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Stochastic Foundation of Finance (FIN 538) Master of Finance Program

Fall 2020 Sample Exam (Duration: 120 min)

金融Final exam代考 Using Ito’s lemma, compute the following Ito integral (that is,give it an expression in term of some Riemann integral over time

Question 1 (20 points in total):  金融Final exam代考

1a. (10 points) Let X_t be a stochastic process

dX_t = e^Xt dt +  log (X_t)dB_t;      t ∈ [0, T ]

where B_t is a Brownian motion. Using Ito’s lemma, compute the following Ito integral (that is,give it an expression in term of some Riemann integral over time, plus some elementary functions of processes X and B)

1b. (10 points) Suppose that the process X_t = e^at+Bt is a martingale. Find a and calculate E[X_t|F_s], s < t for that value of a.

Question 2 (30 points in total):  金融Final exam代考

Let’s consider a world with three dates t = 0, 1, 2. Refer to the graph below. Consider a stock that pays nothing at t = 1, and st t = 2 pays: S_uu = $8 (if state at t = 2 is uu); S_ud = $4 (if state at t = 2 is ud);  S_dd = $0 (if state at  t = 2 is dd).  The  price at t = 0 of this stock is S₀ = $4. The risk-free rate from t = 0 to t = 1 is r₀ = 4%.  If state  at t = 1 is u, the risk-free rate will be r_u = 20%.  If state at t = 1 is d, the risk-free rate will be  r_d = 10%. The risk neutral probabilities are given the graph for the corresponding transitions.

2a. (10 points) Compute the value S_u and S_d of this stock at t = 1

2c. (10 points) Compute the risk-neutral probability x for the transition from t = 0 to state u at t = 1.

2d. (10 points) In the same market, compute the current (at t = 0) price P₀ of a zero-coupon bond of face value  of $132 (that is,  bond pays nothing at t = 1,  and pays $132 in any  state at   t = 2).

Question 3 (40 points in total): Consider a risky firm whose pre-tax value follows the stochas- tic process  金融Final exam代考

Vt = V0e0.5t+Bt

where B_t is a Brownian process. Let’s define effective tax τ_t such that, at any time t, one dollar of pre-tax value corresponds to (1 − τ_t) dollars of after-tax to the firm owner. Let’s assume that the effective tax is random and follows the process

τ = 1 − e(µ− 1 σ²)t+σBt

3a. (10 points) Express after-tax value S_t of the firm at time t in term of pre-tax value V_t and effective tax τ_t.

3b. (10 points) Using Ito’s lemma, express dV_t, and dτ_t in terms of dt and dB_t. 3c. (10 points) Compute the after-tax expected return rate of the firm at time t.

3d. (10 points) Compute the after-tax conditional volatility of the firm at time t. Give economic explanation for all terms in this conditional volatility.

Question 4 (10 points). Calculate V ar(∫ t s²dB ).

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