Vega Risk in RiskManager

Product Technical NoteVega Risk in RM3Jorge Minajorge.mina@riskmetrics.comRevised by George Zhou 1george.zhou@riskmetrics.comRevised Note Version Draft 1.0 March 18, 2005Keywords: Implied Volatility, Volatility Smile, Risk Reversal, Strangle, Delta.Summary: We discuss the implementation of the vega risk module to capture volatility risk.1 IntroductionVega is the risk due to volatility fluctuations (i.e., the volatility of volatility). In previous versions of RMwe have used volatility as a static pricing parameter. The key to the implementation of Vega risk is toallow volatility to fluctuate like any other risk factor. However, the implementation of Vega is somewhatmore complicated than just adding another risk factor because of the volatility smile. The volatility smileis the name given to the empirical observation that options with different strike prices (or moneyness) havedifferent volatilities (as implied by the Black-Scholes model).In this note, we provide some basic background material and explain the implementation of Vega risk withinthe RM framework.2 Important Concepts and Intermediate Results• Black-Scholes Call:c = Se (b−r)T Φ(d 1 ) − Xe −rT Φ(d 2 ) (1)whered 1 = log(S/X) + (b + σ2 /2)Tσ √ Td 2 = d 1 − σ √ T (3)(2)1 Credit is due to Thomas Ta for his work on solving for volatility by rooting finding process.1

12.4Figure 1: USD-JPY implied volatility smile12.21211.811.6σ11.411.21110.810.610.40.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9δ• Black-Scholes Put:p = Xe −rT Φ(−d 2 ) − Se (b−r)T Φ(−d 1 ), (4)• Delta: Delta is the first order derivative of the option price with respect to the underlying.For a call, we have that:while for a putδ c (S, σ) = e (b−r)T Φ(d 1 ) (5)δ p (S, σ) = δ c (S, σ) − e (b−r)T (6)Note that 0 ≤ δ C ≤ e (b−r)T and −e (b−r)T ≤ δ P ≤ 0. Delta is also a good measure of moneyness.For example, an “at-the-money-forward” ATMF call (a call with a strike equal to the forward value ofthe underlying at expiry) has a delta close to 0.5, while an “out-of-the-money” option (an option witha very large strike) has a delta close to 0.• Volatility smile: the volatility smile can be thought of as a one-to-one map from moneyness intovolatility. In other words, the volatility smile gives us the appropriate volatility for a certain moneyness.Delta is the concept of moneyness that we will adopt to describe the volatility smile. Figure 1shows the one-month USD-JPY volatility smile for 1/24/1996.2

• Term structure of volatility: defines the relationship between the time to expiry (tenor) for an optionand the implied volatility. In other words, the volatility for a 3M ATMF option is different from a 1YATMF option.• Volatility surface: a two dimensional relationship that relates moneyness and tenor with impliedvolatility. This is just a combination of the volatility smile and the term structure of volatility.3 Implementation3.1 A Quadratic Model For Volatility SmileThe description of the entire smile in terms of the 25, 50, and 75 delta is done through the strangle and riskreversal prices.The price of a strangle is:str = σ 25 + σ 752where σ 25 , σ 50 , σ 75 are the 25, 50, and 75 delta vols.− σ 50 , (7)The price of the risk reversal isrr = σ 25 − σ 75 . (8)We will also denote the 50 delta vol as atm = σ 50 .Once we have the strangle and risk reversal prices, we can approximate the volatility smile by 2f(δ) ≡ ν(δ; atm, rr, str) = atm − 2rr(δ − 1/2) + 16str(δ − 1/2) 2 . (9)3.2 The DataEach volatility surface will be described by a set of volatility smiles defined on a set of tenors. In otherwords, the volatility surface on a given day will be stored as a two dimensional array.The tenors can be arbitrary, but the volatility smiles will have to be either a one point curve (stored in a50 delta series) or three points (for 25, 50, and 75 delta). The reason behind this restriction is that wecan linearly interpolate between tenors, but the interpolation in the smile has to be done carefully to avoidbreaching arbitrage constraints. If the user enters only one point we will assume that the smile is flat (samevolatility value for every strike).In the data representation we adopt the convention that all deltas quoted are call detlas.The volatility time series will be treated exactly as the other risk factors for simulation purposes.2 For a derivation of this equation see Malz(1997).3

