Request PDF on ResearchGate | FX Volatility Smile Construction | The foreign exchange options Uwe Peter Wystup at University of Antwerp. 20 FX Volatility Smile Construction Dimitri Reiswich, Uwe Wystup September Authors: Dimitri Reiswich Uwe Wystup Research Associate Professor of. The smile construction procedure and the volatility quoting mechanisms are FX Furthermore, we provide a new formula which can be used for an efficient and robust FX smile construction. Uwe Wystup, Dimitri Reiswich; Published

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September, 8th Abstract The foreign exchange options market is one of the largest and most liquid OTC derivative markets in the world. Surprisingly, very little is known in the academic literature about the construction of the most important object in this market: The implied volatility smile. The smile construction procedure and the volatility quoting mechanisms are FX specific and differ significantly from other markets.

We give a detailed overview of these quoting mechanisms and introduce the resulting smile construction problem. Furthermore, we provide a new formula which can be used for an efficient and robust FX smile construction. The volatility smile is the crucial object in pricing and risk management procedures since it is used to price vanilla, as well as exotic option books. Market participants entering the FX OTC derivative market are confronted with the fact that the volatility smile is usually not directly observable in the market.

This is in opposite to the equity market, where strike-price or strike-volatility pairs can be observed. These quotes can be used to construct a complete volatil- Table 1: In the next section volatilihy will introduce the basic FX terminology which is necessary to understand the following sections.

We will then explain the market implied information for quotes such as those given in Table 1. Finally, we will propose an implied volatility function which accounts for this information. By definition, an amount x in foreign currency is equivalent to x S t units of domestic currency at time t.

The term domestic does not refer to any geographical region. By default, FX vanilla options are of European style, since their American style counterparts are too expensive or not worth more. Assuming non-stochastic interest rates and the standard lognormal dynamics for the constructipn exchange rate, at time t the domestic currency constructjon of a vanilla option with strike K and expiry time T is given by the Black-Scholes formula.

We may drop some of the variables of the function v depending on context. Note that v constructipn in domestic currency. The option position, however, may also be held in foreign currency.

We will call the currency in which an option s value is measured its premium currency. The notional is the amount of currency which the holder of an option is entitled to exchange. The value formula applies by default to one unit of foreign notional corresponding to one share of stock in equity marketswith a value in units of domestic currency.

For instance, a delta of 0. In foreign exchange markets we distinguish the cases spot delta for a hedge in the spot market and forward delta for a hedge in the FX forward market. Furthermore, the standard delta is a quantity in percent of foreign currency. The actual hedge quantity must be changed if the premium is paid in foreign currency, which would be equivalent to paying for stock options in shares of stock.

We call this type of delta the premium-adjusted delta. However the final hedge quantity will beEUR which is the delta quantity reduced by the received premium in EUR.

Consequently, the premium-adjusted delta would be The following sections will introduce. Related work can be found in Beier and Renner forthcoming. Unadjusted Deltas Definition 1.

We obtain Vanilla option: The above definition is motivated by the construction of a hedge portfolio using FX forward contracts as hedge instruments. The forward delta gives the number of forward contracts that an investor needs to enter to completely delta-hedge his FX option position; f, therefore, is a number in percent of foreign notional.

This symmetry only works for forward deltas. Premium Adjusted Deltas Definition 3. The premium-adjusted spot delta is defined as. While v is the option s value in domestic currency, v S is the option s value in foreign currency. Equation 9 can also be interpreted as follows: This is equivalent to the domestic units to sell.

We find Vanilla option: The defining equations for premium-adjusted deltas have interesting consequences: While put deltas are unbounded and strictly monotone functions of K.

### EconPapers: FX volatility smile construction

Thus, the relationship between call deltas and strikes K is not one to one. Typical shapes of the spot and premium-adjusted deltas are plotted against the strike in Figure 1 v K S Spot K Spot p. smike

The premium-adjusted forward delta f, pa is the percentage of foreign notional one needs to trade in the forward to be delta-neutral, corrected by the value of the option in foreign currency. Delta Conventions for Selected Currency Pairs The question which of the deltas is used in practice cannot be answered systematically. Wysyup summary of current market conventions can be found in the forthcoming book by Ian Clark. Both, spot and forward deltas are used, depending on which product is used to hedge.

