CRR model: American put option. Consider the CRR model withT = 2 and S0 = 100, Su1 = 120, Sd1 = 90. Assume that the interest rate r = 0.Consider an American put option with reward process g(St, t) = (Lt − St)+ andvariable strike price L0 = 105, L1 = 116, L2 = 111.(a) Find parameters u, d, the stock price at time T = 2, and a martingale measure˜P on (Ω, F2).(b) Compute the price process P a of this option using the recursive relationshipP at = max{(Lt − St)+, (1 + r)−1(˜pP aut+1 + (1 − ˜p)P adt+1)}with the terminal condition P a2 = (L2 − S2)+.(c) Find the holder’s rational exercise time τ ∗0 .(d) Find the replicating strategy for the option up to the exercise time τ ∗0 .(e) Check whether the arbitrage price process (P at ; t = 0, 1, 2) is a martingale or asupermartingale under ˜P with respect to the filtration F
Question
CRR model: American put option. Consider the CRR model withT = 2 and S0 = 100, Su1 = 120, Sd1 = 90. Assume that the interest rate r = 0.Consider an American put option with reward process g(St, t) = (Lt − St)+ andvariable strike price L0 = 105, L1 = 116, L2 = 111.(a) Find parameters u, d, the stock price at time T = 2, and a martingale measure˜P on (Ω, F2).(b) Compute the price process P a of this option using the recursive relationshipP at = max{(Lt − St)+, (1 + r)−1(˜pP aut+1 + (1 − ˜p)P adt+1)}with the terminal condition P a2 = (L2 − S2)+.(c) Find the holder’s rational exercise time τ ∗0 .(d) Find the replicating strategy for the option up to the exercise time τ ∗0 .(e) Check whether the arbitrage price process (P at ; t = 0, 1, 2) is a martingale or asupermartingale under ˜P with respect to the filtration F
Solution
This is a complex question that involves several steps. Here's how you can approach it:
(a) The parameters u and d can be calculated from the given stock prices. u is the up factor and d is the down factor. In this case, u = Su1/S0 = 120/100 = 1.2 and d = Sd1/S0 = 90/100 = 0.9. The stock price at time T = 2 can be calculated by multiplying the initial stock price by the up and down factors. Since the interest rate r = 0, the martingale measure P~ is simply the risk-neutral measure, which is the probability measure under which the discounted price of the risky asset is a martingale.
(b) The price process P_a of the option can be computed using the recursive relationship given. Start with the terminal condition P_a2 = (L2 - S2)+ and work backwards to find P_a1 and P_a0.
(c) The holder's rational exercise time τ*0 is the time at which the holder should exercise the option to maximize their payoff. This can be found by comparing the payoff from exercising the option at each time t with the expected payoff from continuing to hold the option.
(d) The replicating strategy for the option up to the exercise time τ*0 involves holding a certain number of shares of the stock and a certain amount in the risk-free asset at each time t to replicate the payoff of the option. This can be found by solving a system of equations involving the option price and the stock price at each time t.
(e) To check whether the arbitrage price process (P_at; t = 0, 1, 2) is a martingale or a supermartingale under P~ with respect to the filtration F, you can calculate the expected value of P_at+1 given the information available at time t and compare it to P_at. If the expected value is equal to P_at, then the process is a martingale. If the expected value is less than or equal to P_at, then the process is a supermartingale.
Please note that this is a general approach and the actual calculations will depend on the specific values given in the problem.
Similar Questions
Consider the CRR model withT = 2 and S0 = 100, Su1 = 120, Sd1 = 90. Assume that the interest rate r = 0.Consider an American put option with reward process g(St, t) = (Lt − St)+ andvariable strike price L0 = 105, L1 = 116, L2 = 111.(a) Find parameters u, d, the stock price at time T = 2, and a martingale measure˜P on (Ω, F2).(b) Compute the price process P a of this option using the recursive relationshipP at = max{(Lt − St)+, (1 + r)−1(˜pP aut+1 + (1 − ˜p)P adt+1)}with the terminal condition P a2 = (L2 − S2)+.(c) Find the holder’s rational exercise time τ ∗0 .(d) Find the replicating strategy for the option up to the exercise time τ ∗0 .(e) Check whether the arbitrage price process (P at ; t = 0, 1, 2) is a martingale or asupermartingale under ˜P with respect to the filtration F
A stock is currently trading at $50; its annual volatility is 0.40, the risk-free interest rate is15% per annum with continuous compounding, and ∆t is equal to three months. Use thebinomial model to answer the following questions:i. Calculate the price of a 6-month European put option with an exercise price of $105written on this stock.(5 marks)ii. Calculate the price of a 6-month American put option with an exercise price of $105written on this stock.
Compute the price process P a of this option using the recursive relationshipP at = max{(Lt − St)+, (1 + r)−1(˜pP aut+1 + (1 − ˜p)P adt+1)}with the terminal condition P a2 = (L2 − S2)+
Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to maturity is four months. (a) What is the price of the option if it is a European call? (1 mark) (b) What is the price of the option if it is an American call? (1 mark) (c) What is the price of the option if it is a European put? (1 mark) (d) Verify that put–call parity holds. (1 mark)
Consider the following information for an individual stock: Current share price is S10Risk-free rate is 5% pa compounded continuously Volatility of the stock returns (o) is 30% pa Strike price is S12 Time to maturity of the option is 9 mthsThe firm is expected to pay dividend estimated at S1 per share in 2 mths, S1.40 pershare in 8 mths and S1.80 per share in 14 mths. Use the closed-form Black-Scholes model to price the European put option with the abovecharacteristics 3.96a5.13b)1.25cdNone of the above
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.