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\begin{document}

\title{FX Option Trading~~~}



\author[1]{Dean}%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{May 04, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup








\sloppy


\section*{FX \& FX Option Markets}

{\label{640931}}

The FX market is the most liquid capital market in the world. In the
major currencies, there is continuous and liquid trading. Prices are
easily observable and constantly responsive to supply and demand.

Unlike other markets, the FX market is a true 24-hour market. It is open
from 7 a.m. Monday in New Zealand through to 5 p.m. Friday on the West
Coast of the United States.

\section*{}

{\label{640931}}

\section*{Basics of FX Options}

{\label{640931}}\par\null

\subsection*{Call Option or Put Option?}

{\label{412824}}

A foreign exchange (FX) trade is a simultaneous purchase of one asset
and the sale of another. Therefore, it is easy to get confused about
calls and puts. The standard FX terminology where the first currency in
a currency pair (for example, USD in the USD/JPY pair) is the `base'
currency is not always helpful. From a mathematical standpoint, USD in
this case is actually the denominator, not the numerator.

In FX options, a~call on one currency~is always a~put on another. In a
USD/JPY option, a dollar call is always a yen put, and vice versa. To
simply talk about calls would be unhelpful; it would be better to talk
about a dollar call or, better still, a dollar call/yen put.

~

\subsection*{Pricing Terminology}

{\label{492224}}

An \textbf{FX option} \textbf{premium} \textbf{is usually stated as
units of the quoted currency per unit of underlying base currency}.

Let us take the example of a USD/JPY option on an underlying amount of
USD 10 million. The~\textbf{price~}in terms of the~\textbf{quoted
currency is 2.11}. This means that the option costs JPY 2.11 per dollar.
Since the underlying is USD 10 million, the option price is~\textbf{JPY
21.1 million}~(2.11 x 10 million).

It is also possible to quote the option premium as a percentage of the
underlying. In the example given above, \textbf{if the USD/JPY spot rate
is currently 82}, then the option~\textbf{price~}as a~\textbf{percentage
of the underlying~would be~2.11/82 = 2.57\%}. This is either the JPY
21,100,000 referred to previously (allowing for rounding), or 2.57\% of
USD 10,000,000 = USD 257,000.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.42\columnwidth]{figures/kh-201550/kh-201550}
\caption{{This is a caption
{\label{358839}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.42\columnwidth]{figures/kh-201551/kh-201551}
\caption{{This is a caption
{\label{178980}}%
}}
\end{center}
\end{figure}

\subsection*{Review Question: Option
Cost}

{\label{544782}}

A trader buys a 3-month EUR call/USD put option on an underlying amount
of EUR 10 million. The EUR/USD strike is 1.2100 and the current spot
rate is 1.2000. The price is 0.0167.~

How much does the option cost in EUR (to the nearest euro)?

\par\null\par\null

\section*{Breakeven Point}

Using the convention of `quoted currency per unit of base', an option
premium can be thought of as `points' (like the points for a forward FX
trade). It is then simple to derive a breakeven point for the trade --
the exchange rate at expiry where an option owner `breaks even', taking
into account the premium.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.42\columnwidth]{figures/kh-201552/kh-201552}
\caption{{This is a caption
{\label{858110}}%
}}
\end{center}
\end{figure}

Consider a USD call/JPY put option struck at 82. The option price is
2.00 (2 yen per dollar of underlying). The breakeven point will,
therefore, be at a USD/JPY rate of 84 (option price + cost of premium).
The option owner makes money once the payoff exceeds the premium value
(when USD/JPY rises above 84). The graph demonstrates this.

\subsection*{Breakeven Point -- Forward Value of
Premium}

{\label{944438}}

Strictly speaking, the~\textbf{forward value of the premium}~at the
expiry date should be considered. If the USD call/JPY put option was a
3-month option and rates were 4\%, then the future value of the premium
would be 2.02. The breakeven point would, therefore, be~\textbf{84.02}.

