Definitions

Question 1

Risk Free assets

Answer 1

Over the time period considered there is no uncertainty in the future, hence no risk in the return, we model it as Aexp{-r(T-t)}

Question 2

Explain what A, t, r and T mean in terms of a risk free asset

Answer 2

r is the continuous compound interest rate

A is the amount invested at time t

T is the expiry time

Question 3

Risky assets

Answer 3

Stocks or shares usually, traded on the open market. We don’t know what their price will be in the future, so we treat it as random

Question 4

What is arbitrage

Answer 4

An opportunity to make a risk free profit with no initial outlay

This returns a profit regardless of what the asset does in the future

Question 5

No Arbitrage principle (1st version)

Answer 5

There can be no opportunities to make a riskfree profit with no initial outlay.

Question 6

No Arbitrage principle (2nd version)

Answer 6

There can be no riskfree way to get a better rate of return than the bank interest rate r

Question 7

What assumptions are made in the no Arbitrage principles (2 criteria)

Answer 7

i) Traders buy and sell at the same price, so we are ignoring taxes etc.

ii) the market’s adjustment to the pressures of supply and demand does not happen instantaneously

Question 8

Short selling

Answer 8

Here we ‘borrow’ shares and trade with them, but have to then return the same number of shares to their owner

Could mean you have to buy the shares back at the market price later on, could be very expensive

Question 9

Frictionless/Liquidity/Divisibility/Short-Selling Assumption:

Answer 9

An investor can hold any quantity (positive or negative, integer or fractional) of an asset, and at any time can buy and sell any quantity at the prevailing price (with no transaction charges).

Question 10

What is a derivative

Answer 10

Derivative securities are assets which depend on some other (risky) asset, called the underlying asset

Question 11

Give examples of derivatives

Answer 11

Forward contracts & options

Question 12

What is a forward contract

Answer 12

A deal to sell a specified amount of some commodity (shares) at a specified future time T, for a fixed price F.

The contract is assumed to not have any value, though both sides have to buy or sell the agreed contracts at the expiry time, regardless of who wins or loses

Question 13

Give the ‘fair’ price of a forward contract, according to the no arbitrage principle

Answer 13

S_0 exp{rT}, so initial price multiplied by exponential of the product of the interest rate and expiry date.

Question 14

If a forward contract was priced unfairly how could you make an arbitrage opportunities (2 cases)

Answer 14

i) if F>S_0 exp{rT} a guaranteed profit can be made at time T by selling a forward contract and taking out a loan at time 0 to buy the commodity

ii ) If F<S_0 exp{rT} can guarantee profit by buying a forward contract and short selling the commodity at time 0, investing the proceeds risk free until time T

Question 15

What is an option

Answer 15

They are derivative securities that give the holder the option but not the obligation to buy or sell an asset for a specified price at some time in the future

Question 16

Difference between a put and call option

Answer 16

In the call option the person buying the asset gets to choose if they want to, in a put option its the person selling that gets to make the choice at time T

In both cases the person who gets to make the choice is known as the holder of the option, as they are the ones that has paid to have the choice

Question 17

European call option

Answer 17

gives the holder the right to buy one unit of the asset for a price E (the exercise price) at some future time T (the expiry
time).

Question 18

European Put option

Answer 18

gives the holder the right to sell one unit of the asset for the price E at time T

Question 19

What makes American options different to European ones

Answer 19

the holder may choose to exercise them at any time up to T

Question 20

What are the payoff equations for a call option

Answer 20

max(S-E, 0) for final asset price (on the open market) of S and the agreed exercise price E

Question 21

Payoff formula of put option

Answer 21

max(E-S, 0)

Question 22

Put -Call parity formula

Answer 22

P(t) = C(t) + Eexp{-r(T-t)} - S(t)

Question 23

Explain what P, C E, r, T, t and S mean in the put call parity formula

Answer 23

P = value of put option (at t)
C = value of call option
r = compound interest rate
T = expiry time
t = variable time we are interested in
S = market price of the asset

Question 24

What 2 things is it important to understand about the Put-call parity

Answer 24

(i) it does not tell us the actual value of either P(t) or C(t), only how these two are related

