Answered step by step
Verified Expert Solution
Question
1 Approved Answer
Exercise 4 (A 1-step 3-state model). Consider a 1-step 3-state model, where there are three possi- ble time evolution w1,W2,W3 and a stock with current
Exercise 4 (A 1-step 3-state model). Consider a 1-step 3-state model, where there are three possi- ble time evolution w1,W2,W3 and a stock with current price So = 100 and time-l price Swi) = 120, Siwi) = 110, S (W) = 90. Assume one-time interest rate r = 4% during (0,1). Now consider a European call option on this stock with strike price K = 110. (1) Consider the following portfolio [4, shares of the stock] + [Short one call option with strike 110 and maturity 1). (4) Can one find A, such that the above portfolio is perfectly hedged? (ii) Consider the following portfolio [4, shares of the stock] + [x cash). (5) Can one find Ao and x such that the above portfolio replicates one long call option? (iii) Write down the (2 x 3) profit matrix whose first and second row corresponds to one share of the stock and one long call option, respectively. Show that there exists a risk-neutral probability distribution p* = (P1, P2, P3), but it is not necessarily unique. (iv) Based on (i)-(iii), how should one price the call option at time t = 0? Is it possible at all? Give your reasoning S.W) = 120 So = 100 -S.(W) = 110 SW) = 90 T = 4% t=0 t=1 FIGURE 1. A 1-step 3-state model Exercise 4 (A 1-step 3-state model). Consider a 1-step 3-state model, where there are three possi- ble time evolution w1,W2,W3 and a stock with current price So = 100 and time-l price Swi) = 120, Siwi) = 110, S (W) = 90. Assume one-time interest rate r = 4% during (0,1). Now consider a European call option on this stock with strike price K = 110. (1) Consider the following portfolio [4, shares of the stock] + [Short one call option with strike 110 and maturity 1). (4) Can one find A, such that the above portfolio is perfectly hedged? (ii) Consider the following portfolio [4, shares of the stock] + [x cash). (5) Can one find Ao and x such that the above portfolio replicates one long call option? (iii) Write down the (2 x 3) profit matrix whose first and second row corresponds to one share of the stock and one long call option, respectively. Show that there exists a risk-neutral probability distribution p* = (P1, P2, P3), but it is not necessarily unique. (iv) Based on (i)-(iii), how should one price the call option at time t = 0? Is it possible at all? Give your reasoning S.W) = 120 So = 100 -S.(W) = 110 SW) = 90 T = 4% t=0 t=1 FIGURE 1. A 1-step 3-state model
Step by Step Solution
There are 3 Steps involved in it
Step: 1
Get Instant Access to Expert-Tailored Solutions
See step-by-step solutions with expert insights and AI powered tools for academic success
Step: 2
Step: 3
Ace Your Homework with AI
Get the answers you need in no time with our AI-driven, step-by-step assistance
Get Started