Extracting Market Expectations from Currency Options’ Risk Reversals

the future price movement of the underlying asset. the the application of foreign-exchange options’ 25-delta risk reversals skewness of market expectations future in the States subprime crisis, the collapse of


Introduction
Currency option market has developed strongly since breakdown of the Bretton Wood agreement in the early 1970s. Options have begun to be an alternative risk management tool to cope with the high exchange rate volatility. After the stock market crash in 1987 market participants could have observed some new phenomena in option prices. The so-called volatility smile appeared to violate the famous Black-Scholes model [Black, Scholes, 1973]. It is believed that the volatility smile (skew) might have reflected market participants' fear of another stock market crash [Wang, 2008]. Since that time, many researchers have focused their effort to extract information embedded in option market prices. There are numerous implications of their studies. Options market where: (1) where: S -the price of underlying asset at time t K -strike price r -risk-free interest rate t T − -time to maturity ) (x N -cumulative distribution function of the standard normal distribution σ -expected volatility of underlying instrument Since r S t , are observable in interest rate and underlying instrument market, t T K , , are specified in contract, the only unknown variable is volatility σ . It can be either estimated from time series return of underlying instrument (e.g. standard deviation) or calculated from the market prices of traded options by reversing the Black-Scholes formula. The volatility obtained from BS model is called implied volatility [Dumas, Fleming, Whaley, 1998]. The Black-Scholes model assumes that prices of underlying instrument follow a Geometric Brownian motion with constant volatility.
It means that implied volatility should be the same for all options on the same underlying instrument but with different strikes and expiries. It is one of assumptions that usually turns out to be inappropriate [coleman, li, verma, 2001]. it is observed that implied volatilities change with both maturity and strike prices. Firstly, options with the same exercise prices but with different maturities mark substantial differences in implied volatility [Heynen, kemna, vorst, 1994]. Due to the fact that implied volatility of long-term options differs from the volatility of short-term options, the term structure of options' implied volatility is curved. Secondly, observed implied volatility is very often a convex function of strike prices. This phenomenon is called "volatility skew" or "volatility smile" [Mayhew, 1995]. It means that usually out-of-the--money and in-the-money options display higher volatilities than at-the-money options.
The Black-Scholes model assumes constant volatility and lognormal distribution of underlying asset. However, the major avenue of research shows that the prices of underlying asset are usually skewed (negatively or positively) with higher kurtosis (fat tails) compared to the Black-Scholes log-normal distribution. Moreover, both kurtosis and skew of distribution may be changing over time. The analysis of kurtosis and skewness may provide important information about market participants' expectation concerning future price movements of underlying asset. The paper is focused on currency options market. The most common currency options pricing model was developed by Garman and Kohlhagen [1983]. They extended the Black-Scholes formula (1) to deal with the presence of two interest rates (one for each currency). The trading conventions are very specific for currency market and significantly different from the trading procedures in bond or equity options market. Currency options are traded mostly in terms of volatilities, not in terms of option's premiums. over-the-counter currency option quotes are made on Garman-Kohlhagen implied volatilities. Moreover, implied volatilities are not quoted at a fixed strike price, but at a garman-kohlhagen delta. The implied volatility quotes are mainly available in three types of options strategies: the delta-neutral straddle, the risk reversal, and the butterfly spread [Beber, Breedon, Buraschi, 2010]. The foreign exchange option's implied volatilities can be written as follows: where: ATM σ -at-the-money (ATM) implied volatility with an approximate delta of 0.5 RR σ -risk reversal implied volatility BF σ -butterfly spread's implied volatility ∆ -the rate of change of option price with respect to changes in the underlying instrument A straddle is a combination of a call and put option with the same strike price and the same maturity, assuming delta for both options equals 0.5. When a strike price is close to a forward price, the straddle is used as a measure of at-the-money (ATM) implied volatility with an approximate delta of 0.5. The second strategy, risk reversal, measures the difference between implied volatility of an out-of-the-money (OTM) call option and out-of-the-money (oTM) put option. in regard to Black-Scholes option's delta, the moneyness level is usually set at 25-delta. The 25-delta risk reversal consists of a long position in a 25-delta call option and a short position in a 25-delta put option. As far as a 25-delta put and a 25-delta call options are concerned, the strike prices are such that about 25% of the distribution lies to the right of the call's strike and 25% to the left of the put's strike [campa, chang, reider, 1998]. The risk reversal is applied to measure the slope of implied volatility smile. The butterfly spread strategy is a combination of four option contracts with the same expiration dates but three different Pobrane z czasopisma Annales H -Oeconomia http://oeconomia.annales.umcs.pl Data: 05/06/2020 21:53:55 U M C S strike prices. it is quoted as the average of two oTM options' volatilities minus the aTM volatility (2). The butterfly spread is perceived as a measure of curvature of the implied volatility smile [Deuskar, Gupta, Subrahmanyam, 2008].