(ii) it will be true whatever assumptions we make about the possible future values of the asset price (although these assumptions will affect the actual values of P(t) and C(t)). In particular, it will be true whether we treat the share price as a discrete time stochastic process or a continuous time stochastic process

Question 25

Whats the lower bound for the price of a call option

Answer 25

C(t) >= S(t) - E exp{-r(T-t)}

Question 26

What are the assumptions of the One-Step Binomial Model

Answer 26

The share has initial price S(0) at t=0 and the share price is given at S(T) by:

S(T) = S+ with prob p or S- with prob 1-p

S(^-)exp{-rT} <S(0) <S(^+)exp{-rT}

Question 27

Give an example of a payoff of a One-Step binomial model

Answer 27

To give a specific example, suppose that X is a call option with expiry time T and exercise price E, where S − < E < S+, then X+ = S + − E and X− = 0, since the payoff of the option is max(S(T) − E, 0). Similarly for a put option, we would have X+ = 0 and X− = E − S −. Note that we do not necessarily have X+ > X−

Question 28

State the One-Step Binomial theorem giving the intial value of the option

Answer 28

X(0) = ((S^+X^- - S^-X^+)/(S^+-S^-))exp{-rt} + ((X^+-X^-)/(S^+-S^-))S(0)

Question 29

What is the expected payoff of the one step binomial model

Answer 29

E[X(T)] = pX^(+) + (1-p)X^(-)

Question 30

Whats the expected discounted payoff of the claim

Answer 30

E[exp{-rT}X(T)] = exp{-rT}(pX^(+) + (1-p)X^(-))

Question 31

Why is the expected discounted payoff not the correct price

Answer 31

It depends on probability which can lead to an arbitrage oppourtunity

Question 32

Write the risk neutral probability

Answer 32

p_(*) = (S(0)exp{rT) - S^(-)) / (S^(+) - S^(-))

Question 33

Why is the risk neutral probability importatn

Answer 33

When you use it to calculate the discounted expected payoff of the claim you get the correct arbitrage value of the claim, something you don’t get when using the actual market free value of p

Question 34

What is a stochastic process

Answer 34

A sequence of random variables, X_0, X-1,…. representing observations of some random system at successive times 0,1,…

The probability distribution of the general X_n can depend on n and its history up to n, they can also be finite of infinite

Question 35

Whats the random walk

Answer 35

stochastic process with probability to move forward of backward at each time step, can be written interatively of explicitly

Question 36

How do your represent a random walk with drift iteratively

Answer 36

S_0 = 0;
S_n = S_n-1 + mu + sigmaY_n for n>= 1
with Y_n iid random variables with equal probability of generating {1,-1}

Question 37

Find E[Y_i], Var[Y_i], E[S_n], Var[S_n] for a symmetric random walk with drift

Answer 37

E[Y_i] = 1/2(1^2) + 1/2(-1^2) = 0
Var[Yi] = 1/2(+1)^2 +1/2(−1)^2 − 0^2 = 1.

E[Sn] = nµ + σSum{Y_i} = nmu
Var(S_n) = sigma^2 SumVar(Y_i) = nsigma^2

Question 38

What does central limit theorem say about the distribution of S_n for large n

Answer 38

It will have normal distribution with mean and variance:

S_N ~ N(n mu, n sigma^2)

Question 39

What is a Markov process

Answer 39

A stochastic process which is memoryless, future evolution only depends on the state it is currently in

Question 40

What is the filtration of a stochastic process

Answer 40

F_m is the set of all events which are determined by time m, and the family (F_m)_(m>=0) is a filtration of the underlying probability space

Question 41

What are the three rules of conditional expectation

Answer 41

i ) Time zero rule: E[Z|F_0] = E[Z]

ii ) Tower law: E[E[Z | Fn] | Fm] = E[Z | Fm]

iii) taking out a known factor: If Y is a random variable whose value is already determined by time m then;
E[Y Z | Fm] = Y E[Z | Fm]

Question 42

What is a Martingale

Answer 42

A discreter time stochastic process (X_n)_(n>=0) such that

i ) E[Xn | Fm] = Xm for all n > m
ii ) E[ |Xn| ] < infinity for all n.

Question 43

What does the risk neutral probability do if we consider the one step binomial method as a stochastic process

Answer 43

It makes it a martingale