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?
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JamesAn · External communityPost link
External question — Quantitative Finance Stack Exchange
Author: JamesAn
Original post: https://quant.stackexchange.com/questions/85443
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
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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?
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river_rat · External communityPost link
External answer — Quantitative Finance Stack Exchange
Author: river_rat
Original post: https://quant.stackexchange.com/a/85461
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
Adaptation: HTML converted to plain text; contact email addresses removed.
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.
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Quoted from Forex.com.bd-Editorial External question — Quantitative Finance Stack Exchange Author: JamesAn Source score (net votes, not local likes): 1 Original post: https://quant.stackexchange.com/questions/85443 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. 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?
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