Nemeth Revision Notes

7/29/2019 Nemeth Revision Notes

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MSM 3G11/MSM4G11Mathematical Finance revision

1 Introduction1.1 Stocks, shares and equities

What is a share?You should be able to explain what is a share,

What is a value of a share?You should be able to explain what is the actual value of a share and how the valueof a share is influenced by paying dividends to shareholders.

1.2 Supply and Demand

You should be able to explain how the value of shares are determined by the laws ofsupply and demand.

1.3 Long selling and short selling

What are terms long and short refer to. You should be able to explain the practice ofshort selling and the motivation behind it.

1.4 ArbitrageHow the laws of supply and demand prevent an instantenous risk free profit?

You should be able to explain what is the notion of arbitrage, or equivalently why isimpossible to make an instantenous risk free profit.

You should be able to use the notion of arbitrage when needed (for example when youdeduce the Black-Scholes equation).

1.5 Risk

What is risk?You should be able to explain what is the notion of risk.What are the different types of risk?You should be able to explain what is the difference between specific and non-specific

risk and give examples.

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1.6 Risk-free investments and arbitrage

You should be able to explain what is a hypothetical risk-free investment, when shouldyou invest in such and investment and when should you keep your money in the bank.You should be able to explain how the market forces (based on the laws of supply and

demand) make your hypothetical profit to be equal to the profit obtained by depositingmoney in the bank.

1.7 Hedging

The term hedging is used to refer to investment strategies that are designed to reduceor minimise the risk to a portfolio of investments. You should be able to explain this inmore detials and give examples for hedging strategies.

2 What are Financial Options?

You should be able to explain what are options in general, what are the European Call andPut options and what is the difference between a European Call Option and a EuropeanPut Option.

You should understand and be able to explain what is the value cost of an option.You should understand and know the pay-off functions for all of the European Options

considered in this chapter.You should also be able to draw the pay-off diagrams for these options, or any other

European option whose pay-off function is given.

3 Random walk model of asset prices

It is very important to know the stochastic differential equation

dS= S(dX+ dt) .

You should be able to explain the meaning of this equation and what happens when = 0.You should be able to explain what is a random walk and give numerical examples for

random walks. You should be able to understand (NOT TO DRAW) the correspondingdiagrams for random walks. However, you WILL NOT BE ASKED any MatLAB CODES,or other computer codes for the exam.

You should know the following two approximations for the drift and volatility:

m = 1ndt

n1i=0

Si+1 SiSi

and

2 2 = 1(n 1)dt

n1i=0

Si+1 Si

Si mdt

2

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but you will NOT be asked to derive them.NOTHING will be asked FROM the section 3.1 The size of random fluctua-

tions.You will be asked to understand, be able to deduce and to use the Black-Scholes

equation:

1

22S2

2V

S2+V

t rV + rSV

S= 0.

For example you have to use Black-Scholes equation for deducing the boundary condi-tions for the European vanilla put. Another example where you have to use Black-Scholesequation is the Put-call parity for European options.

The full deduction of Black-Scholes equation cannot be asked (because its length), butparts of it can be asked. However, you will not be asked to justify why (dX)2 can bereplaced by dt.

You have to MEMORIZE the Black-Scholes equation. In general you have to MEMO-RIZE EVERY FORMULA of the lecture notes UNLESS YOU ARE EXPICITLY TOLDin the revision notes that a given formula does NOT have TO be MEMORIZEd. However,in case of some problems some of the formulas might be given to you.

You should be able to discuss the Black-Scholes equation and the corresponding DeltaHedging strategy (section 3.4).

You are not required to know how to deduce the diffusion equation from the Black-Scholes equation using transformation of variables (and fuction), but you are required toknow these transformations. Note that this deduction was Question 1(*) from ExerciseSheet 4. None of the problems from the exercise sheets denoted by (*) will be asked.

The diffusion equation has the solution

u =1

2

u0 (s)exp

1

4(x s)2

ds

It is important to know this formula. Based on this formula you can deduce the valueof several European options. This means that you can be asked to solve the Black-Scholesequation for all the considered European options, or other options whose pay-off functionis given. This means here that you have to be able to deduce u0(s) for these options.

