Vega risk and the smile

Transcription

1 The RiskMetrics Group Working Paper Number Vega risk and the smile Allan M. Malz This draft: April 2000 First draft: September Wall St. New York, NY

2 Vega risk and the smile Allan M. Malz April 2000 Abstract Vega risk can be a large part of the risk of a portfolio containing options. Any market participant owning option positions should be able to compute that risk. Vega risk is analytically easy to nest into the standard risk management framework. The treatment of vega risk in portfolios is, however, often impeded by the lack of availability of data on option implied volatilities. Vega risk is also complicated by the prevalence of volatility smiles and term structures in most option markets. Volatility smiles, in spite of their occasionally treacherous effects on option books, are often neglected by risk managers. This paper provides a guide to incorporating vega risk into a classical value-at-risk (VaR) model. The paper includes a tractable approach to capturing the effects of the volatility smile and term structure on vega risk and their interaction with other risk factors. In our discussion, we will present several examples using a high-quality database of foreign exchange implied volatilities.

3 1 Introduction Vega risk can be a large part of the risk of a portfolio containing options. Any market participant owning option positions should be able to compute that risk. Vega risk is analytically easy to nest into the standard risk management framework. The treatment of vega risk in portfolios is, however, often impeded by the lack of availability of data on option implied volatilities. Vega risk is also complicated by the prevalence of volatility smiles and term structures in most option markets. Volatility smiles, in spite of their occasionally treacherous effects on option books, are often neglected by risk managers. This paper provides a guide to incorporating vega risk into a classical value-at-risk (VaR) model. The paper includes a tractable approach to capturing the effects of the volatility smile and term structure on vega risk and their interaction with other risk factors. In our discussion, we will present several examples using a high-quality database of foreign exchange implied volatilities Definition of vega risk Option positions are exposed to a range of market risks. Delta and gamma risk are the exposures of an option position to changes in the prices of the underlying assets. Vega is the exposure of an option position to changes in the implied volatility of the option: vega = option value implied volatility. (1) The change in the option value is defined as a partial derivative, that is, it assumes all other factors determining the option value, such as the current level of the underlying asset price and the remaining time to maturity, are held constant. Vega is measured in dollars or other base currency units. Implied volatility is generally measured in percent per annum. Units of implied volatility are often called vols, so dollar-yen might have 1 The source of the database is the J.P. Morgan foreign exchange desk. The database includes implied volatilities for a wide range of currency pairs and maturities, and includes extensive coverage of volatility smiles and is available from the RiskMetrics Group. 1

4 an implied volatility of 12 percent or 12 vols. In this document, we will always express implied volatility as a decimal, so one vol equals Implied volatility is a measure of the general level of option prices. As the name suggests, an implied volatility is linked to a particular option valuation model. In the context of the valuation model on which it is based, an implied volatility has an interpretation (under the model s assumptions) as the market-adjusted or risk neutral estimate of the standard deviation of returns on the underlying asset over the life of the option. The most common option valuation model is the Black-Scholes model. This model is now familiar enough that in some over-the-counter option markets, dealers quote prices in terms of the Black-Scholes implied volatility. Vega risk can be thought of as the own price risk of an option position. Since implied volatility can be used as a measure of option prices and has the interpretation as the market s perceived future volatility of returns, option markets can be viewed as markets for the commodity asset price volatility. Exposures to volatility are traded and volatility price discovery occur in option markets. Vega risk is unique to portfolios containing options. In her capacity as a pure market maker in options, an option dealer maintains a book a portfolio of purchased and written options and delta hedges the book, that is, purchases or sells the underlying assets so as to insulate the book against short-term fluctuations in the price of the underlying. This leaves the dealer with risks which are unique to options: gamma and vega. 3 The nonlinearity of the option payoff with respect to the underlying price generates gamma risk. The sensitivity of the option book to the general level of option prices generates vega risk. Because implied volatility is defined only in the context of a particular model, an option pricing model is required to measure vega. Vega is then defined as the partial derivative of the call or put pricing formula with respect to the implied volatility. In the commonly-used Black-Scholes model, the expression for the vega, denoted κ, of a call or put option on a dividend-paying asset is 2 The terminology can be a bit confusing, since an implied volatility itself may also be referred to as a vol, as in equity index vols have been rising of late. 3 Theta is not a risk, since it is not random. Rather, it is a cost of holding options. 2