The paper is focused on option's risk reversal strategy. in respect of the foreign exchange option market, risk reversals reflect the cost of buying insurance against foreign currency appreciation, financed by providing insurance against foreign currency depreciation [Brunnermeier, Nagel, Pedersen, 2008]. Moreover, risk reversal can be applied to assess market perception of risk associated with high appreciation or high depreciation of the currency. When risk reversal is large and positive, it suggests that higher probabilities are attached to large appreciation of base currency against quoted currency. On the other hand, when risk reversal is large and negative, it implies that higher probabilities are attached to large depreciation of base currency against quoted currency [Campa, Chang, Reid, 1998].
risk reversals are believed also to reflect the changing risk appetite or risk perception of market participants. Gagnon and Chaboud [2007] analysed US dollar to Japanese yen 25-delta risk reversals and they found that risk reversals may be very useful to show changes in carry traders' risk perception. The higher is the value of risk reversal, the higher probabilities are attached to large Japanese yen appreciation, which is identified with possible carry trade losses. campa, chang and reider [1998] found that the stronger is the currency, the more expectations are skewed towards a large appreciation of that currency. Risk reversals were applied by them as a measure of skewness. According to Campa, Chang and Reider [1998], when e.g. US dollar is strong, market participants may attach greater probabilities to states of the world in which US currency becomes much stronger. They claim that the relatively high valuation of these states may result from market participants' risk aversion. Kohler [2010] applied risk reversals to analyse situation on currency market during turbulent times in financial market. She demonstrated that during financial market crises in 1997-1998 and 2008-2009, market participants disproportionately tried to hedge against an appreciation of funding carry trades' currencies (e.g. Japanese yen and Swiss franc) or to hedge against a large depreciation of less actively traded currencies (e.g. South African rand).

Extracting market expectations from currency options' 25-delta risk reversal
Foreign exchange market has been surveyed every three years since 1986 by the Bank for International Settlements. Last survey, that took place in April 2016, shows that daily average turnover in FX market amounted to 5. JPY. Another very important currency is British sterling (GBP) that accounted for about 13% of market share. The paper is focused on Japanese yen to Australian dollar (JPY/AUD), American dollar to British pound (USD/GBP) and Polish zloty to euro (Pln/Eur) exchange rates. in the exchange rates, the first mentioned currency is a quoted currency, and the second currency is a base currency. Exchange rates are depicted in proper mathematical expression where base currency is a denominator and quoted currency is a numerator. An increase of exchange rate is associated with the appreciation of base currency and depreciation of quoted currency. The empirical analysis is based on data sets that cover 25-delta risk reversals daily quotes for currency pairs JPY/AUD, USD/GBP, and PLN/EUR during the period from January 2005 till January 2017. Prices are expressed in volatility points. Options on each currency pair have 3 months fixed time to maturity.
The paper is focused on Japanese yen, as a popular carry trades' funding currency that is believed to reflect changes in market participant's risk perception, especially during turbulent times in financial market. The 2016 Brexit referendum impact on market participants' currency risk perception is presented on the basis of uSD/gBP exchange rate. The paper studies also PLN/EUR exchange rate market in order to show the influence of 21 st -century financial crisis on market expectations concerning Polish currency.
One of the most popular investment strategies in currency market is so-called carry trade. In the most basic form of the strategy, investors and traders borrow low-yielding currencies in order to finance a purchase of currency with a relatively high interest rate [Liu, Margaritis, Tourani-Rad, 2012]. The prolonged low interest policy of the Bank of Japan brought about Japanese yen to be the one of the most popular funding currencies [Czech, 2016]. During the period from January 2005 to June 2007, carry traders activities contributed to the significant increase in supply of Japanese yen which had a profound impact on the yen depreciation. The situation has changed when the sub-prime mortgage crisis began (Fig.1, I).
In the aftermath of events I and II (Fig. 1) market participants expectations changed dramatically. 25-delta risk reversals reached as low level as -12. When the risk reversal quote is large and negative it implies that higher probabilities are attached to large depreciation of base currency (Australian dollar). Thus, market participants expected Japanese yen to appreciate against Australian dollar. The same results have been obtained by Brunnermeier, Nagel and Pedersen [2008], among others. It is worth stressing the fact that, between July 2007 and February 2009, exchange rate USD/JPY went down from 123.86 to 87.32.
