ψ1 and ψ2 are prices. What do they price? ψ1 and ψ2 are state prices.Actually it also can be interpreted as the probabilities of up and down movement in the price of the underlier asset. The no- arbitrage price of the derivative is decided by these two prices. If we already got the price for the derivative and no ψ1 and ψ2 can satisfy the equations in question 2 that’s mean arbitrage opportunities exist.
What is the essential condition that ψ1 and ψ2 must satisfy to prevent arbitrage opportunities between the securities in a state pricing representation?
Here we assume :
B(t) is the price of riskless asset at time t
S(t) is the price of the underlier asset at time t f (t) is the price of the derivative at time t
1+r is the payoff from riskless asset in state 1and 2 at time t+1
S1(t+1) is the price of the underlier asset in state 1 at time t+1
S2(t+1) is the price of the underlier asset in state 2 at time t+1 f1(t+1) is the price of the derivative in state 1 at time t+1
F2(t+1) is the price of the derivative in state 2 at time t+1
We can derive three three equations from the matrix multiplication above.
B(t) = (1+r)ψ1 + (1+r)ψ2
S(t) =S1(t+1)ψ1+ S2(t+1)ψ2 f(t) =f1(t+1)ψ1+f2(t+1)ψ2
If ψ1and ψ2 can be found , then there is no arbitrage opportunity.When the solutions of these three equations don't exist that mean there is arbitrage opportunity.
The current traded price of an underlier is $10, and it will be either $5 or $15 at the end of the period. Riskless borrowing and lending for the period is 10% p.a. A European vanilla Put option on the underlier with strike price of $9 is trading in the market. What is the no-arbitrage value of the European vanilla Put option?
S(t)=$10, S1(t+1)=$5, S2(t+1)=$15,then f1(t+1)=4, f2(t+1)=0
We assume B(t) which is risk risk free asset at time t is 1, then
B(t) = (1+0.1)ψ1 +