3.3 Interpolation Between SmilesSince the volatility smiles are only given at a given set of tenors. Interpolation is needed to get the smile atother option expiries.We use linear interpolation. Specifically, for time t, assume that t 1 , t 2 are the closest pair of nodes thatbrackets t (t 1 < t < t 2 ). Then we interpolate the 25, 50, and 75 deltas at t 1 and t 2 to get the correspondingdeltas at t, namelyσ δ (t) = (t 2 − t)(t 2 − t 1 ) σ δ(t 1 ) + (t − t 1)(t 2 − t 1 ) σ δ(t 2 ), δ = 25, 50, 75 (10)For any t before the smallest node or after the largest node, take the smile at the corresponding end node asthe smile at t.3.4 Calculating Option Volatility From A SmileSuppose we have an option which expires at time t, and would like to calculate a volatility for the optionfrom the smile at t.If the volatility smile at t is flat, then volatility from the smile has to be the same value, namely the ATMvol.If the volatility smile at t is not flat, then we need to calculate the volatility from the smile. One can writeequation (9) asatm − 2rr[δ(S, σ) − 1/2] + 16str[δ(S, σ) − 1/2] 2 − σ = 0 (11)So calculating the volatility from a smile is equivalent to finding the root of the above equation. Iterativetype of algrothims such as Newton-Raphson method can be employed for the root finding.3.5 Safe Volatility Smile CalculationNote that the quadratic volatility smile, as a function of delta, is not guaranteed to be positive over the deltavalue range. Thus measures have to be taken to guard against negative volatility numbers.First, let’s denote by δ u the maximum value of call delta, δ 0 the delta value on which the quadratic smilereaches maximum or minimum when str ≠ 0. It is easy to see that the two values are given byNow define a modidifed quadratic volatility smile as:δ u = e (b−r)T (12)δ 0 = 1 2 + rr for str ≠ 0 (13)16str4

⎧⎪⎨ f(δ) if δ ∈ [25, 75]f m (δ) = f(25) if δ < 25⎪⎩ f(75) if δ > 25(14)We can now define a safe volatility smile function as follows:step 1 If str > 0, go to step 2;else, go to step 3;step 2 If f(δ 0 ) > 0, let f s (δ) = f(δ);else:(a) if δ 0 is beyond [0, δ u ], go to step 3;(b) if δ 0 is within [0, δ u ] but beyond [25, 75], let f s (δ) = f m (δ);(c) if δ 0 is within [25, 75], let f s (δ) = atmstep 3 If (f(0) > 0)&(f(δ u ) > 0), let f s (δ) = f(δ);else let f s (δ) = f m (δ)The part of using flat ATM vol, as currently implemented, can be improved. One possible idea is to fitpiece-wise linear functions between the three deltas (and take 25 and 75 deltas for extropolation).To apply the safe volatility smile function, an iterative algorithm mentioned in last section can be modifiedas follows:1. If any intermediary volatility in the iterative algorithm is negative, stop the iterative algorithm andreturn f s (δ(S, atm)), namely calculate δ with ATM vol then calculate the safe smile function giventhis delta.2. If the iterative algorithm exits normally, return f s (δ(S, σ)), where σ is the volatility returned by theiterative algorithm.3.6 User Specified Implied VolatilityIf the user specifies an implied volatility directly or indirectly (by specifying an option price and we calibratethe implied volatility), we do not want to replace the calibrated volatility with the simulated one, but rathershock the calibrated volatility on each scenario using proportional vol changes.Specifically, let σ imp , f s (δ base )f s (δ new ) denote the user-specified implied vol, the volatilities calculatedfrom the smile in the base senario and a new scenario, respecitively. Then the option volatility to be used inthe new scenario isσ new = f s(δ new )f s (δ base ) σ imp (15)5

3.7 Stress TestingEach volatility time series should be a risk factor available in the stress test module. The user should be ableto modify each point in the volatility surface separately.3.8 Instrument CoverageWe will initially support Vega only for those instruments that are priced within the Black-Scholes framework.In other words,• Equity options and options on equity futures• Commodity options and options on commodity futures• FX options• Swaptions, caps/floors/collars, options on interest rate futures.In the meantime, we will start doing research on the use of local volatility surfaces to price exotic derivatives(e.g., barrier options, AROs).3.9 Mapping Volatility Surfaces to Instruments• EquityFor options on individual equities and options on equity futures, we map each volatility surface to itscorresponding equity or equity index time series.• CommodityFor commodity options and options on commodity futures, we map each volatility surface to itscorresponding commodity time series.• FX OptionsFX options are always written on a currency pair. For each currency pair there are only two possibilities:either a call on the first currency and a put on the second one, or a put on the first currency anda call on the second one. For example, USD/JPY options are either USD call / JPY put or USD put /JPY call. Note that an option to buy one dollar for 120 yen (USD call) is equivalent to an option tosell 120 yen and receive one dollar (JPY put).Since the volatility of a 25 delta call is (approximately) equal to the volatility of a 75 delta put, wehave that the volatility of a 25 delta USD call / JPY put is equal to the volatility of a 75 delta USD put /JPY call. This is important because we will only store one volatility smile per currency pair. Figure 26