5 Figure 1: Option value as a function of implied volatility. Dollar-yen calls, premium in premium 4 3 at the money out of the money in the money vol shock Ν κ = τs t e dτ φ ln ( S t ) ) X + (r d + σ 2 τ 2 σ, (2) τ where σ is the implied volatility, S t is the current underlying price, X the exercise price, τ is the option s remaining time to maturity, r is the financing or money market rate with the same term as the option, d is the dividend, interest or other cash flow the underlying asset is expected to return over the term of the option, and φ is the standard normal density. 4 Some properties of vega in the Black-Scholes model are illustrated in Figures 1 and 2: The value of a put or call increases monotonically with the implied volatility. The option vega is therefore always positive. Vega is at a maximum for an at-the-money option. The value of an option is therefore almost linear in the implied volatility for an at-the-money option. 4 In contrast to other option sensitivities, vega is not a Greek letter. Vega is a star in the constellation of Lyra and an automobile once manufactured by the Chevrolet division of Ford. It is usually represented by the Greek letters κ or λ. 3

6 The vega of a long call or put option is positive and the vega of a short call or put option is negative. The vega of a longer-term option is greater than that of a shorter-term option (provided all other arguments in the Black-Scholes pricing formulas are held constant). Κ Figure 2: Vega of a 1-month dollar-yen option forward rate S t The vega exposure of an option position is defined as the change in the value of the position if the implied volatility rises one vol. It is calculated as the option vega times the underlying amount of the option (the number of options ) times the increment in implied volatility: option value underlying amount implied volatility implied volatility underlying amount κ implied volatility, with implied volatility usually set to 0.01, so implied volatility goes from, say, 15 to 16 percent (see Figure 1). In the case of a foreign exchange option on the dollar denominated in foreign currency units, we can convert the exposure back to U.S. dollars or other domestic base currency by dividing by the spot exchange rate: 4

7 underlying amount κ implied volatility. S t This expression can also be applied to convert the vega of any currency option with a premium denominated in foreign currency units to a base currency (for example, an option on sterling or the Euro denominated in dollars, where S t is the sterling or Euro exchange rate). Example 1.1 (Foreign exchange) Consider a short position in a one-month at-the-money forward dollaryen put. (An at-the-money forward option has an exercise price equal to the current forward price of the asset.) The option is a put on the dollar, with premium and exercise price denominated in yen. The underlying amount is USD 1,000,000, the current spot rate is 120, the financing rate, that is, the yen-denominated Treasury bill or money market rate, is 50 basis points, the U.S. money market rate is 5 percent, and the implied volatility is 15 percent. If implied volatility rises by one vol (0.01), the value of the put option rises by , so the initial option vega is κ = = The underlying amount must be multiplied by ( 1), since it is a short option 0.01 position. The vega of the position is κ = 137,591 or S t κ = USD 1, Example 1.2 (Option on Eurodollar futures) The Black-Scholes formulas can be applied to any asset. There is a simplified formula applicable to options on forwards, where all cash flows occur at the maturity of the contract. 5 As an approximation, the formula is also applied to futures: ( ) κ = τf t,t φ ln Ft,T + σ 2 X 2 τ σ, (3) τ 5 The formula is often called Black 76 after its original publication in (Black 1976) 5

