Econ task
econ代写 | 经济代写 | 金融代写 – 这是一个经济方面的practice, 考察经济概念的理解, 包括了经济等方面
Final Review
- We construct a binomial tree with 3 periods for the parametersr= 0 .25,d= 0.5 andu= 2 and for the current stock priceS 0 = 8, for computing the priceC 0 of a call option at time 0 with payoff
C 3 =S 3 min
i{ 0 , 1 , 2 , 3 }
Si
Please considerra simple interest rate
(a) Specify the risk neutral probability weights.
(b) Compute the priceC 0 of the call at time 0
(c) Give also the hedging strategy 0 at time 0.
(d) Is there any arbitrage opport unity in this models? Justify your
answer.
- Starting at some fixed time, letS(n) denote the price of a certain secu- rity at the end ofnadditional weeks,n1. A popular model for the evolution of these prices assumes that the prices assumes that the price ratios
S(n)
S(n1)
forn1 are independent and identically distributed
lognormal random variables. Assuming this model, with lognormal
parameters= 0.0165 and= 0.0730, what is the probability that:
(a) the price of the security increases over the next week.
(b) the price at the end of two weeks is higher than it is today?
- Show that ifXnsatisfies the recursion equation
Xn+1=Xn+Vn;X 0 = 0
andVnis a process where valuesVn(ti) andVn(tj),ti 6 =tjare uncorel-
lated, thenXnis a Markov sequence.
- W(t), t 0 is a standard Brownian motion process. Show that {Y(t),t 0 }is a martingale when
Y(t) =W^2 (t)t
What isE(Y(t))?
- Letrandfdenote the domestic and foreign risk-free rates, respectively. LetStbe the exchange rate. that is the price of a unit of foreign cur- rency in terms of the domestic. Assume a geometric Brownian motion forSt, dSt= (rf)Stdt+StdWt, whereW is a standard Brownian motion under a measureP.
(a) Show thatSt=S 0 e(rf
(^2) /2)t+Wt (b) Is the process Steft S 0 ert =eWt (^2) t/ 2 a martingale underP?
- Suppose thatV(t) is a normal process withE(t) = 3 and C(t 1 ,t 2 ) = 4e^0.^2 |t^1 t^2 |. Find the probability thatV(7)<2.
- LetW(t) be a Wiener process. Which of the following processes is also a Wiener process:
(a) U(t) = 3W(t)
(b) V(t) =W(t) + 4
- The price of a share of stock,S(t), satisfies the Stochastic Differential Equation dS(t) =rS(t)dt+S(t)dW(t) whereW is a standard Brownian motion,r >0 is the risk-free rate (short rate) and >0 is the volatility coefficient. Next, we consider the following strangle that pays at the maturity date the payoff
C(T) =
A-S(T), S(T)< A;
0 , AS(T)B;
S(T)-B, B < S(T).
where 0< A < B
(a) Give a closed form formula for the stock price at timeT, in terms
of the stock price at time t,S(t).
(b) Write the risk-neutral pricing formula of the pricec(t,s), at time
t, for the current stock priceS(t) =s, of the strangle.
(c) Derive an analytical formula for the pricec(t,s) of the strangle.
- LetXt=eg(t)[eW(t)+eW(t)] whereg(t) is a function oft.
(a) Use the Ito formula to computedXt
(b) Find all functionsg(t) for whichXtis a martingale.
- LetN(t) be a Poisson Process with rate, (E[N(t) =t). Fort > S, computeP(N(t)> N(s)).
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