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?
Compute risk neutral branching probability
In the Sn/Mn = qS(n+1) + (1-q)S(n+1)