8 where F t,t is the time-t price of a forward or futures contract expiring at time T. Note that the domestic interest rate and the dividend rate are absent from the formula. Instead, they are embedded in the futures price itself via the relationship F t,t = S t e (r d)τ. We can apply this formula to measure the vega risk of a Eurodollar futures option. Eurodollar futures settlement prices are expressed as indexes. If the 3-month Libor rate on the maturity date of the futures is r T, then the contract settlement price is 100(1 r T ). The cash settlement on the contract is designed to replicate the difference between 3 months interest at the entry rate minus the settlement rate, times the size of the contract, and is set at (1 r T ) times the number of contracts. The value of a tick or basis point is therefore USD 25. Because the underlying factor is 3-month USD Libor, which we wish to bound above zero, the Black 76 formulas for puts and calls on Eurodollar futures, caplets and floorlets are usually expressed in terms of the Libor rate, so that F t,t represents the implied future Libor rate in equation 3. The vega of a Eurodollar futures option position in dollars is then 25 number of contracts κ implied volatility. 1.2 Limitations of the Black-Scholes model The Black-Scholes model assumes that asset returns follow a random walk with a constant volatility. The Black-Scholes implied volatility is the market estimate of that volatility. Since volatility is a constant, vega risk does not exist in the Black-Scholes model. In real-life markets, both realized and implied volatilities fluctuate, as illustrated in Figure 3 for dollar-yen. Vega risk is meaningful only in the context of a model in which volatility is random. There is therefore a contradiction, which we will overlook for now, in using the Black-Scholes implied volatility and the Black-Scholes vega to measure vega risk. The Black-Scholes model is only a useful first approximation to the true model governing volatility. From this agnostic viewpoint, the Black-Scholes 6

9 Figure 3: Dollar-yen spot exchange rate and 1-month at-the-money forward implied volatility. Source: DataMetrics 140 spot left axis vol right axis Jan 94 Jan 95 Jan 96 Jan 97 Jan 98 Jan implied volatility is no longer necessarily a correct measure of anticipated volatility. It is rather a marketadjusted parameter in the Black-Scholes option pricing formulas which is closely related to but not identical with anticipated volatility. The variation of implied volatility over time is one of several important respects in which the behavior of observed option prices differs from that predicted by the Black-Scholes model. In real-life markets, the implied volatility for a given asset at a given time is not an identical constant for all options on a given asset observed at a given time, but differs for options with different exercise prices and different times to maturity. These patterns of implied volatility are known as the volatility smile and the term structure of volatility: 6 Smile The volatility smile describes the characteristic shape of the plot of implied volatilities of options of a given time to expiry against the delta or against the exercise price: out-of-the money options often have higher implied volatilities than at-the-money options. 6 On the smile and term structure, see for example (Heynen 1994), (Xu and Taylor 1994), (Dumas, Fleming and Whaley 1998) and (Peña, Rubio and Serna 1999). 7

10 Smirk and sneer The volatility smile is often skewed, so that out-of-the money call options have implied volatilities which differ from those of equally out-of the money put. The lopsided pattern of implied volatility across exercise prices is often called the smirk or sneer by market participants who need a laugh. Term structure The term structure of implied volatility describes the pattern of options with the same exercise price but different maturities, which generally have different implied volatilities. The volatility smile is typically represented as a schedule of implied volatilities of options on the same underlying and with the same maturity but different exercise prices, as in Figure 4. Figure 4: The volatility smile in the S&P index option market. Implied volatilities on Sept. 11, 1998 of options on Dec. 98 futures Ν 33 futures price X In the equity index smile depicted in Figure 4, the skewness of the smile is so pronounced as to swamp the curvature. This is not always the case. These limitations of the Black-Scholes model provide a road map for this paper. As noted, if the Black- Scholes model were in perfect accord with reality, vega risk would not exist and this paper would have no motivation. In the next section, we will focus on the vega risk generated by the time-variation of implied 8

