Consider a two-step binomial tree, where a stock that pays no dividends has current price 120, and at each time step can increase by 20% or decrease by 10%. The possible values at times T = 2 are thus 144, 108 and 81. The annually compounded interest rate is 5%. a) Write down the value of the money market account Mm at all states of the tree
Consider a two-step binomial tree, where a stock that pays no dividends has
current price 120, and at each time step can increase by 20% or decrease by 10%. The
possible values at times T = 2 are thus 144, 108 and 81. The annually compounded interest
rate is 5%.
a) Write down the value of the money market account Mm at all states of the tree
b) By using the martingale condition for Sm/Mm, find the risk-neutral probabilities with
respect to the money market numeraire Mm at each node of the tree.
c) By using the the martingale condition for Z(m, 2)/Mm show that
Z(0, 2) =
65
63 11.12 .
d) Use (c) and an appropriate martingale condition to prove that the risk-neutral probability,
with respect to the numeraire Z(m, 2) of the stock having value 120 at T = 1 is 44/65.
Hence show that the risk-neutral probabilities of this state, with respect to the money
market account and the ZCB with maturity T = 2, differ by 2/195. Do you want to
revisit your comments in Question 2 (b)?
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