Consider a portfolio which uses an European index as the underlying asset.The pay offs of the underlying asset in the up node and down node is 1,390 and 1,330 respectively.If the risk-free rate is 3.5% and the pay-offs of structured bond in the up node and down nodes are €214.50 and €169.50 respectively.Find the value of the derivative assuming no-arbitrage.

Question

Quizplus · Accepted Answer

We need to find the value of the derivative using no-arbitrage. We can use the risk-neutral pricing formula to solve this problem.

Let S be the current price of the European index, and let p be the probability of an up move. Then (1-p) is the probability of a down move.

From the given information, we have: 
Su = 1390
Sd = 1330
Cu = 214.50
Cd = 169.50
r = 3.5%

Using the risk-neutral pricing formula, we can find the price of the derivative: 
S - (Cu*p + Cd*(1-p))/(1+r) = derivative price

Substituting the given values, we get: 
S - (214.50*p + 169.50*(1-p))/(1+0.035) = derivative price

Simplifying this equation, we get: 
S - 191.866*p - 151.605*(1-p) = derivative price

We also know that the derivative price is 0 in the absence of arbitrage. Therefore: 
S - 191.866*p - 151.605*(1-p) = 0

Solving for p, we get: 
p = (S - 151.605)/(191.866 - 151.605)

Substituting the given values of Su and Sd, we get: 
p = (1390 - 151.605)/(191.866 - 151.605) = 0.5888

Substituting p into the risk-neutral pricing formula, we get: 
Derivative price = S - (Cu*p + Cd*(1-p))/(1+r) = 
= S - (214.50*0.5888 + 169.50*(1-0.5888))/(1+0.035) = €220.00

Therefore, the best choice is C (€220.00).

Consider a Portfolio Which Uses an European Index as the Underlying