Given the currency pair , GBPUSD with
spot price as $S_t$ at time $t$, Strike price as $K$, $I$ is an indicator function indicating if GBPUSD is below the "Knock-in-Rate" at expiry, $L$ denotes the leveraged ratio and the spot price at time to maturity $T$ is $S_T$. Note that the knock-in-rate given is less than the strike rate.
If the Notional is given in USD, $Notional_{\scriptsize{USD}}$. Then how do you convert this into a GBP Notional for the payoff equation and what is the reason for this way of converting, i.e,
There are two conditions for the payoff:
- If spot at maturity is greater than the strike rate, then we have the option to buy GBPUSD at the given strike rate
- If the $S_{T}$ is lesser than the knock-in at maturity, then we are obliged to buy GBPUSD at strike rate with the leveraged Notional. Note that this will always produce a negative payoff since the given knock-in is lesser than strike.
$$ Payoff(t,T) = \frac{Notional_{\scriptsize{USD}}}{?}( ( S_T - K )^{+} - L.(K-S_{T} ).I) $$
I am basically trying to price this exotic FX option using Monte Carlo simulation and need to get the payoff right to get the correct price in the end.
Similarly If we go one step ahead and try creating a payoff for an FX forward at time $t$ of the same currency pair, where now K is the forward rate, would the payoff look like, $$ \frac{Notional_{\scriptsize{USD}}}{?} (S_{T}-K) $$ For FX forward, I reckon it will be $\frac{Notional_{\scriptsize{USD}}}{K}$, since notional exchange happens at the forward rate at maturity. However, should it not be the same or the FX Option where the currencies are bought or sold at the strike rate at maturity.