\par\null

\subsection*{Review Question: Breakeven for EUR Put/USD
Call}

{\label{231980}}

A 6-month EUR put/USD call, struck at 1.2000 (EUR/USD), is priced at
0.0338 USD per underlying euro. Assume interest rates are
0\%.~\textbf{What would be the breakeven EUR/USD rate for this trade
(correct to four decimal places)?}

\par\null

\subsection*{Pricing Vanilla European FX
Options}

{\label{712366}}

Vanilla European FX Options can be valued using the Garman-Kohlhagen
model. The formula is identical to the orginal variation of
Black-Scholes devised for a dividend paying stock except that the
dividend is replaced by the continuously-compounded risk-free rate of
the asset (the currenty being bought or sold). If the FX world becomes
confusing, it helps to think of one of the currencies as an asset,
paying a return, which is being bought using the domestic currency.

\par\null

Let us look at the Garman-Kohlhagen formula for valuing a call and put:

\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.56\columnwidth]{figures/2018-06-12-21-22-14-Formula/2018-06-12-21-22-14-Formula}
\caption{{This is a caption
{\label{516987}}%
}}
\end{center}
\end{figure}



\par\null\par\null

\textbf{Risk-Free Rate~}

The risk-free rate, in practice, is the deposit rate converted to a
continuously compounded basis, of the appropriate maturity for the
currency concerned.

\subsection*{Spot \& Forward Delta}

{\label{307682}}

The exposure from an option trade is a forward exposure; a trader is
effectively long/short a forward FX transaction. A forward FX position
can be replicated with three components:

\begin{enumerate}
\tightlist
\item
  spot FX position
\item
  long interest rate position in one currency
\item
  short interest rate position in the other currency
\end{enumerate}

It is common in FX trades to hedge the spot exposure immediately and the
interest rate exposures separately. The forward delta of an option is
treated in the same way. However, option models typically generate a
forward delta initially. This must be converted into a spot equivalent
to get the correct hedge for the spot market. This is done by taking the
currency's forward value and discounting it back to the present.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/2018-06-12-21-23-27-Intuition/2018-06-12-21-23-27-Intuition}
\caption{{This is a caption
{\label{312179}}%
}}
\end{center}
\end{figure}

\par\null

Calculating Spot Delta

Consider a 1-year option in USD/CHF on an underlying of USD 1 million.
The spot rate is 0.9668 and the forward delta is 50\% (USD 500,000).
1-year deposit rates in USD are 4\%, while 1-year deposit rate of CHF
are 5\%. Both are quoted on an actual/360 basis.

\textbf{What is the present value of the USD 500,000?}

\par\null

\textbf{Spot delta of USD}

1. The forward value needs to be discounted through the standard
equation:

\(e^{-rt}\) where

\texttt{r=risk-free\ rate\ for\ domestic\ currenty}

\texttt{t=time\ to\ expiry\ of~option}

2. The continuously compounded equivalent of an annual 1-year 4\%
desposit rate is 3.975\%.

\par\null

The following formula is used to transform deposit rates to
continuousley compounded equivalents:

\(\ln\left(1+\left(\frac{r}{m}\right)\right)^m\) where

\texttt{r\ =\ periodic\ interest\ rate}

\texttt{m\ =\ number\ of\ periods\ (for\ instance,\ two\ for\ semi-annual)}

where money market rates are quoted on an actual/360 basis, r needs to
be adjusted by multiplying by 365/360. Therefore, in given example

\(\ln\left(1+\frac{0.04\left\{\frac{365}{360}\right\}}{1}\right)^{^1}=0.039754759=3.975\%\)

\par\null

Therefore,~\(e^{-rt}=0.96103\).

3. The forward delta of 50\% is transformed to a spot delta
of~\(0.50\ \times0.96103=0.4805\%\) or USD 480,500.

\par\null\par\null

\textbf{Spot delta in CHF}

The present value calculated as USD 480,500 can be checked by
discounting the forward value of the CHF.

1. Using the rates above, the 1-year forward FX rate is 0.9762.
Therefore,~\(e^{-rt}=0.95175\)

The forward FX rate is calculated as follows:

\par\null

\begin{align*}
\text{Forward rate} F &= \text{Spot rate} \times [(\frac{(1+\text{CHF interest rate} \times Days/360)}{(1+\text{USD interest rate} \times Days/360)})] \\
&= 0.9668 \times [(1+(0.05x365/360))/(1+(0.04x365/360))]\\
&= 0.9668 \times 1.009744 \\
&= 0.9762
\end{align*}

So~\(0.9762\ \times0.95175=0.9291\)

\par\null

2. One dollar in the future is worth 0.96103 today. At the USD/CHF
current spot rate, USD 0.96103=0.9291 CHF.

\par\null

So the two discounted amount (0.9291 and 0.9291) are equivalent
(allowing for rounding).

