FX Option Pricing with Settlement and Delivery Dates

# Is $S_0$ actually $F(0, 2)$ in pricing formulas for forex derivatives?


# Is $S_0$ actually $F(0, 2)$ in pricing formulas for forex derivatives?












I am reading a book that says the value of an FX option is given by a formula involving $S_0$.

The same formula can be found in the Wikipedia article on foreign exchange options in which $c$, the domestic currency value of a call option into the foreign currency is given as:

$c=S_{0}e^{-r_{f}T}{\mathcal {N}}(d_{1})-Ke^{-r_{d}T}{\mathcal {N}}(d_{2})$

where

${\displaystyle d_{1}={\frac {\ln(S_{0}/K)+(r_{d}-r _{f}+\sigma ^{2}/2)T}{\sigma {\sqrt {T}}}}},$

$d_{2}=d_{1}-\sigma {\sqrt {T}},$

and

$S_0$ is the current spot price,

$K$ is the strike price,

${\mathcal {N}}(x)$ is the cumulative normal distribution function,

$r_d$ is the domestic risk free simple interest rate,

$r_f$ is the foreign risk free simple interest rate,

$T$ is the time to maturity, and

$\sigma$ is the volatility of the FX rate.

My question is, given the settlement lag, is $S_0$ actually the exchange rate today, or $F(0, s)$, the forward exchange rate where $s$ is the settlement lag, which could be 0 days, 1 day, 2 days, etc?

The reason I'm confused is that on one hand, it seems to be that $F(0, s)$ is the correct number to use, since that is what I will actually receive if I exercise the option.

But on the other hand, the book I am reading says that I can use $S_0$ to convert the value of my option from one currency to another by simply multiplying it .... but clearly if $S_0 = F(0, s)$ then this is not true, then I'd be getting the value of my option in another currency $s$ days from now, not today.

So which one of these conflicting pieces of information is correct?

## Answer by river_rat (score 3)

https://quant.stackexchange.com/a/85461

So you typically need to keep track of 5 dates to get FX pricing correct in practice (ignoring crosses, where you may need a whole lot more dates).

- Pricing Date - $t$

- Premium / Value Date - $t_v$

- Spot Date @ Value / Pricing Date - $t_s$

- Option Expiry Date - $T_e$

- Delivery Date (which is typically the spot date at expiry but doesn't need to be) - $T_d$

The important thing to remember is that for vanilla options we don't price against spot. We use Black-76 so we are worried about the delivery date forward price. So we actually use this equation to price $$ DF(t, t_v, T_d)\times\left(F(t, t_s, T_e)N(d_1)-KN(d_2)\right)$$ This also explains some of the weird idiosyncrasies of the FX market like the Wednesday effect, weekend theta bleeds, the sawtooth vol pattern as these dates don't move in lockstep.