11 volatility and ignore the volatility smile and term structure. In Section 3, we incorporate the smile and term structure into our measure of vega risk. 2 Vega risk of a portfolio containing options A simple and tractable way of calculating vega risk is to treat implied volatility analogously to other market risk factors, such as equity, foreign exchange, and interest rates of different maturities, to which a portfolio is exposed. The RiskMetrics approach is to assume that proportional changes returns in implied volatility follow a random walk with a mean of zero. The implied volatilities of different assets are correlated with one another and with the remaining factors in the portfolio. Under this assumption, the levels of implied volatilities are jointly lognormally distributed and logarithmic changes in implied volatility are normally distributed with mean zero and a constant standard deviation. The assumption of lognormality is analogous to that frequently made for interest rates. It is a simple approach and avoids assigning a positive probability to negative implied volatilities. We will use the RiskMetrics maturity vertices and cash flow mapping. All Value-at-Risk (VaR) numbers will assume a one-day holding period and a 95 percent confidence level, corresponding to a z-value of In this section, we maintain the Black-Scholes hypothesis that implied volatility is constant across different exercise prices for a given asset and option maturity. This captures exposure to the general level of the implied volatilities of the options in the portfolio, including the correlations of implied volatility with returns on the underlying market factors. A more precise methodology, which captures the differences in vega risk among options on the same asset with different exercise prices and maturities, will be discussed in the next section. To calculate VaR, we need the standard deviation of the implied volatility over a sample period, which we denote ν vol and express as a one-day rate. 7 The term vol-of-vol is often used to describe the volatility of implied volatility. We will use a two-year sample period and no RiskMetrics decay factor in our examples. 7 We are reserving notation involving σ for implied volatilities and using ν to represent the vol of vol. We are also risking confusion by expressing implied volatility at an annual rate and the vol of vol at a daily rate. 9

12 Under our assumptions, the logarithm of tomorrow s implied volatility is normally distributed with a mean of zero and a standard deviation ν vol τ, where τ is the VaR horizon in days. 8 interval for next-day implied volatility is then 1.65σe ±ν vol. The 90 percent confidence 2.1 Parametric VaR If the only market risk of an option were vega, the VaR of a portfolio containing a single option would be underlying amount κ 1.65σ(e ν vol 1) underlying amount κ 1.65σ ν vol. This expression is similar to that shown above for the vega exposure of the option, but now the increment to implied volatility has been changed from the generic implied volatility = 0.01 to the one-day/95 percent shock 1.65σ(e ν vol 1). In the case of foreign currencies, it may be necessary to divide by the exchange rate to convert to domestic currency units. Example 2.1 (Example 1.1 continued) We can illustrate these concepts by calculating the standard deviations and correlations of the spot exchange rate and its implied volatility. The data are displayed in Figure 3. The standard deviation of one-day logarithmic changes in one-month implied volatility is ν vol = (5.62 percent). The VaR arising from vega risk, disregarding other market risks, is =USD 1, Note that we have divided by the spot rate of An option position is exposed to delta and gamma risks as well as vega, and these must also be accounted for in VaR. For expository purposes only, we will outline a simplified parametric VaR which incorporates exposure to both the underlying price and to implied volatility. In the examples, we will use the Monte Carlo method with full repricing to compute VaR for portfolios containing options. 8 To be precise, the expected value of the logarithm of the next-day implied volatility is σ ν2 vol 2. We will ignore the second term, as it is negligible in short-term analyses. 9 This understates the vega risk compared with the more exact calculation (e ) =USD 1,

13 Denote the option delta by δ. The delta exposure, also called the delta equivalent, is calculated as the option delta times the underlying amount of the option times the increment in the the underlying price: underlying amount The parametric VaR of a simple option position is option value underlying amount δ. underlying price [ 1.65 underlying amount [ δsν spot κσν vol ] 1 ρ spot,vol ρ spot,vol 1 ][ δsνspot κσν vol ] Example 2.2 (Example 1.1 continued) The dollar-yen put option delta is Because we are dealing with a short put position, the delta equivalent is a long dollar position of =USD 489,320. The daily spot rate volatility is ν spot = The VaR of the delta equivalent, ignoring vega as well as the gamma and higher-order nonlinear exposures to the underlying spot rate, is =USD 7, The correlation of dollar-yen returns with logarithmic changes in the one-month implied volatility is 0.395: there has been a tendency for a weakening dollar to coincide with a rise in option prices. The negative correlation might be due to an increased desire of Japanese exporters to hedge their foreign exchange earnings if a weaker dollar appears likelier. The portfolio on which this simple parametric VaR is calculated consists of a long dollar exposure against the yen and a short dollar-yen one-month vega exposure. The parametric option position VaR is [ ] [ ][ ]. or USD 8, The correlation benefit reduces VaR by =USD