Moreover, Figure 1 presents another two different events that brought about substantial decrease in 25-delta risk reversals. In May 2010 (Fig. 1, III), higher market distress driven by a European sovereign debt crisis led to a large increase of global risk aversion [Botman, Carvalho Filho, Lam, 2013]. Market participants again attached higher probabilities to Japanese yen appreciation rather than depreciation. It means that out-of-the-money put options on JPY/AUD were, on average, Pobrane z czasopisma Annales H -Oeconomia http://oeconomia.annales.umcs.pl Data: 05/06/2020 21:53:55 U M C S more expensive than corresponding OTM call options (in comparison to the value predicted by Black-Scholes model). Market expectations were skewed in favour of a large decrease of JPY/AUD exchange rates.
another event that highly affected the value of Japanese currency was Pacific typhoon (Fig. 1, iv) that took place in September 2011. Despite the natural disaster in Japan, market participants placed higher probabilities to large appreciation of Japanese yen. as a result, the currency became stronger although there were no sufficient fundamental reasons for it. It is worth emphasizing that, since the mid-1990s, there have been 12 episodes during which the value of Japanese yen has increased by more than 6% (in nominal terms) within one quarter [Botman, Kang, 2015]. Moreover, these periods often coincided with events outside Japan. It shows that the value of Japanese currency is highly affected not only by economic, fundamental factors, but also by global changes in investors' mood, risk perception and global risk aversion.   (Fig. 2). First drop of USD/GBP risk reversals were observed after the bankruptcy of Lehman Brothers in 09.2008. The larger the negative value of 25-delta risk reversal is, the higher probability is attached to depreciation of base currency (British pound). Another decrease of 25-delta risk reversal for USD/GBP resulted from the European sovereign debt crisis in 2010. However, it needs to be emphasized that the biggest drop in risk reversals of USD/GBP occurred in June 2016, when British citizens decided to cut a long-term relationship with the European Union. In effect of Brexit referendum, the risk reversal value plunged the depths it had reached even during the height of financial crisis (Fig. 2). The result of Britain's referendum drew substantial changes in market expectations concerning the value of British pound. Market participants were skew in favour of a large drop in the USD/GBP exchange rate market.  It needs to be emphasised that risk reversals may vary substantially over time. Carr and Wu [2007] have shown that risk reversals may be highly changing, even from negative to positive values. This is a unique feature of currency options contracts. Equity options implied volatility skewness also varies over time. However, in comparison to currency options, it stays highly negative across most sample periods [Foresi, Wu, 2005].

Conclusions
Measures of volatility implied in option prices can provide important insight into market participants' perception about the future price movement of the underlying asset. The paper is focused on foreign exchange options risk reversals volatility. Risk reversals reflect the cost of buying insurance against foreign currency appreciation, financed by providing insurance against foreign currency depreciation. risk reversals are applied to assess how market participants perceive the balance of risks between a large decrease and a large increase in the exchange rate market. When the risk reversal is large and positive it implies that market participants place higher probability to large appreciation of base currency (depreciation of quote currency). On the other hand, when risk reversal is large and negative, it suggests that higher probabilities are attached to large depreciation of base currency (appreciation of quote currency).
Japanese yen is considered to be one of the most popular funding currencies in carry trade strategy. During the period from January 2005 to June 2007, carry traders activities contributed to significant increase in supply of Japanese yen which had significant impact on the yen depreciation. The situation has changed when the sub-prime mortgage crisis began (07.2007). It has been shown that during high-volatility and turbulent periods in financial markets, market participants placed higher probability on Japanese yen appreciation than its depreciation against Australian dollar. This phenomenon was observed even after Pacific typhoon in 2015. Despite the natural disaster in Japan, market participants placed higher probabilities to large appreciation of Japanese yen.
As far as the British pound is concerned, the biggest drop in risk reversal value was observed after Brexit referendum (June 2016), when higher probabilities were attached to large sterling depreciation. Risk reversals implied volatilities plunged the depths it had reached even during the financial crisis (2008)(2009)(2010).
Last but not least, 25-delta risk reversals of PLN/EUR were positive for the whole sample period. The biggest increase of their values occurred at the height of the global financial crisis, after the bankruptcy of lehman Brothers. During that time, market expectations were skewed in favour of large Polish zloty depreciation against euro currency.