\par\null

\subsection*{Volatility Quotations}

{\label{982417}}

There is general agreement about the calculation of option prices for
vanilla European FX options. Therefore, it is common to quote volatility
levels between traders rather than specific option prices. This is
particularly useful when the underlying spot is moving rapidly (an ATM
forward FX rate at 10 a.m. may be inappropriate at 11 a.m.). Rather than
quote individual option prices, traders talk to each other in terms of
implied volatility. This will usually change less rapidly than the other
variables.

Due to the importance of the \textbf{smile}
(Fig.~{\ref{873349}}) in FX option volatility (to be
covered later), it is common for traders to also quote the level of
delta associated with an option volatility as a guide to the moneyness
of the option strike. The implied volatility for a 50 delta option would
therefore be volatility for an ATM option, as ATM delta is always
roughly 50\%. Implied volatility for a 25 delta call or a 25 delta put
would focus on the volatilities for the respective OTM options.

\textbf{Smile}: A smile refers to a curve shaped like a smile that
results from plotting the strike price and implied volatility of a group
of options with the same expiration date.

\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/2018-06-12-21-56-30-Intuition/2018-06-12-21-56-30-Intuition}
\caption{{Smile
{\label{873349}}%
}}
\end{center}
\end{figure}



\subsection*{Review Question:~Delta \&
Moneyness}

{\label{894299}}

A 3-month EUR call/USD put has a 25\% delta. The current EUR/USD 3-month
forward rate is 1.2000. Which of the following is true in relation to
the strike of the option?

\begin{enumerate}
\tightlist
\item
  The strike is above 1.2000
\item
  The strike is below 1.2000
\item
  Information given is insufficient to determine the strike
\end{enumerate}

Answer:~

1~ Correct. The option is 25\% delta. Therefore, it must be OTM. As it
is an EUR call option, the OTM strike price allows us to
buy\textbf{~fewer}~euro with dollars than would be the case if we traded
in the market at the moment. Currently, we would need to give up 1.2000
dollars to buy 1 euro. If the strike were below 1.2000, then the call
would allow us to buy~\textbf{more}~euro than is currently the case,
which cannot be correct. The strike of the option must be above 1.2000.

\textbf{Review Question:~Call \& Put}

Consider two associated 1-year EUR/USD FX options. \textbf{}\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
Spot rate & 1.2050\tabularnewline
\midrule
\endhead
Option strike & 1.1800\tabularnewline
Volatility for 1-year options at that strike & 10\%\tabularnewline
1-year deposit rates in both EUR and USD markets & 5\%\tabularnewline
Price of 1st option & 2.85\% (percentage of principal)\tabularnewline
Price of 2nd option & 4.83\% (percentage of principal)\tabularnewline
\bottomrule
\end{longtable}

Which of the following statements is correct?

\par\null

\begin{enumerate}
\tightlist
\item
  The option which costs 2.85\% is an EUR put
\item
  The option which costs 4.83\% is a USD call
\item
  The option which costs 2.85\% is an EUR call
\end{enumerate}

Answer:

1~ Correct. The spot rate in EUR/USD is expressed as number of dollars
per euro. With equal interest rates, the forward rate of EUR/USD will be
identical to spot. The option with a price of 2.85\%, the cheaper
option, must be OTM. At 1.1800, euro would be worth less (USD 1.1800)
than it is now (USD 1.2050). Consequently, the cheaper of the two
options is the right to sell euro for dollars at a worse price than is
currently available. Therefore, the option that costs 2.85\% is an EUR
put/USD call.

