in five years’
The company you work for is considering acquiring several of the latest generation desktop computers for use in an office. The company expects the equipment to have a useful life of three years. The
You have the choice of investing a sum of money in four alternative schemes:(a) one which will pay 10 per cent per annum interest compounded daily;(b) one which will pay 10.25 per cent per annum
A utility company is currently reviewing its manpower requirements in the domestic-appliance servicing department. One feature of the servicing work relates to domestic central-heating systems. For
You are currently employed by a high street bank and have been asked to investigate the bank’s policy in one particular area of its operations, that relating to small loans to individual customers.
An investigation is being carried out into a local authority’s housing policy. The local authority has a limited supply of housing which it owns and lets. It receives requests for housing from a
Call up the second simulation at the DOS prompt by typing the command:and then pressing the ENTER key. The program should then start up and you can follow the on-screen instructions. Run the
Call up the first simulation from the DOS prompt by typing the command:and then pressing the ENTER key. The program should then start up and you can follow the on-screen instructions.(a) Run the
For a particular project the information in Table 13.17 has been obtained.(a) Construct a network diagram for this project showing EFTs, LFTs and the critical path.(b) The company concerned has
A firm is making minor alterations to its automated stock-handling system in one of its warehouses. The relevant information relating to this project is shown in Table 13.16.The table shows the
A project comprises the activities shown in Table 13.15, which also shows the duration of each activity.(a) Construct a network diagram for this project and determine the expected completion time
You have been appointed secretary of a student group which has an annual programme of invited speakers from the business world. The group has suffered from a membership decline over the last few
In the project detailed in Table 13.13, activity labels indicate the major activities to be completed in a project.492 13 PROJECT MANAGEMENT ATM project Normal Crash Preceding duration Normal
A large banking organisation is installing a network of automated teller machines(ATMs) for use by its customers in a large town. These machines are installed in an exterior wall of the bank and
2 A business organisation is trying to improve staff morale and motivation and has decided to replace the existing staff canteen with a modern cafeteria. The works department has drawn up a detailed
Consider the situation of waking up in the morning and going to the kitchen to make yourself a cup of tea or coffee. Identify the various activities you would need to complete for this project and
A large construction company has recently won the contract for routine inspection and maintenance of the Channel Tunnel linking the UK to France. Obviously, routine maintenance and repair work,
A domestic appliance manufacturer is evaluating its delivery system from the production plant to showrooms around the country. One item used in the transportation of large appliances is a
Following internal reorganisation in a large brewing company, a comprehensive review of stock-control procedures is currently being undertaken. You have been asked to investigate stock control in the
A local authority operates over a large geographical area. The Education Department is responsible for ensuring that stocks of a certain item are kept for distribution to schools through the area on
A company requires a particular item from an outside supplier for its production process. Annual demand for the item is 3000 units. Order costs are estimated at £15 and holding costs at 20 per cent
Describe the quantitative techniques that are available to assist a manager with stock-control responsibilities in an organisation.
What are the basic decisions that a manager with stock-control responsibility faces and what are the likely objectives of such a manager in this context?
A small electrical repair shop finds it has two main types of work. Customers bring items in for repair: TVs, videos, etc. require major repair; other repairs are classed as minor. At present the
A small engineering company makes two products for export, A and B. Both products go through a four-stage process: turning, grinding, polishing, finishing.The two products are produced in batches of
For the problem in Exercise 1 assume the firm now wishes to maximise revenue rather than profit.(a) Formulate this problem in terms of determining the revenue-maximising combination of the two
A manufacturer is trying to determine the combination of two products, A and B, that should be produced. Each product passes through a three-stage production process: Stages I, II and III. For
A senior police manager is reviewing manpower allocation of police officers to a number of geographical districts which fall under her responsibility. Data has been collected on a number of
A training agency is currently reviewing part of its operations. It runs one particular scheme whereby young people who are unemployed are put through a training course which provides them with a
A large retail organisation has a number of home-furnishing outlets in a particular region, selling products such as carpets, curtains, domestic furniture and fittings.The organisation has been
As manager of a local leisure centre you are investigating the frequency of use of the centre by its members. You hope that by understanding some of the key factors that influence how often members
The corporate planning department in a large brewing company has been trying to develop a medium-term forecasting model relating to the demand for one of their products: a low-alcohol lager. The
A small engineering company has been monitoring its total costs over the last few years on a quarterly basis together with the number of employees and production level. In order to assist the company
Table 10.15 shows data on two variables for the UK from the Central Statistical Office for the period 1970 to 2003.Cars New registrations of cars shown in thousands as a monthly average.Income Real
For the following data calculate the correlation coefficient and plot the data on a scatter diagram. Do you think it measures the strength of the relationship accurately?x 1 2 3 4 5 6 7 y 8.25 7 6.25
Investigate sources of published statistics that show a seasonal pattern in publications such as Monthly Digest of Statistics, Economic Trends. For a variable which is published in both unadjusted
Working with a small group of colleagues, develop a consensus view of private car sales in Western Europe in five years time. In order to do this you will need to:● collect historical data on car
A large retail organisation is currently investigating a number of areas of its operations with a view to improving performance in terms of efficiency, sales and profits. The organisation has been
Table 9.20 shows the official seasonally adjusted series for the consumer expenditure categories shown in Table 9.19. Compare these with your own calculations. Comment on any differences.EXERCISES
The following exercises all relate to the development of a suitable time-series decomposition model for the data shown. In each case you should adopt the following procedure:(a) Plot the original
Return to the unit sales data used in this chapter. Apply the exponential-smoothing model to this data using values for α of 0.15, 0.2, 0.25 and 0.3. Assess the accuracy of each choice of α and
A public transport organisation has had a reputation for many years of caring little for its customers – the passengers it carries. In particular, the enquiry office has had a particularly poor
A hospital laboratory has the task of screening blood samples of patients and producing a summary report for clinicians. Because of the limitations of equipment, staff training and time, the
A large manufacturing firm is concerned about lost production: that is, production capability that was not utilised for a variety of reasons. One of the causes of such lost production has been
Return to Exercise 10 in Chapter 4. Test to see whether there is a significant difference between mean customer spend in the two stores. Draft a short report explaining your conclusion in
In Chapter 1 reference was made to a survey undertaken by the author into the use of quantitative techniques by businesses. One of the questions in the survey asked whether businesses were using such
The author has been involved in research into the use of something known as compensation orders in Scottish courts (The Use of the Compensation Order in Scotland, J Hamilton and M Wisniewski,
An airline operating between London and Aberdeen knows that a large part of its customer base represents business executives. The airline is considering offering a limousine service tofrom the
Recent legislation requires adherence to detailed and complex food-labelling requirements on products sold to consumers. A recent survey examined whether the food labelling on supermarket own-brand
An international company routinely rotates its senior managers every five to six years around its activities worldwide. Staff are financially compensated for such moves but with many staff buying a
A company employs largely part-time staff, and in the past has found staff turnover to be 23 per cent each year (that is 23 per cent of staff in any one year leave the company). This is felt to be
The Inland Revenue has introduced a self-assessment system for tax payments whereby citizens themselves calculate the amount of tax to be paid on income each year. Naturally, the Inland Revenue

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