Options Lattices and Cox Ross Rubinstein Models Flashcards
Given Values at t=1 of all states how to find constituents of portfolio
Value at t=1 matrix in different states * Inverse of value of bond and shares at t=1 in different instances
Value of put option and call option on expiry
Put: strike-spot
Call: spot-strike
How does a change in probability of entering a state impact the replication argument for finding option values
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
F under put call parity
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
How to check if option prices satisfy put call parity
P-C=Z*(K-F)
AD prices calc
Future value in future states matrix inverse * PV of assets now according to rf or q
Why for high strikes might European and American call options have the same price by a binomial model
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
Whats a bermudian option
Can be exercised early but only one specific dates
Why does PC parity not relate to american options
In theory we would not expected american options to satisfy PC parity as the argument relies on options being exercised at maturity only
Why is american options always more valuable than european one
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.
Price call and put option 15% out of the money
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
How to find price defaltors
Kernel of state/probability of being in state
kernel is arrow debreu prices
How should the price deflator behave?
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
Continue from Q160
—-need to add in—–
Arrow and debreu prices goal
To get payoff of 1 in upstate for ADup and payoff of 1 in downstate for ADdown : general premise
Define deflator
ZCB price * risk neutral probability for a state
Whats the difference between real world vs risk neutral world
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.
Arbitrage constraints in binomial model
Forward exp(r-q)h lies between d and u
ensures positive pricing kernels and correct range for the risk neutral probabilities
Capital indices vs total return idnex
Capital indices dividends are ignores, generally bank pockets dividends. Total return index assumes dividends immediately reinvested
What is a power option
Derivative whoe payoff is based on a power of the share price
Vt-1 inductive formula for european options
kdVd+KuVu
Why would we use utility theory to price options and what additional assumptions are needed
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
What are the first and second fundamental theorems of asset pricing
- Discrete market is arbitrage free iff there is one risk netural probability law ie: existence of AD prices
- Arbitrage free market is complete iff there is a unique risk neutral law ie. uniqueness of AD prices
Whats a check when valuing options
Always ensure put call parity holds
Relate knock in, knock out and vanilla
Knock in option = vanilla option-knock out option
Whats a knock out structure
Contract automatically canclled if price of underlying falls in certain range
What’s a knock in structure
If underlying asset price falls into a certain range a standard or vanilla call or put option comes into being.
When might an American call or put option be exercised - really generally
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.
Why is binomial interest rate model more complicated than share model
Many bonds in issue at one time with different terms
Interest rates rise means bond prices decrease so modelling bonds means modelling interest rates
If yield curve was flat for each valuation date what arbitrage opportunity woudl exist
Hold long and short bonds and sell the medium term so every time yield curve moved between valuation dates you get a free lunch
What is m in recurrence relation for bond lattices
M in number of upsteps in bond prices (ie downsteps in interest rates)
Relate ZCB to continuously compounded yield
Exp(yield t*n)=ZCBt
How do we get that the binomial tree recombines?
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.