\par\null

\section*{FX Volatility Smiles}

{\label{434868}}

Volatility smile in the FX option market is largely based on the
fat-tails phenomenon. There are more extreme price movements than are
suggested by the distribution in the standard pricing models. Volatility
smiles are so deeply embedded in the FX world that the option market has
two prices describing the smile for each maturity. ~

\par\null\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\begin{minipage}[b]{0.44\columnwidth}\raggedright\strut
Price 1\strut
\end{minipage} & \begin{minipage}[b]{0.50\columnwidth}\raggedright\strut
Price 2\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.44\columnwidth}\raggedright\strut
The degree of smile is given by the difference between the average
implied volatility for 25\% delta options and the implied volatility for
ATM options. This is the 25\% butterfly/strangle value.\strut
\end{minipage} & \begin{minipage}[t]{0.50\columnwidth}\raggedright\strut
The second price looks at any asymmetryin the smile between OTM calls
and OTM puts. This is the value for the 25\% risk reversal (RR). An RR
here is a combination of a 25\% delta OTM call option and a 25\% delta
OTM put option.\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

\textbf{Risk Reversal}

The term risk reversal will mean different things in different asset
classes. Usually, it refers to a strategy of simultaneously selling an
OTM put and buying an OTM call. More broadly, it can refer to any
position where the gamma/delta of an option strategy switches from
positive to negative at some point.

\par\null

\subsection*{Volatility Smiles - An
Example}

{\label{784646}}

Let us assume that implied volatility for a 3-month EUR/USD ATM option
is 9.20\%. The spot rate is 1.2084. The 25\% delta options have strikes
of 1.1761 (put) and 1.2524 (call). Implied volatilities are 9.51\% and
9.21\% respectively.

What are the values for the 25\% delta butterfly and 25\% delta RR?

\par\null

\textbf{25\% butterfly}

The size of the smile can be determined by finding the difference
between the average of the implied volatilities and the ATM implied
volatility.

\par\null

\begin{align}
(9.51+9.21)/2=9.36\%  \\
ATM = 9.20\% \\
25\% \text{ butterfly} = (9.36-9.20) = 0.16%
\end{align}

\textbf{25\% RR}

The smile is asymmetric - volatility in this case is higher for 25\%
delta puts than it is for 25\% delta calls. This asymmetry or value for
the 25\% RR, is the difference between OTM implied volatilities of the
two option types. Conventionally, the value~ of the put is subtracted
from that of the call

\par\null

\begin{math}
25\% RR = (9.21-9.51) = -0.30\%
\end{math}

\subsection*{Volatility Smiles for Different Maturities \&
Strikes}

{\label{201409}}

The volatility smile varies according to option maturity and strike
level. The graph below, with shows the price difference between an
option priced using ATM volatility and one priced using the smile,
demonstrates this.

\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/2018-06-13-00-00-46-Printable-Topic/2018-06-13-00-00-46-Printable-Topic}
\caption{{Premium over ATM volatility in bps for EUR/USD (Spot 1.2038)
{\label{478528}}%
}}
\end{center}
\end{figure}

\subsection*{\texorpdfstring{{}Smile -- Pricing versus
Hedging}{Smile -- Pricing versus Hedging}}

{\label{993591}}

The presence of strike-dependent smiles only affects the implied
volatility of an option as regards its valuation and pricing. Due to the
shortcomings of the Black-Scholes equation, implied volatility is best
thought of as a price parameter for options -- it is not necessarily a
good description of the underlying price evolution process.

Delta and vega sensitivities that are `thrown out' by the adjusted
volatilities are not the best guides to a hedging strategy. Option
expert Riccardo Rebonato points out that an adjusted, `smiley',
volatility is ``the wrong number to put in the wrong formula to get the
right price of plain-vanilla options''.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.42\columnwidth]{figures/2018-06-13-00-03-15-Intuition/2018-06-13-00-03-15-Intuition}
\caption{{Smile -- Pricing versus Hedging
{\label{768870}}%
}}
\end{center}
\end{figure}

\section*{}

\subsection*{Volatility Risks}

{\label{492237}}

{The way in which the volatility smile affects the prices at all strike
levels of options in major currencies adds a particular risk to a
trader's portfolio.}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/2018-06-13-00-04-45-Intuition/2018-06-13-00-04-45-Intuition}
\caption{{Volatility Risks
{\label{784290}}%
}}
\end{center}
\end{figure}

\selectlanguage{english}
\FloatBarrier
\end{document}

