Week 3 Flashcards
Definition of arbitrage in multiperiod model
Self financing portfolio definition
A self financing portfolio is a sequence of positions in 2 assets {α} and {β} which does not require any injection of value after time 0
No arbitrage in multiperiod binomial 2 asset model requires
Which must hold for every single 1 period sub tree
Introduce a probability measure for a multi period 2 asset binomial model
Where n is time subscript and m is value subscript
Cn / Bn = ?with prob measure defined by B
define a stock under Risk neutral measure with 1 step
Denoted Q and Using MMF as numeraire
With S is stock
Denote MMF
Growing with deterministic rate r
We have
No arbitrage for risk neutral measure (both ways)
How to choose u, d and p for risk neutral measure
Moment matching
Or you can derive u, d and p from the E and V
And u*d = 1
Derive EQ[s1/s0] and VQ
where we enforce this E and V (for some reason)
Introduce a 3rd asset C (with payoff Ψ(SN at time T) to the risk neutral model with a stock S
express expectation of c/m at time N given information up to time n
Find C0 using the martingale condition
Taylor expansion of p and (risk neutral prob) q up to deltaT term
Derive EP[σ sqrt(Δt) sum(xk) ]
if xn = -1 this is a down branch, equiv for + 1 we arent using clt for triangular arrays until we combine both var and E
CLT for TA
converges in distribution and under measure P
Use below to derive BS European call option formula
Fair value of one call and one short (European) under BS?
Value of portfolio?
call put parity?
