TH6154财政计算方法论

本课程是金融数学导论，是三门课程中的第一门模块（以及金融数学II和资产管理人的数学工具-共同为现代的理论和实践打下了坚实的基础金融数学。虽然本课程主要涉及离散金融数学时间，FMII将在连续时间内涵盖金融数学，MTAM将专注于风险管理和投资组合理论。那么，金融数学究竟是什么呢？一个简单的定义是它是用途金融工具建模和定价的数学技术。这些财务仪器TH6154: Financial Mathematics 1Term 1, 2022-23

1.Preface

This course is an introduction to financial mathematics, and is the first in a series of threemodules (along with Financial Mathematics II and Mathematical Tools for Asset Man-agement1) that together give a thorough grounding in the theory and practice of modernfinancial mathematics. While this course primarily deals with financial mathematics in discretetime, FMII will cover financial mathematics in continuous time, and MTAM will focus onrisk management and portfolio theory.So what exactly is financial mathematics? A simple definition would be that it is the useof mathematical techniques to model and price financial instruments. These financialinstruments include: Debt (also known as ‘fixed-income’), such as cash, bank deposits, loans and bonds; Equity, such as shares/stock in a company; andDerivatives, including forwards, options and swaps.As this definition suggests, there are two central questions in financialmathematics:1Formally known as Financial Mathematics III.2What is a financial instrument worth? To answer this, we will develop principled waysin which to:Compare the value of money at different moments in time; and Take into account the uncertainty in the future value of an asset.2. How can financial markets usefully be modelled? Note that modelling always involves atrade-off between accuracy and simplicity.This course introduces you to the basics of financial mathematics, including: An overview of financial instruments, including bonds, forwards and options. An analysis of the fundamental ideas behind the rational pricing of financial instruments,including the central ideas of the time-value of money and the no-arbitrage princi-ple. These concepts aremodel-free, meaning that they apply irrespective of the chosenmethod of modelling the financial market; An introduction to financial modelling, including the use of the binomial model toprice European and American options.The course culminates in a derivation, via an approximation procedure, of the celebratedBlack-Scholes formula for pricing European

2.options

Interest rates and present value analysisThe value of money is not constant over time – a thousand pounds today is typically worthmore than a contract guaranteeing ￡1,000 this time next year. To explain this, consider that If ￡1,000 was deposited into a bank account today, it would accumulate interest by thistime next year; Inflation may reduce the purchasing power of ￡1,000 over the course of the year.In this chapter we explore how to take into account interest rates and inflation when comparingassets that generate cash at different moments of time, using the concept of present value;this gives us a way to make a principled comparisons between investments. Later in the coursewe will use these ideas to assign a fair price to fixed-income securities,2 such as bonds.An important theme in this chapter is the notation of standardisation, i.e. adjustinginterest rates and rates of return so that different rates can be compared fairly.1.1 Interest ratesSuppose we borrow an amount P , called the principal, at nominal interest rate r. Thismeans that if we repay the loan after 1 year, we will need to repay the principal plusanextrasum rP called the interest, so in totalP + rP = P (1 + r).interest rate r, then in a year’stime the account value will grow to P (1 + r). Note that here we have assumed that thenominal interest rate is annualised, meaning that interest is calculated once per year; unlessexplicitly specified otherwise interest rates will always be assumed to be annualisedin this course.Sometimes the interest is not calculated once per year, but instead is compounded every1nth of a year. This means that every 1n-th of a year you are charged (or, in the case of a bankaccount, gain) interest at rate r/n on the principal as well as on the interest that has alreadyaccumulated in previous periods. Continuing the example of the loan above, this would meanthat after one year we would owe. (2)If the interest rate is ‘annualised’ then it is compounded annually (i.e. once per year), whichcorresponds to n = 1 and gives the same result as in (1).Example 1.1. Suppose you borrow an amount P , to be repaid after one year at interestrate r, compounded semi-annually. Then the following will happen sequentially throughoutthe year. After half a year you will be charged interest at rate r/2, which is added on to theprincipal. Thus, after 6 months you owePWe can generalise the notion of present value to a cash-flow stream a = (a1, a2, . . . , an)that pays ai at the end of year i, for i = 1, . . . , n. The present value.To see this, observe that the cash-flow stream a = (a1, a2, . . . , an) can be replicated byfirst splitting the stream into the individual payments a1, a2, . . ., and then depositing thecorresponding amounts PV (a1), PV (a2), . . . needed to replicate these payments. SincePV (a) = PV (a1) + PV (a2) + . . .+ PV (an),the total amount you need to deposit to replicate the cash-flow stream is PV (a).Remark 1.13. This is our first example of an argument that uses the idea of replicationto assign a value to a financial instrument. Later in the course we will formalise this type ofargument by using the no-arbitrage assumption.Example 1.14. You are offered three different jobs. The salary paid at the end of each year(in thousands of pounds)

A 32 34 36 38 40
B 36 36 35 35 35
C 40 36 34 32 30

Which job pays the best if the interest rate is (i) r = 10%, (ii) r = 20%, or (iii) r = 30%?4The term net present value (NPV) is sometimes used when discussing cash-flow streams as opposed toa single payment, but we will not use this term.10Solution. We shall compare the present values of the cash-flow streams. The present valuefor job A isWTo see this, observe that the cash-flow stream a = (a1, a2, . . . , an) can be replicated byfirst splitting the stream into the individual payments a1, a2, . . ., and then depositingthecorresponding amounts PV (a1), PV (a2), . . . needed to replicate these payments. SincePV (a) = PV (a1) + PV (a2) + . . .+ PV (an),the total amount you need to deposit to replicate the cash-flow stream is PV (a).Remark 1.13. This is our first example of an argument that uses the idea of replicationto assign a value to a financial instrument. Later in the course we will formalise this type ofargument by using the no-arbitrage assumption.Example 1.14. You are offered three different jobs. The salary paid at the end of each year(in thousands of pounds)

A 32 34 36 38 40
B 36 36 35 35 35
C 40 36 34 32 30

Which job pays the best if the interest rate is (i) r = 10%, (ii) r = 20%, or (iii) r = 30%?4The term net present value (NPV) is sometimes used when discussing cash-flow streams as opposed toa single payment, but we will not use this term.10Solution. We shall compare thepresent values of the cash-flow streams. The present valuefor job
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