# econ代写 | 经济代写 | 金融代写 – 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
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1. 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 }
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``````Si
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``````Please considerra simple interest rate
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``````(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
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1. 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)
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``````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:
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``````(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?
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1. Show that ifXnsatisfies the recursion equation
``````Xn+1=Xn+Vn;X 0 = 0
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``````andVnis a process where valuesVn(ti) andVn(tj),ti 6 =tjare uncorel-
lated, thenXnis a Markov sequence.
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1. 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
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``````What isE(Y(t))?
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1. 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
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(^2) /2)t+Wt (b) Is the process Steft S 0 ert =eWt (^2) t/ 2 a martingale underP?

1. 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.
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
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1. 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

#### S(T)-B, B < S(T).

``````where 0< A < B
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``````(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.
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``````(c) Derive an analytical formula for the pricec(t,s) of the strangle.
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1. 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.
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1. LetN(t) be a Poisson Process with rate, (E[N(t) =t). Fort > S, computeP(N(t)> N(s)).