Risk reversals and strangles are described in Section 2.3.2 as packages of calls and puts with strikes on either side of the forward. Using the approximation to the SABR formula per Section 6.2.3, compute the implied vols for strikes 0.8 F, F, and 1.25F : where F = 1 is the initial value of the forward. Take σ₀ = 15%, β = 1, ρ = −30%, and ν = 50% initially. Try varying each of σ₀, ρ, and ν one at a time and see how the smile changes. Hence, convince yourself that a mapping between σ_0, ρ, ν, and atm option prices, risk reversal prices, strangle prices is meaningful and stable.

Section 2.3.2

Section 6.2.3

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The FX markets have developed a significant body of conventions over the years. Typically, options are not quoted via strikes but by deltas instead. The most liquid options tend to be at-the-money, 25-delta strangles and risk reversals and 10-delta strangles and risk reversals. We consider these below. At-the-money options These are not as innocent as they sound. Specifically, at-the-money typically means delta-neutral except for long expiries (e.g. over 10 years in USD/JPY) where it means the strike corresponds to the forward FX rate. Delta-neutral refers to a call and put having the same delta. (Obviously, we mean call delta = put delta.) Specifically, the Black-Scholes formula for a call is C =Xe"fTN(dı) — KenTN(d2) with d₁ Ac d₂=d₁ - o√T, where X is spot FX, K is strike, T is expiry, r is domestic rate, rf is foreign rate, and o is vol. Thus, the domestic spot delta of a call is given by log()+ (r-rf + 1/0²) T αντ > = Ə (XefT TN(d₁) - Ke TN (dz)) Əx 1 =enf7 N(d1) + Xe-"Tn(d1) -Ke-T 'n(d₂). 'Χοντ =e="fT N(d₁) + Xe-¹₁Tn(d₂)e-d₂0√T-10²T_ - Ke-Tn(d₂). with n(x) n(d₂)e = e¯¹₁¹ N(d₁) + = e-rf™ N(d₁) = -d₂0√T-10²T e 1 Χοντ ², so that n(d₁) A n(d₂) {e(_r+r₁) XOVE (Xe - IT Kel-T+7)T - KerT) Χοντ X 48= 2x (x)=xax-X² 1 = Similarly, we can show that the domestic spot delta for a put is Ap = efT (N(d₁) 1). So, delta neutrality requires e −fTN(d1)= -e-T (N(d₁) 1), i.e. we need to choose a strike such that d₁ N-¹ (), i.e. K = X exp(-o√/TN−¹(½) + (r − rƒ + 1/0²)T). However, in FX, we can see the price of an option from either currency. Specifically, suppose we have an option to buy one unit of currency A (e.g. US dollar) for K units of currency B (e.g. yen). The Black-Scholes formula gives its price C in terms of currency B. But say the majority of market participants function in currency A,¹ then they would like to hedge in currency A. If they had sold an option and received currency B, then this option premium (which can be valued in currency A) serves as a natural hedge for the con- tract. This leads to the concept of a foreign delta, and some currency pairs have their at-the-money options quoted as being delta-neutral based on the foreign delta (in currency A) = Notice that in currency B terms, the value is now ac C C əx X X = Ac 1 Χοντ √ (d₂+0√T) ² e √2π 1 Χσντ C On top of this, delta-neutral can be based on spot delta or forward delta. Forward delta is where the derivative of price is with respect to the forward rather than the spot. It is typically used for options whose expiries are not too short (e.g. over one year). It is beyond the scope of this book to discuss all the conventions in FX. So it is better to conclude here, but it is certainly useful to make the reader aware that there are more complications than one would have initially guessed. Strangles Strangles are where you are long a call and a put both being out- of-the-money to a similar extent. It is thus a measure of volatility smile (i.e. how much higher out-of-the-money implied volatility is compared to at-the-money implied volatility). In FX, the market quotes market strangles. Specifically, the quotes are for 25-delta¹² and 10-delta strangles, and the strikes are chosen as follows: Let q = σA Atm be the quote of the A-delta strangle (e.g. where A = 25). Then given the at-the-money volatility Atm, we can determine the volatility of the strangle A. The strikes of the strangle K and Kp are such that Ac = efTN (d₁ (Kc)) and Ape TT (1- N (di (Kp))). Notice that whilst this does not tell us the volatilities at strikes Kc and Kp, it imposes the key constraint C (Kc) + P (Kp) C (Kc,σA) + P (Kp,σA), i.e. the price of the strangle based on the true volatilities of call and put options agrees with the price of the strangle if both call and put options are priced with the volatility σ. Risk reversals A risk reversal can be described as being long a call and short a put with the same moneyness or vice versa. In this way, a risk reversal is a measure of skew (i.e. how much more out-of-the-money calls are worth vis-à-vis out-of-the-money puts). Typically, quotes are for 25-delta and 10-delta risk reversals. Specifically, the quote is of the form QA = oc-op, where ac and op are implied volatilities for the strikes Ke and Kp for the appropriate delta. The trouble is: What are these strikes? Since strikes for mar- ket strangles are not obtained from deltas computed from the true volatilities at those strikes, their strikes are different from those of risk reversals. As such, the problem is not uniquely determined in that some form of model dependent volatility interpolation over strikes, together with the quotes for market strangles is necessary, in order to obtain the strikes for the risk reversals since all we have is the constraint that the volatilities at this strike must satisfy.