Options Lattices and Cox Ross Rubinstein Models

Question 1

Given Values at t=1 of all states how to find constituents of portfolio

Answer 1

Value at t=1 matrix in different states * Inverse of value of bond and shares at t=1 in different instances

Question 2

Value of put option and call option on expiry

Answer 2

Put: strike-spot
Call: spot-strike

Question 3

How does a change in probability of entering a state impact the replication argument for finding option values

Answer 3

Nothing changes. For two states replication argument works in both with probability of 1 (when probabilities add to 1) so the individual probabilities of the two states do not enter into the calculation or determining the value at time 0 of an option

Question 4

F under put call parity

Answer 4

Forward value of share = to the value fo the share rolled up at risk free rate and accunting for dividends yield at a future time point

Question 5

How to check if option prices satisfy put call parity

Answer 5

P-C=Z*(K-F)

Question 6

AD prices calc

Answer 6

Future value in future states matrix inverse * PV of assets now according to rf or q

Question 7

Why for high strikes might European and American call options have the same price by a binomial model

Answer 7

Optimal early exercise occurs when spot>strike price of option by a long way
The theoretical optimal exercise boundary here is above the top fo the lattice so the lattice does not detect it so prices resulting are the same.
With BScholes limit of closer lattice points an dsteps a difference should appear as American calls should still be strictly more valuable than European, even by a little bit. However BS only used for european

Question 8

Whats a bermudian option

Answer 8

Can be exercised early but only one specific dates

Question 9

Why does PC parity not relate to american options

Answer 9

In theory we would not expected american options to satisfy PC parity as the argument relies on options being exercised at maturity only

Question 10

Why is american options always more valuable than european one

Answer 10

Assuming optimal exercise of an american option is must at least be as valuable as the corresponding european options because the american option has an additional right to exercise early if the borrower wills it.

Question 11

Price call and put option 15% out of the money

Answer 11

Can assume the market level without loss of generality generally for Qs. So say market level is 100 then both options would have a strike of 85

Question 12

How to find price defaltors

Answer 12

Kernel of state/probability of being in state

kernel is arrow debreu prices

Question 13

How should the price deflator behave?

Answer 13

Should be a decreasing function of market level, representing a marginal utility. If this didn’t hold and there was higher deflator in up state than mid state might suggest investors are not maximizing utilities or an error in calculation

Question 14

Continue from Q160

Answer 14

—-need to add in—–

Question 15

Arrow and debreu prices goal

Answer 15

To get payoff of 1 in upstate for ADup and payoff of 1 in downstate for ADdown : general premise

Question 16

Define deflator

Answer 16

ZCB price * risk neutral probability for a state

Question 17

Whats the difference between real world vs risk neutral world

Answer 17

No risk premium so current share value si discoutned at risk free rate. Implied in risk neutral law that investors are indifferent to levels of risk and values of cash flows are purely based on their expectations.

Risk neutral probabilities used for pricing and tell you nothing about likelihood of how much you expect to earn by investing. In real world amount earned by investment would increase as you take extra risks investing in shares.

Question 18

Arbitrage constraints in binomial model

Answer 18

Forward exp(r-q)h lies between d and u
ensures positive pricing kernels and correct range for the risk neutral probabilities

Question 19

Capital indices vs total return idnex

Answer 19

Capital indices dividends are ignores, generally bank pockets dividends. Total return index assumes dividends immediately reinvested

Question 20

What is a power option

Answer 20

Derivative whoe payoff is based on a power of the share price

Question 21

Vt-1 inductive formula for european options

Answer 21

kdVd+KuVu

Question 22

Why would we use utility theory to price options and what additional assumptions are needed

Answer 22

Use if replicating protfolios do not work - example incomplete market. Need additional assumptions:
Real world probabilities
Utility function of average agenct
Need agents optimal investment portfolio to have a positive solution for all assets

Question 23

What are the first and second fundamental theorems of asset pricing

Answer 23

1. Discrete market is arbitrage free iff there is one risk netural probability law ie: existence of AD prices
2. Arbitrage free market is complete iff there is a unique risk neutral law ie. uniqueness of AD prices

Question 24

Whats a check when valuing options

Answer 24

Always ensure put call parity holds

Question 25

Relate knock in, knock out and vanilla

Answer 25

Knock in option = vanilla option-knock out option

Question 26

Whats a knock out structure

Answer 26

Contract automatically canclled if price of underlying falls in certain range

Question 27

What’s a knock in structure

Answer 27

If underlying asset price falls into a certain range a standard or vanilla call or put option comes into being.

Question 28

When might an American call or put option be exercised - really generally

Answer 28

Call: if asset has massive dividend yield or if asset value is so large that even small div yield is more lucrative
Put: time value decreases as you get close to expiry.

Question 29

Why is binomial interest rate model more complicated than share model

Answer 29

Many bonds in issue at one time with different terms
Interest rates rise means bond prices decrease so modelling bonds means modelling interest rates

Question 30

If yield curve was flat for each valuation date what arbitrage opportunity woudl exist

Answer 30

Hold long and short bonds and sell the medium term so every time yield curve moved between valuation dates you get a free lunch

Question 31

What is m in recurrence relation for bond lattices

Answer 31

M in number of upsteps in bond prices (ie downsteps in interest rates)

Question 32

Relate ZCB to continuously compounded yield

Answer 32

Exp(yield t*n)=ZCBt

Question 33

How do we get that the binomial tree recombines?

Answer 33

As a result of assumption of parallel yield curve spacing of g and continuously compounding yields. - no realistic as yield curve would move up and down in reality but more complex model which would fix this issue wouldn’t recombine so.