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# Mathematics of Finance and Valuation

2020/2021
Academic Year
ENG
Instruction in English
5
ECTS credits
Course type:
Elective course
When:
4 year, 1, 2 module

### Course Syllabus

#### Abstract

This is an introduction to an exciting and relatively new area of mathematical application. It is concerned with the valuation (pricing) of ‘financial derivatives’. These are contracts which are bought or sold in exchange for the promise of some kind of payment in the future, usually contingent upon the share-price then prevailing (of a specified share, or share index). The course reviews the financial environment and some of the financial derivatives traded on the market. It then introduces the mathematical tools which enable the modelling of the fluctuations in share prices. Inevitably these are modelled by equations containing a random term. It is this term which introduces risk; it is shown how to counterbalance the risks by putting together portfolios of shares and derivatives so that risks temporarily cancel each other out and this strategy is repeated over time. As this procedure resembles hedging a bet – that is, betting both ways - one talks of dynamic hedging. The yield of a temporarily riskless portfolio is equated to the rate of return offered by a safe deposit bank account (that is a riskless bank rate) which is assumed to exist; this equation assumes that the market which values shares and derivatives actually is in equilibrium and hence eliminates the opportunities of ‘arbitrage’ (such as making sure profit from, say, buying cheap and selling dear). The no-arbitrage approach implies in the continuous time model that the price of a derivative is the solution of a differential equation. One may either attempt to solve the differential equation by standard means such as numerical techniques or via Laplace transforms, though this is not always easy or feasible. However, there is an alternative route which may provide the answer: a calculation of the expected payment to be obtained from the contract by using what is known as the synthetic probability (or the risk-neutral probability. One proves that, regardless of what an investor believes the expected growth rate of the share price to be, the dynamic hedging acts so as to replace the believed growth rate by the riskless growth rate. Though this may seem obvious in retrospect it does require some careful reasoning to justify. The course considers two approaches to risk-neutral calculation, using discrete time and using continuous time. Continuous time requires the establishment of a second-order volatility correction term when using standard first-order approximation from calculus. This leads to what is known as Ito’s Rule. Finite time arguments need some apparatus from Linear Algebra like the Separating Hyperplane Theorem. We enter the subject from the discrete time model for an easier discussion of the main issues. #### Learning Objectives

• introduce the main mathematical ideas involved in the modelling of asset price evolution and the valuation of contingent claims (such as call and put options) in a discrete and in a continuous framework #### Expected Learning Outcomes

• Explain the basic financial derivative contracts, how and when they are used
• price contingent claims in the model with one risky asset and two random outcomes (states).
• price contingent claims in the model with several risky assets and one period of time.
• explain the concept of a complete financial market and the absence of arbitrage in the one-period model.
• price contingent claims in the model with one risky asset and two random outcomes (states)
• explain the concept of a complete financial market and the absence of arbitrage in a multi-period model
• Apply the required concepts from the theory of random sequences: filtration, adapted and predictable sequences, conditional expectation, martingales, martingale measures.
• apply the general results of the multiperiod model in the binomial model
• explicitly compute the equivalent martingale measure in the binomial model
• explain the concept of a random process, and, in particular, the Brownian motion and the geometric Brownian motion
• apply basic rules of the Ito calculus for stochastic processes
• to apply the Black-Schles formula for European options
• compute implied volatility
• price perpetual American put options in the Black-Scholes model #### Course Contents

• Financial environment
1.1 Present value of future income 1.2 Bonds, stocks, derivative securities 1.3 Arbitrage arguments 1.4 Hedging
• One risky asses and two states
2.1 The simplest market model 2.2 Valuation of forward and futures contracts 2.3 Valuation of options 2.4 Valuation by expectation
• One period many assets
3.1 Description of the model 3.2 The notion of arbitrage 3.3 Fundamental Theorem, of Asset Pricing
• Multi-period models
4.1 Information trees and related notions 4.2 Valuation of contracts in the multiperiod models 4.3 Arbitrage in the multiperiod model 4.4 Risk-neutral measures
• The binomial model
5.1 The T-period binomial model 5.2 Valuation of different contracts
• Continuous-time modelling
6.1 The discrete random walk and Brownian motion 6.2 Stochastic differential equations 6.3 Itô’s formula
• Black-Scholes model
7.1 Geometric Brownian motion as a stock price model 7.2 The Black-Scholes equation 7.3 The Black-Scholes formula for option prices
• Perpetual options
8.1 The notion of a perpetual option 8.2 The Cauchy-Euler equation and the value of a perpetual option #### Assessment Elements

• examination
• home assignments
• UoL exam
UoL exam is not counted in overall grade
• October test
• class activities #### Interim Assessment

• Interim assessment (2 module)
0.29 * home assignments + 0.01 * class activities + 0.5 * examination + 0.2 * October test #### Recommended Core Bibliography

• Hull, J. C. (2017). Options, Futures, and Other Derivatives, Global Edition. [Place of publication not identified]: Pearson. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1538007

#### Recommended Additional Bibliography

• Stochastic calculus for finance. Vol.1: The binomial asset pricing model, Shreve, S. E., 2004
• Stochastic calculus for finance. Vol.2: Continuous-time models, Shreve, S. E., 2004