14 For currency options on the dollar against other currencies, the delta equivalent is in dollars, so it need not be converted back. For other currency pairs such as the exchange rate of the Euro and sterling against the dollar, the exchange rate, the option premium, and the vega exposure are all denominated in dollars, but the delta equivalent is denominated in Euros or pounds. 2.2 Monte Carlo VaR In principle, exposure to volatility is a nonlinear risk, like the exposure to the underlying asset price. The return distribution to options is therefore skewed rather than symmetrical. This implies that to correctly measure its risks in the RiskMetrics framework, we ought to use the Monte Carlo approach with full repricing: applying normally distributed shocks to the implied volatility, repricing the option, and computing the 5-percent quantile. If we restrict attention to at-the-money options, vega risk is almost exactly linear, as seen in Chart 1. Parametric computations of vega VaR are therefore a good approximation for at-the-money options, but may be a poor approximation for in- and out-of-the-money options. Even for at-the-money options, parametric VaR computations are not advisable, as the exposure to the underlying is highly nonlinear, so the total VaR of a portfolio containing options will be inaccurate. The Monte Carlo computation of VaR proceeds as follows: Variance-covariance matrix Estimate the variance-covariance matrix of shocks to the underlying price and the implied volatility. This step is also required for parametric VaR computation. Scenario generation Generate a large number of multivariate normal shocks with mean zero and a variancecovariance equal to. Valuation Perturb the spot rate and the implied volatility by each shocks and revalue the position. Tabulate Tabulate the changes in position value and calculate the lower 5 percent quantile. 12

15 In the earlier examples, most of the risk was delta risk, that is, exposure to the price of the underlying asset. In some option portfolios, vega risk can be much greater than delta risk. Some examples of portfolios of option which are high in vega risk but low in delta risk are a delta-hedged option position, a long (short) straddle, which consists of a long (short) call and a long (short) put with the same exercise price, and astrangle, which consists of a long (short) call and a long (short) put with different exercise prices. Example 2.3 (Example 1.1 continued) We will now elaborate our standing dollar-yen example by adding to it a delta hedge. The portfolio consisting of the option and the delta hedge will generally have much smaller fluctuations in value than the naked option. The remaining discrepancies between fluctuations in the value of the option and its hedge generate gamma risk. Monte Carlo is recommended to compute the VaR of a hedged position, since gamma risk has a nonlinear relationship to fluctuations in the underlying price. Our portfolio now consists of a short put position together with a short dollar position of USD 489,320 (see Example 2.2). Setting aside vega risk for a moment, this portfolio can only lose value as the exchange rate fluctuates. (If we were dealing with a long option position, the portfolio could only gain value as the exchange rate is perturbed.) This is another aspect of the option gamma: it results in a highly skewed distribution of gains and losses with a mean well below zero and a long adverse tail. To illustrate, consider the Monte Carlo results for the VaR of the hedged option portfolio. First, we generate n independent realizations u i,i = 1,...,nfrom a normal distribution with mean zero and standard deviation The perturbed value of next-day dollar-yen is Se u i. According to the Black-Scholes model, the concomitant changes in value of the short put option, expressed in dollars, are 1 S p(s, τ, X, σ, r, r ) 1 Se u p(seu i,τ,x,σ,r,r ), i 13

