You are an associate at a Sydney commercial bank. One of the bank's commercial clients has been looking into FX options and came across reporting conventions for OTC FX options and options strategies that they did not understand. The client has asked you to write a brief note explaining how implied volatilities can be used to summarize information about option prices. You need to write a clear description of how it is possible to represent options prices and strike prices through information about implied volatilities and option deltas. Your report is targeted to a relatively non-technical reader – / 2 someone who has a very rudimentary understanding of the Black-Scholes-Merton formula and option payouts on expiration. You find some examples of implied volatilities for European options and for European option positions in the British Pound (GBP) in term of US dollars (USD) over a one-week period (trading days from 14 January through 21 January). These implied volatilities are computed from option prices using the Black-Scholes-Merton model for currency options. The first table of data gives the spot price of one British Pound in terms of US Dollars as well as the implied volatilities of USD-denominated options written on the GBP during this period: Date Spot Rate Implied Volatilities 1 Week 1 Month 3 Month 6 Month 1 Year 2 Years 14-Jan USD1.9584 8.9% 9.18% 9.18% 8.97% 8.94% 8.81% 15-Jan USD 1.9708 8.78% 9.16% 9.16% 8.93% 8.88% 8.80% 16-Jan USD 1.9664 9.10% 9.31% 9.31% 9.07% 8.97% 8.85% 17-Jan USD 1.9753 8.96% 9.32% 9.32% 9.13% 9.01% 8.89% 18-Jan USD 1.9547 9.05% 9.33% 9.33% 9.21% 9.13% 8.98% 21-Jan USD 1.9456 9.56% 9.89% 9.89% 9.49% 9.28% 9.07% The volatilities quoted above apply to so-called "50-delta calls," which means to say they are the volatilities of USD-denominated European calls on the GBP whose strike prices are chosen so that their deltas are 0.5. The second table of data you use in your illustration gives market implied volatilities for two different option strategies. The first strategy is known as a "25-delta reversal" and is a combination of a long out-the-money European call with a delta of 0.25 and a short out-themoney European put with a delta of -0.25. The second strategy is a "25-delta strangle" which is a combination of a long out-the-money European call with a delta of 0.25 and a long outthe-money European put with a delta of -0.25. The following table provides market implied volatilities for USD-denominated 25-delta reversals and 25-delta strangles on the GBP with different maturities, over the period 14 to 21 January: Date 25-Delta Risk Reversals 25-Delta Strangles 1 Month 3 Months 1 Month 3 Months 1 Year 14-Jan -0.88% -0.94% -0.63% 0.24% 0.31% 0.38% 15-Jan -0.78% -0.83% -0.40% 0.24% 0.31% 0.38% 16-Jan -0.73% -0.84% -0.54% 0.25% 0.31% 0.38% 17-Jan -0.71% -0.81% -0.53% 0.25% 0.31% 0.38% 18-Jan -0.76% -0.81% -0.59% 0.25% 0.31% 0.38% 21-Jan -0.82% -0.91% -0.59% 0.25% 0.31% 0.38% The following market conventions apply to the implied volatilities quoted above:  the listed volatility of a 25-delta reversal is the volatility of the underlying call minus the volatility of the underlying put;  the listed volatility of a 25-delta strangle is the average of the volatilities of the underlying call and put minus the volatility of the corresponding 50-delta call. / 3 Further, assume that the continuously compounded risk-free interest rates in the UK and the US are 4% and 2%, respectively for all maturities. You begin by converting the data from the first table into a table outlining the option strike price associated with each implied volatility value.1 Be sure to explain to your client what you have done, in addition to presenting the strike price information. Can you also compute the option price for each of these strike prices? Next you move to the second table and compute the strike price for each call and put used in the 25-delta reversals and 25-delta strangles for each date. Be sure to explain to your client what you have done, in addition to presenting the strike price information.2 You illustrate how this information can now be used to value other options with the following example. Suppose on 14 January you sold a three-month European call on 100,000 GBP with a strike of USD 1.98 per GBP. Show how, by using linear interpolation across strikes, one can estimate the implied volatilities for this option on that day.3 You finish your report by plotting the volatility surface for each of (implied volatility as a function of the time to expiration and the strike price, for each of the two calls and the put option. Are there any noticeable patterns or surprises in these surfaces, either on or between days?