## FX volatility smile construction

Generally, forward hedges are popular to capture rates risk besides the spot risk. So naturally, forward hedges come up for delta-one-similar smlle or for long-term options. In practice, the immediate hedge executed is generally the smil, because it has to be done instantaneously with the option trade. At a constructiln time the trader can change the spot hedge to a forward hedge using a zero-cost FX swap.

Using forward deltas as a quotation standard often depends on the time to expiry T and on whether the currency pair contains at least one emerging market currency.

If it does, forward deltas are the market default. The premium-adjusted delta as a default is used for options in currency pairs whose premium currency is FOR.

We provide examples in Table 2.

Typically, the premium currency is taken to be the more commonly Table 2: It is the attempt to specify the middle of the spot distribution in various senses. In addition to that, the notion of ATM involving delta will have sub-categories depending on which delta convention is used. ATM-spot is often used in beginners text books or on term sheets for retail investors, because the majority of market participants is familiar with it. ATM-fwd takes into account that the risk-neutral expectation of the future spot is the forward price 1which is a natural way of specifying the middle.

It is very common for currency pairs with a large interest rate differential emerging markets or long maturity. Choosing the strike in the ATM-deltaneutral sense ensures that a straddle with this strike has a zero spot exposure which accounts for the traders vega-hedging needs.

We summarize the various at-the-money definitions and the relations between all relevant quantities in Table 3. The resulting problem is to find a strike, given a delta and a volatility. The following sections will outline the algorithms that can be used to that end.

As the spot and forward deltas differ only by constant discount factors, we will restrict the presentation to the forward versions of the adjusted and unadjusted deltas. Conversion of Forward Delta to Strike The conversion of a non-premium-adjusted delta to a strike is straightforward. This is a straightforward procedure for the put delta, which is monotone in strike.

This is not the case for the premium-adjusted call delta, as illustrated in Figure 1. It is common to search. Strike region for given premium-adjusted delta. The right limit K max can be chosen as the strike corresponding to the non premium-adjusted delta, since the premium-adjusted delta for a strike K is always smaller than the simple delta corresponding to the same strike.

For example, if we are looking for a strike corresponding to a premium-adjusted forward delta of 0. It is easy to see that the premium-adjusted delta is always below the non-premiumadjusted one.

Discounting the last inequality yields the Black-Scholes formula, which is always positive. One can solve this implicit equation numerically for K min and then use Brent s method to search for the strike in [K min,k max ].

The resulting interval is illustrated in the right hand side of Figure 2. This knowledge is crucial to understanding the volatility construction procedure in FX. In FX option markets it is common to use the delta to measure the degree of moneyness.

Consequently, volatilities are assigned to deltas for any delta typerather than strikes. For example, it is common to quote the volatility for constructon option which has a premium-adjusted delta of These quotes are often provided by market data vendors to their customers.

However, the volatility-delta version of the smile is translated by the vendors after using the smile construction procedure discussed below. Other vendors do not provide delta-volatility quotes. In this case, the customers have to employ the smile construction procedure.

### FX Volatility Smile Construction – Dimitri Reiswich, Uwe Wystup – Google Books

Unlike in other markets, the FX smile is given implicitly as a set of restrictions implied by market instruments. This is FX-specific, as other markets quote volatility versus strike directly. A consequence is that one has to employ a calibration procedure to construct a volatility vs. This section introduces the set of restrictions implied by market instruments and proposes a new method which allows an efficient and robust calibration. Before starting the smile construction it is important to analyze the exact Table 4: In particular, one has to identify first 1 We will take a delta of 0.

For example, Figure 3 shows two market consistent smiles based on the EURUSD market data from Table 4, assuming that this data refers to different deltas, a simple or premium-adjusted one. It is obvious, that the smiles can have very different shapes, in particular for out-of-the-money and in-the-money options.

Misunderstanding the delta type which the market data refers to would lead to a wrong pricing of vanilla options. Both currency pairs use the forward delta neutral at-the-money quotation.

The next subsections explain which information these quotes contain.