16 where p(s, τ, X, σ, r, r ) is the Black-Scholes value of a put and r is the foreign money market interest rate (which in this context takes on the role of d, the dividend rate). 10 The VaR of the hedge is calculated by tabulating e u i 1 δ, e u i where δ<0 is the initial delta of the put option. This expression is derived by noting that to put on the hedge, the dealer initially sells δ dollars against the yen. The resulting yen-denominated cash flow is Sδ. The next day, the short dollar/long yen position is unwound or rolled over at a rate Se u i : the yen cash flow is +Se u i δ. The mark-to-market gain or loss of S(e u i 1)δ is converted to dollars at today s exchange rate Se u i. 11 The market risk of the hedge is linear and therefore has a symmetric distribution centered about a mean of zero. 12 The value under each scenario of the option-plus-hedge portfolio, still taking into account only spot fluctuations, is computed by summing these two components: 1 S p(s, τ, X, σ, r, r ) 1 Se u p(seu i,τ,x,σ,r,r ) + eui 1 δ. i e u i To bring vega risk into the picture, we generate jointly normally distributed shocks to both the cash price or spot rate and implied volatility. That is, we take n observations (u i,spot,u i,vol ) from a bivariate normal distribution with a mean of zero and a variance-covariance matrix equal to 10 We have ignored the aging of the option. 11 We have ignored the forward points paid to carry the long dollar position overnight. Even with a large interest rate differential such as 5 percent favoring the yen, the points add about 2 basis points to the daily cost of holding yen against dollars, negligible compared with the standard deviation of 1 percent. 12 See (Mina and Ulmer 1999) for more detail on linear and nonlinear risks. Note that the VaR on the delta hedge is not centered exactly at zero. Rather, because we convert to dollars by dividing by the exchange rate, there is a slight downward bias: if the average shock u i is zero, the average of the e u i is slightly less than zero. 14

17 Table 1: VaR of hedged dollar-yen option 0.05 quantile 0.95 quantile mean median unhedged option, spot only delta hedge hedged option, spot only option, vol only unhedged option, spot and vol hedged option, spot and vol quantile VaR ( 1) = = [ ][ νspot 0 1 ρ spot,vol 0 ν vol ρ spot,vol 1 [ ] ][ νspot 0 0 ν vol ] The vega risk in isolation is nearly perfectly linear, since the option is at-the-money. The distribution of returns will still be quite skewed, since the variation in portfolio value arising from spot fluctuations is skewed. The portfolio is revalued using 1 S p(s, τ, X, σ, r, r ) 1 Se u i,spot p(seui,spot,τ,x,σe u i,vol,r,r ) + eui,spot 1 e u δ. i,spot Table 1 shows results of one run of the Monte Carlo simulation with n =10,000. Incorporating vega risk gives a more accurate picture of the risks of the hedged position, raising the VaR from USD 1,701 to USD 2,589. The difference between the median and mean of the simulated results is a back-of-the-envelope measure of the skewness in the distributions of returns. As the examples make clear, vega risk can add considerably to the VaR of a portfolio containing options. One reason for this is the typically high vol of vol. Far from being a constant, as the Black-Scholes model 15

18 assumes, implied volatility fluctuates widely, as can be seen from Figure 3. In Example 2.2, the vol of vol was about 5.6 percent per day, or 80 percent per annum. The vol of vol itself also fluctuates widely: for example, the vol of vol of dollar-yen has ranged between 60 and 140 percent in 1998 and Vega risk in the presence of a volatility smile and term structure Portfolios containing options are exposed not only to changes in the level of implied volatility, but to changes in the curvature and skewness of the smile, and changes in implied volatility along the smile. changes in the slope of the term structure of volatility, changes along the term structure of volatility as the option ages, We will examine each of these in turn, focusing on foreign exchange markets. 3.1 The volatility smile in the foreign exchange markets In foreign exchange markets, options are traded among dealers using the option delta as a metric for exercise price and the Black-Scholes implied volatility as a metric for price. The volatility smile can then be represented as a schedule of implied volatilities of options on the same underlying and with the same maturity but different deltas. This convention is unambiguous, since a unique exercise price corresponds to each call or put delta and a unique option premium in currency units corresponds to each implied volatility. The most liquid option markets are for at-the-money forward and for 25-delta calls and puts. There are also somewhat less actively traded markets in 10-delta calls and puts. 13 The liquidity of out-of-the-money options in foreign exchange markets has led to the widespread trading of option combinations, in particular the straddle, the strangle, and the risk reversal. The straddle combines 13 See (Malz 1997) for detail. 16