The value of an European vanilla call equalsSN(d1)

Eexp(

r(T

t))N(d2) (see problem sheet 4, question 3(*)), where

N(q) =12

q

exp

1

2s2ds,

d1 =log(S/E) + (r + 2/2)(T t)

T t

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and

d2 =log(S/E) + (r 2/2)(T t)

T t

.

You DONT HAVE TO MEMORIZE the latter formulas for the value of an Europeanvanilla call (or similar formulas for an European put). You are NOT REQUIRED TODEDUCE these formulas neither, but you might be asked to use these formulas.

4 Solving The Black-Scholes Equation

All formulas and proofs of this chapter can be asked. The chapter include the followingtopics:

Boundary and final conditions of European vanilla call and put options,

Put call parity for European options,

Continuous and discrete dividends,

Forward contracts and futures

Next, some more details:

Please note that for deducing the boundary conditions you need to use both thestochastic equation from the random walk model, the Black-Scholes equation and thepay-off functions.

You should know the portfolio V = S+ PC corresponding to a put-call parity andthe value S+ P C= Eer(tT) of the portfolio. In fact this latter equation is called theput-call parity. You are required to understand and deduce this equation.

You should be able to compare the put-call parity and the delta hedging.You should know the modified version of the Black-Scholes equations for continuous

dividends. You should be able to explain how this equation can be deduced; that is,to show what are the differences of this deduction from the deduction of the classicalBlack-Scholes equation.

You can be asked to transform the modified version of the Black-Scholes equations forcontinuous dividends into the classical Black-Scholes equation for a transformed option

(Exercise Sheet 4, Question 6).You should be able to explain what are forward contracts and futures and what is the

difference between a forward contract and an option.You should be able to deduce the equation

V

t+

1

22F2

2V

F2 rV = 0.

for the value V(F, t) of an option on futures (Exercise Sheet 4, Question 10).

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5 American options

You should be able to explain what is an American option, what is the difference betweenEuropean and American Options and derive the models for an American vanilla put andAmerican vanilla call. It is important to know the boundary conditions.

It is especially important to MEMORIZE Figure 11, which contains the pay-off ofa Vanilla put and the corresponding values of the European and American Vanilla putoptions at some time t.

It is important to know what is the meaning of the optimal exercise boundary. Inparticular cases you can be aked to deduce the optimal exercise boundary (see Question2 of Exercise Sheet 5 for an American perpetual call with a constant dividend yield).

You should be able to rewrite the model for an American vanilla put in linear com-plementarity form using the differential operator for the diffusion equation. You shouldknow the linear complementarity form because it is important for the numerical solutionsof American options.

It is important to know the effect of continuous dividends on American options andthe models for perpetual american options.

6 Summary of main financial models

Chapter 6 is a summary of the main financial models. It is important to MEMORIZE andto be able to deduce all formulas of this chapter. In the cases of dividends and perpetualoptions the classical models should be adapted using sections 5.5.2 and 5.5.3.

7 Numerical MethodsYou should be able to explain and know:

the finite difference approximations,

the numerical grid,

explicit finite difference scheme,

the implicit finite difference schem.

You should be able to compare the explicit and implicit finite difference schemes basedon stability and efficiency. It is important to know the criteria for stability. However, theempirical arguments about stability (section 7.4.1) will not be asked. Computer examplesare not asked neither (section 7.7.2).

You can be asked to work out numerical problems for European and American vanillacalls and puts.

Therefore, it is important to know where the initial conditions are implemented onthe grid and how to use the linear complementarity model for the American options.

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Exercises

You can be asked to solve exercises similar to the ones presented during the lectures andto the ones of the exercise sheets. Questions of the excersise sheets denoted by a (*) WILLNOT BE ASKED.

Important remarks

Anyting which has been omited from this revision, either accidentally or because of lackof time can be asked during the exam. This includes everything presented during thelectures and all questions from the problem sheets exept the ones denoted by a (*).

Questions which are ONLY SIMILAR to the questions from lectures and problemsheets CAN BE ALSO ASKED.

ALL FORMULAS have to be MEMORIZED except the ones you were told not to in

this revision notes.Please check the electronic files of my presentations to see if anything has been omittedfrom this revision.

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