IEOR 4700 Homework 10: Wednesday April 7 2021 金融工程作业代写 Problem 2. [25 points] Assume the continuously compounded spot rates of Problem 1. Find the value of an FRA that enables the h...View details
E4700 Final Examination
170 points total
金融工程考试代考 You can make use only of your lecture notes but no other sources of information or help in completing this exam. You may use only a hand held
You can make use only of your lecture notes but no other sources of information or help in completing this exam. You may use only a han held calculator. No Excel or any computer math programs are allowed.
Problem 1 [30 points]
Consider a European put option on a stock with expiration 5 months, strike $40. The continuously compounded annual interest rate is 2%. The current stock price is $40, and its annual volatility is 25%.
(a) Suppose the underlying stock pays no dividend. Use the Black-Scholes formula to find the value of the put. [10 points]
(b) Suppose the underlying stock pays a continuous dividend of 1% annually. Use the Black-Scholes formula to find the value of this put option. [10 points]
(c) Suppose the underlying stock pays one dividend of $3 at the end of the second month and one dividend at the end of the sixth month. Use the Black-Scholes formula to find the value of this put option. [10 points]
Note: You should use a table of N(x) at the end of the Formulas sheet to evaluate any cumulative standard normal distribution functions.
Problem 2 [30 points] 金融工程考试代考
A 3-month American call option on the value of the British Pound in US Dollars has a strike price of $1.56 per British Pound. The current exchange rate is $1.58 per British Pound and the annual volatility is 25%. The continuously compounded US interest rate is 3% annually and the continuously compounded British interest is 4% annually.
(a) Use a three-period CRR binomial tree to find the option price at every node and label the nodes at which early exercise occurs. [20 points]
(b) Suppose you sell one option. Describe the hedging strategy if the exchange rate goes up in the first period, up in the second period, and up in the third period. [10 points]
Problem 3. [30 points] 金融工程考试代考
A financial institution entered into an interest rate swap with company X today. Under the terms of the swap, it receives 10% fixed rate and pays 6-month LIBOR on a principal of $10 million.
The payments are made semiannually and the rates are quoted annually with semiannual compounding. The remaining life of the contract is 3 years. Suppose that the discount factors are given as follows:
(a) What is the value of the swap from financial institution’s perspective on the day that the swap is entered into? [20 points]
(b) What is the swap rate in the market today? [10 points]
Problem 4 [30 points] 金融工程考试代考
Consider a stock with price S, with volatility σ , that undergoes geometric Brownian motion with drift μ so that
dS(t) = S(t)[μdt + σdZ(t)]
The riskless return is r, continuously compounded annually. The stock’s dividend yield is zero.
A company issues a European call option with price D(S,t) on the stock S at time t that is a bit different from an ordinary call option: it pays continuous dividends consisting of more identical call options with the same strike and expiration at a continuous rate ε per year. In essence, owning the call pays a dividend of εdt identical calls during every instant dt.
(a) Suppose you want to hedge a long position in this call by shorting stock. Consider the value of the hedged portfolio π(S,t) = D(S,t) – ΔS at time t when the stock price is S. Use Ito’s Lemma to calculate the change in the value of the hedged position when time dt passes, and make the hedge riskless in the usual way. Show that the call satisfies the PDE
Problem 5 [20 points]
is the current price of a European put option with strike 90.
is the current price of a European put option with strike 100.
is the current price of a European put option with strike 130.
All puts have the same expiration. Show that there is a no-arbitrage relation, independent of any model, that requires that
P(100) ≤ ¾P(90) + ¼P(130)
Note: There is no need to use the Black-Scholes formula.
Problem 6. [30 points] 金融工程考试代考
Here is a tree of future one-year rates (in percentage points, annually compounded) from a stochastic interest rate model, starting with 3% today:
(i) Below is the tree of prices of a four-year zero coupon bond B of face value $100 calculated by using the BDT valuation formula, assuming risk-neutral probabilities of 1/2 for up and down moves. Some values already filled in. Complete the tree by finding the values at the nodes with the X. [10 points]
(ii) Find the annually compounded annual yield today corresponding to today’s price for the fouryear zero coupon bond. [5 points]
(iii) Using the price tree for the 4-year bond, find today’s value of a three-year European put P on the 4-year bond with a strike price of $95. [10 points]
(iv) According to the BDT model, how many four-year zero-coupon bonds must you instantaneously own today in order to delta-hedge a long position in one three-year put? [5 points]
For options on stocks that pay dividends, replace the stock price S by the adjusted stock price S*. For a continuous dividend yield d,S* = Se–d(T-t). For discrete cash dividends, S* = S – PV (dividends)
Put-Call Parity on a non-dividend-paying stock:C – P = S – Ke–r(T – t)