19 a call and a put with the same exercise price, usually the current forward outright rate. The strangle and the risk reversal combine an out-of-the-money call and an out-of-the-money put with the same delta, usually 25 percent (less often 35, 30, or 10 percent), and the same maturity. The exercise price of the call component is higher than the current forward outright rate, and the exercise price of the put is lower by the approximately the same proportional amount. Figures 5 and 6 illustrates the payoffs at maturity on these instruments. The strangle is the simpler of the out-of-the-money combinations. It consists of a long out-of-the-money put and call. The dealer either sells the combination to the counterparty or buys it from her. The risk reversal consists of a long out-of-the-money call and a short out-of-the-money put. 14 The dealer exchanges one of the options for the other with the counterparty. Because the put and the call are generally not of equal value, the dealer pays or receives a premium for exchanging the options. Figure 5: 25δ risk reversal payoff at maturity call strike 25 call strike 4 forward S T As noted, the prices of these option combinations are expressed in vols rather than currency units in over-thecounter market parlance. The implied volatility of the put component of an at-the-money forward straddle is identical to that of the call (by virtue of put-call parity) and is referred to as the straddle or at-the-money 14 A foreign exchange option can always be described as both a call and a put. For example, a call on the dollar denominated in yen is also a put on the yen denominated in dollars. 17

20 Figure 6: 25δ strangle payoff at maturity 75 call strike 5 25 call strike 3 1 forward S T forward volatility. It is a measure of the general level of implied volatility for options of a given maturity on a particular currency pair, or a measure of location of the smile. The prices of strangles and risk reversals are also quoted by dealers in vols. Strangle prices are sometimes quoted as the average of the implied volatilities of its put and call components, but more often as the spread between the average and the at-the-money forward vol. Risk reversal prices are quoted as the spread between the call and the put vol. Denoting the 25-delta call and 25-delta put (75-delta call) volatilities by σ 25δ t σ 75δ t and the strangle spread, risk reversal spread, and at-the-money forward volatilities by str t, rr t, and atm t, the midpoint of the time-t strangle price can be expressed as str t = 0.5(σ 75δ t risk reversal price as rr t = σ 25δ t and + σt 25δ ) atm t and the σt 75δ. 15 Using these definitions, the market quotes for the strangle price, risk reversal price, and at-the-money volatility can be solved for σ 25δ t and σ 75δ t : 15 In this notation, we have conveniently mapped both the call and put vols into the call delta and denoted the vol of the 25-delta put by σt 75δ. However, the call delta of a 25-delta put is not exactly 0.75, but the slightly smaller quantity e r τ 0.25, where r is the domestic money market rate and e r τ is the present value of a foreign currency unit deliverable at the option maturity date. 18

21 σt 25δ = atm t + str t + 0.5rr t (4) σt 75δ = atm t + str t 0.5rr t To close a straddle deal, the counterparties use the Black-Scholes formulas to translate the straddle price in vols into currency units. To close a strangle or risk reversal deal, the exercise prices of the individual components must also be set, which in turn requires the counterparties to agree on σt 25δ volatilities of the 25 delta call and the 25 delta put. 16 Black-Scholes delta of a call option and σt 75δ, the implied These are substituted into the expression for the e r τ S t ln ( S t ) ) X + (r r + σ 2 τ 2 σ, (5) τ which is set equal to 0.25 (or e r τ 0.25) to solve for the exercise price of the call (or put). They then translate the price in vols into currency units using the Black-Scholes formulas. The 25-delta risk reversal and strangle prices contain a great deal of information about the shape of the volatility smile: the strangle price is an indicator of the degree of curvature of the smile, while the risk reversal price is an indicator of the degree of skewness of the smile. In Figure 7, we have interpolated between the observed at-the-money forward and 25- and 75-delta implied volatilities and have extrapolated the smile out to arrive at a smooth and continuous function mapping an implied volatility to every delta on [0,e r τ ]. The particular interpolation scheme used in Figure 7 is to pass a polynomial through the observed volatilities. Generally, n volatilities can be fitted exactly with a polynomial of degree n 1, so a quadratic is required to fit the 25-, 50-, and 75-delta implied volatilities. In the typical case in foreign exchange option markets in which 25-, 50-, and 75-delta implied volatilities can be extracted from risk reversal and strangle prices, the smile has an intuitively appealing closed form parametrization: 16 There is a negligible difference between the value of the strangle calculated by applying the 25-delta call and put and by applying a single average strangle vol to both the put and the call. Dealers correct for this by adjusting the strangle vol so as to get the correct strangle value. 19