shows the DataMetrics coverage for FX implied volatilities. For each currency pair and tenor we havean ATM, a 25 delta call, and a 25 delta put (75 delta call) implied vols. It is important to note thatthe information provided in Figure 2 is ambiguous because the call currency is not specified. Forexample, if we have a 75 delta USD call / JPY put we would not know whether to use the vol labeled25 delta call or the one labeled 25 delta put. Therefore, each FX volatility surface must be identifiedby a call currency and a put currency. For example, in the RM file format we can adopt the conventionthat the first currency is always the call currency.In order to map an FX option to a volatility surface, the system has to know that all options on thesame currency pair are mapped to the same surface (regardless of whether it is a call or a put). Inaddition, we need to make sure that in the analytics we pick the right delta depending on whether weare dealing with a call or a put. For example, if we have a 75 delta EUR call / GBP put and we knowthat in the format file the first currency is the call currency, then we would use the 25 delta put columncorresponding to the right tenor in Figure 2. On the other hand, if we have a 75 delta EUR call / USDput, we would use the 25 delta call column in Figure 2.• Caps, floors, swaptions, and options on interest rate futures• Interest Rates1. For swaptions we map each swap curve to its corresponding swaption vol surface in Figure 3.2. For caps and floors we map each money market and swap curve to its corresponding cap/floorvol surface in Figure 3.3. For options on interest rate futures we map each money market and swap curve to its correspondingcap/floor vol surface in Figure 3. This is a temporary solution until we get exchangetraded interest rate futures implied vols.Note that for interest rates we sometimes need to map two volatility surfaces to the same yield curve.For example, we would tie the USD swaption vol to the swap curve if we are doing a swaption, butwe would use the USD cap/floor vol if we are doing a cap, floor, or collar.It is also important to note that implied vols for caps/floors, swaptions, and options on interest ratefutures are quoted as rate vols and NOT price vols. Our current pricing formulas are expressed interms of price vols. My recommendation is to either re-write or add an additional pricing functionfor each of these three instruments. I think it is better to offer two alternatives since replacing pricingfunctions always raises a lot of questions and support issues from clients.7

Figure 2: FX implied volatility coveragePage 16FX & COMMODITY IMPLIED VOLITILITIESThis dataset is available in DataMetrics onlyATM 25 Delta Call 25 Delta PutCur. Cur. 1M 2M 3M 6M 9M 12M 18M 2Y 3Y 4Y 5Y 1M 2M 3M 6M 9M 12M 18M 1M 2M 3M 6M 9M 12M 18MAUD JPY 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3EUR AUD 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3EUR CSK 3 3 3 3 3 3EUR GRD 3 3 3 3 3 3EUR JPY 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3EUR NOK 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3EUR PLZ 2 2 2 2 2 2EUR SEK 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3EUR CHF 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3EUR GBP 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3CHF JPY 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3CHF NOK 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3CHF SEK 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3GBP JPY 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3GBP NOK 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3GBP SEK 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3GBP CHF 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3USD AUD 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3USD BRC 3 3 3 3 3 3USD CAD 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3USD CHF 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3USD CNY 3 3 3 3 3 3USD CSK 3 3 3 3 3 3USD DKK 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3USD EUR 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3USD GBP 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3USD HKD 3 3 3 3 3 3USD INR 3 3 3 3 3 3USD JPY 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3USD KRW 3 3 3 3 3 3USD MXP 3 3 3 3 3 3USD NZD 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3USD NOK 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3ATM 25 Delta Call 25 Delta PutCur. Cur. 1M 2M 3M 6M 9M 12M 18M 2Y 3Y 4Y 5Y 1M 2M 3M 6M 9M 12M 18M 1M 2M 3M 6M 9M 12M 18MUSD PHP 2 2 2 2 2 2USD PLZ 2 2 2 2 2 2USD SEK 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3USD TWD 3 3 3 3 3 3USD THB 3 3 3 3 3 3USD ZAR 3 3 3 3 3 3USD GLO 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3USD SLO 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 38

Figure 3: OTC interest rate implied volatility coveragePage 14INTEREST RATE PRODUCTSThis dataset is available in DataMetrics onlyCap/Floor Market VolsSwaption VolsCountry Region ISOAustralia Asia AUD 2 2 2 2 2 2 2