# How to solve these SDE Problems

Quuestion1.

I make a solution $$r(t)$$ used by Ito's lemma

$$r(t)=e^{-a t}r(0)+\int _{0}^{t}e^{a (s-t)}\theta (s)ds+\sigma e^{-a t}\int _{0}^{t}e^{a u}\,dB^{1}(u)$$

Is this right?

and I try to make solution of $$P(t)$$ and $$X(t)$$.But because of my lack of understand, I couldn't solve that.

Question2.

I cannot understand problem not at all. what is mean of $$\frac{Z_T}{\beta(T)}|F_t$$ and $$\frac{W_T}{\beta(T)} | F_t$$

Question 1.

Let $$(Ω, F, P)$$ be a complete probability space, and let $$B (t) = (B^1 (t), B^2 (t))$$, $$t \in [0, T]$$ be a two-dimensional standard Brownian motion. Let $$(F_t)_{ t \in [0, T]}$$ be a fit that satisfies the standard condition that $$B (t)$$ generates. A US dollar-denominated, default-free zero-coupon bond that pays a repayment of US \$1 at maturity $$T$$, but gives the price $$P (t)$$ at time $$t \in [0, T]$$ as the solution to the next stochastic differential equation It is assumed that $$dP(t)=(r(t)+\lambda b(t))P(t)dt - b(t)P(t)dB^1(t),$$ $$P(T)=1$$ Where $$b (t)$$ is a deterministic positive value function of time $$t, r (t)$$ is a short rate of US dollars, and the following Hull-White model is used. $$dr(t)= (\theta(t)-a\cdot r(t))dt +\sigma \cdot dB^1(t),$$ $$r(0)>0$$,Constant $$\theta (t)$$ is a deterministic function of time$$t$$, and $$a, \sigma$$ are positive value constants. The spot exchange rate $$X (t)$$ for the US$ -Yen is assumed to be the solution of the following stochastic differential equation.

$$dX (t)= \mu Χ(t)dt + \sigma_1 X(t)dB^1(t) + \sigma_2 Χ(t)dB^2(t),$$

$$X (0 ) =Χ_0 > 0.$$

Here, $$\mu$$, $$\sigma_1$$, and $$\sigma_2$$ are positive constants. Use the Ito formula and answer the following. Please also explain the calculation process.

(1) Suppose that a Japanese investor has invested in the top zero coupon bond. This investor recognizes the market value on a yen basis. Calculate $$dS (t)$$ for the value $$S (t) = X (t) P (t)$$ for this investor. (Please express the last equation by $$dt, dB^1 (t), dB^2 (t)$$.)

Question 2.

Let $$(Ω, F, P)$$ be a complete probability space, and let $$B (t) = (B^1 (t), B^2 (t))$$,$$t ∈ [0, T]$$be a two-dimensional standard Brownian motion. The filtration generated by this Brownian motion and satisfying the standard condition is $$(F_t)_{ t \in [0, T]}$$. Suppose$$F = F_T$$. Suppose that the prices $$\beta(t), X (t)$$ and $$Y (t)$$ of each of the deposit, stock $$X$$ and stock $$Y$$ are given by the solution of the following stochastic differential equation.

$$d\beta(t) =r \beta(t)dt,\beta(0)=1.$$

$$dX(t)=\mu_x X(t)dt+\sigma_x X(t)dB^1(t),X(0)=x>0.$$

$$dY(t)=\mu_y Y(t)dt+\sigma_y Y(t)dB^2(t),X(0)=y>0.$$

The parameters are $$r, x, \mu_X, \sigma_X, y, \mu_Y, \sigma_Y> 0$$. Let $$Q$$ be the equivalent Martingale measure with $$\beta (t)$$ as the numeraire. Let $$Q-(F_t)-$$standard Brown's motion be $$\hat{B} (t) = (\hat{B}^1 (t), \hat{B}^2 (t))$$.

(1)Payoff at time $$T$$

$$W_T:=\sqrt{X(T)}$$

The price at time t of the derivative that gives

$$\Pi (t)=\beta (t) E^Q [\frac{W_T}{\beta(T)} | F_t]$$

Find the ratio of $$\sqrt{X(t)}$$,$$\Pi(t)/\sqrt{X(t)}$$.

(2)Payoff at time $$T$$

$$Z_T := X(T) \cdot Y(T)$$

The price at time $$t$$ of the derivative that gives

$$\Phi (t) =\beta(t)E^Q[\frac{Z_T}{\beta(T)}|F_t]$$

Find the ratio of $$Z(t)= X (t) \cdot Y(t)$$ , $$\Phi(t)/Z(t)$$ .

Cautions 1. The settings for this problem generally do not have a ratio of 1.

• Do you really need to solve for $P_t$ and $X_t$ in the first question? It only asks you to compute $dS_t = d(X_t P_t)$ which is a simple application of Ito's lemma. – Freelunch Jun 14 '19 at 7:56
• you mean i don't need to know $S(t)$ to compudte $dS(t)$?? – ddss321 Jun 14 '19 at 8:01
• No, using Ito's lemma we have $dS_t = dX_t P_t + X_t dP_t + dX_t dP_t$. Assuming $B^1_t$ and $B^2_t$ are uncorrelated the $dB^1_tdB^2_t$ terms can be set to zero. I also don't see the need for an explicit expression for $r_t$. – Freelunch Jun 14 '19 at 8:35
• Thnks for your help. I think that the answer of problem 1. is that $dS(t)=(\mu+r(t)+\lambda b(t))S(t)dt +$$(\sigma_{1}-b(t))S(t)dB^{1}(t)+\sigma_{2} S(t)dB^{2}(t)$. – ddss321 Jun 14 '19 at 10:47
• Looks like you forgot the cross term, $dX_tdP_t = -b(t) \sigma_1 S_t dt$. – Freelunch Jun 14 '19 at 12:03