Pricing of a Foreign Exchange Vanilla Option

Question

To understand how Bloomberg prices foreign exchange vanilla options , I extract the following screenshot from its OVML function.

The Black-Scholes formua for vanilla options are \begin{split} & P=\phi\big(Se^{-R_fT}N(\phi d_1)-Xe^{-R_dT}N(\phi d_2)\big) \\ & d_1 = \frac{\ln(\frac{S}{X})+(R_d-R_f)T+0.5\sigma^2T}{\sigma\sqrt{T}} \\ & d_2 = d_1-\sigma\sqrt{T} \end{split} where

$\phi$: 1 for call; -1 for put

$S$: Spot rate

$X$: Strike rate

$R_d$: Domestic interest rate

$R_f$: Foreign interest rate

$\sigma$: Volatility

$T$: Time horizon

From the screenshot, I get

\begin{split} & S = 1.3347 \\ & X = 1.3338 \\ & T = \frac{22}{252} = 0.08730 \text{ yrs} \\ & \sigma = 0.0655 \end{split}

I also look up that the $R_{USD} = 0.75$ and $R_{CAD}=0.50$. Plugging these numbers in, I get

$d_1 = \frac{\ln(\frac{1.3347}{1.3338})+(0.75-0.50)\times 0.08730+0.5\times 0.0655^2 \times 0.08730}{0.0655 \sqrt{0.08730}} = 1.5580$

$d_2=1.5580 - 0.0655*\sqrt{0.0873} = 1.5386$

and $P = 1.3347 \times e^{0.50\times 0.0873}\times N(1.5580) - 1.3338\times e^{-0.75\times 0.0873}\times N(1.5386) = 0.03864$, which is nowhere close to Bloomberg's result of 0.07452.

I also tried multiplying volatility by $\sqrt{12}$, assuming that the volaility they gave is monthly not annualized. The resulting price is 0.05624, which doesn't match either.

I also tried changing 0.75 and 0.5 into 0.0075 and 0.005, assuming that the interest rates are in percentages. The resulting price is 0.01688, which also doesn't match.

What am I missing?

Answer 1

Be careful of your rate conventions!

The issue here is that all your rates are expected to be in units of domestic vs 1 unit of foreign. So for example USDCAD is 1.3347, you really need to be using 1/1.3347 = 0.749 USD per 1 CAD.

So, your inputs need to be $$ \begin{align} S &= 1 / 1.3347 \\ X &= 1 / 1.3338 \\ R_d &= 0.75\% \\ R_f &= 0.50\% \\ \sigma &= 6.55\% \\ T &= 22/252 \end{align} $$ If you do this calculation you will get $$ P = 0.005614 $$ Now, the output of this is also in units of domestic per 1 unit of foreign notional (as pointed out by noob2). e.g. 0.005614 USD per 1 CAD notional. To get from CAD notional to USD notional, divide by $X$. $$ P_d = P / X = 0.7488\% $$ You can compare this to the $0.7452\%$ from your Bloomberg screenshot.

Answer 2

The answer given is mostly wrong: @msitt uses a convoluted way without explicitly mentioning it (put-call symmetry) to actually give the price of a USD Put, not of a USD Call as requested. Here is a more direct and correct approach.

I will consider, as mentioned by @FinanceGuyThatCantCode, that the volatility convention is ACT/365, which is standard.

I will also use the straightforward measures for USD/CAD: domestic=CAD, foreign=USD. The Black-Scholes price in the domestic measure (CAD) uses simply $S=1.33347, X=1.3338$. There are however two maturities to consider: $T_p=\frac{31}{365}$ (spot to payment) and $T=\frac{29}{365}$ (time to maturity). Then use $T_p$ to compute the forward $f=S e^{(r-q)T_p}$ and the discount factor $B(0,T)=e^{-r T_p}$, and $T$ to compute the total variance $\sigma^2 T$, with $\sigma=0.0655$. You then apply the Black-76 formula $$C_{CAD} = B(0,T) \left[f N(d_1)-X N(d_2)\right]$$ This price is in CAD. Now just convert it to USD by dividing by $S$. $$C_{USD}=\frac{C_{CAD}}{S}\,.$$ With $r=0.005$ and $q=0.0075$, this gives $C_{USD}=0.75922$%.

Now the rates are probably given in a specific different daycount convention. Using the rates $r=0.00269, q=0.00815$ from @FinanceGuyThatCantCode, which he implied from Bloomberg, I obtain $$C_{USD}=0.7465\%\,.$$ For USD 1 million, we have $C= 7465$ USD.

Answer 3

You can have a look here and there. The latter will show what BBG does with implied rates (or forward - which is a setting). This will, apart form some daycount differences (which are explained in the links) match BBG 100%.