Suppose there are two currencies X and Y , with corresponding interest rates RX and RY. Define S as the exchange rate, number of units of Y you can buy per unit of currency X. We generally treat a currency as an asset that provides a yield at the appropriate risk-free rate. a) Intuitively argue that the traditional risk-neutral process for the exchange rate S is;
$$dS=(RY-RX)Sdt + \sigma SdW$$
b) Naturally 1/S is also an exchange rate. This time representing the units of X that can be purchased per unit of Y . Using Ito’s lemma, show that
$$d(1/S)=(RX-RY+\sigma^2)(1/S)dt-\sigma (1/S)dW$$
c) These results lead to a paradox. Since the expected growth rate of S is rY ¡rX in arisk-neutral world, symmetry suggests that the expected growth rate of 1/S should berX ¡rY . How do you explain this paradox? Can you offer an explanation and possible solution?
Hey Guys so briefly I have the question up here. I tried using ITO's lemma as shown below.
S= W(t) so,
$$d(W(t))= (RX-RY)W(t)dt +\sigma W(t)dW$$
Then I have;
$\frac{∂f}{∂t} =\frac{RY}{RX} \frac{d}{dx}$
$\frac{∂f}{∂x} =\frac{RX}{RY} \frac{d}{dy}$
$\frac{∂^2f}{∂x^2} =\frac{RX}{RY} \frac{d^2}{d^2y}$
Then RY/RX= S= A
$$df(A,t)= \frac{∂f}{∂t}A(t)dt + \frac{∂f}{∂x}AdA(t) + \frac{1}{2} \frac{∂^2f}{∂x^2}(A(t))\sigma^2dt$$
Do you think that if I plug in the same thing for 1/S and put my numbers right I can manage to offer an explanation for this? I lready tried but could not actually, so I am poen to different solutions. Thanks in advance.