22 Figure 7: The volatility smile in the foreign exchange option market on February 8, 1999 Ν call vol call vol ATM forward vol call σ(δ) = atm t 2rr t (δ 0.50) + 16str t (δ 0.50) 2. (6) The smile in terms of delta can be readily translated into a smile in terms of exercise price by substituting equation (5) into equation (6) and solving for σ as a function of X, as in Figure The smiles displayed in both Figures 4 and 8 are skewed toward lower values of the underlying asset. This is commonly called a put skew, since out-of-the-money puts have higher vols than equally out-of-the-money calls. This is something of a misnomer, since the smiles of puts and calls are identical when graphed against exercise price. If the put and call smiles differed, it would violate put-call parity and open up arbitrage opportunities. 17 There are two trivial approximations involved in the interpolation scheme. First, the delta of an at-the-money forward option is close to, but not precisely 0.5. Second, as noted, the implied volatility of a 25-delta put is identical to that of a call with a delta slightly lower than

23 Figure 8: The volatility smile as a function of strike on February 8, 1999 Ν call strike forward 25 call strike X 3.2 Fixed volatility smile In the simpler approach to incorporating the smile, we ignore changes in the shape of the volatility smile and take into account only parallel shifts in the volatility smile and changes in implied volatility along the volatility smile as the spot rate changes. We use Monte Carlo computation, as in Example 2.3 above. The only new wrinkle here is that implied volatility scenarios are determined not only by the normally-distributed shocks u i,vol to the general level of implied volatility, but by a displacement along the volatility smile. The displacement along the smile is determined by the shock to the cash price, as shown in Figure 9. In the fixed-smile approach, we assume the shock to implied volatility shifts the entire volatility smile up or down, but does not change the shape of the smile. The smile shifts in exercise price-vol space, but not in delta-vol space. 18 The fixed smile approach partially compensates for one of the shortcomings of conventional VaR, the fact that the lognormality assumption is only an approximate description of the actual behavior of cash prices 18 (Derman 1999) labels this approach the sticky smile approach to implied volatility. 21

24 and implied volatilities. The volatility smile is one residue of this non-normality. The fixed smile approach lets us take non-normality into account using standard VaR computations. Example 3.1 (Example 2.3 continued) We return to the example of a delta-hedged short at-the-money dollar-yen option and continue to assume that the at-the-money volatility is 15 percent. In addition, we will assume that the risk reversal and strangle are -2.5 percent and 0.5 percent. Typically, the volatility skew in the dollar-yen market favors the yen, that is, the average risk reversal is negative. A risk reversal of would be considered high, but not unusually so. Similarly, a curvature of is high, but not extreme, for dollar-yen. These data are close to those observed on February 8, 1999 and displayed in Figures 7 and 8, but are rounded off in the example for ease of exposition. Figure 9: Perturbation of spot and vol under the fixed smile approach Ν 0.18 spot shock 0.17 vol shock 0.16 spot shock X A positive shock to the underlying shifts the smile to the right, so that the at-the-money implied volatility now applies to a higher exercise price. An existing put (call) option which yesterday was at-the-money is now slightly out-of-the-money (in-the-money). It is revalued at the implied volatility corresponding to its new, lower (higher) delta. For the dollar-bearish volatility smile of our example, this means that a positive 22

25 shock to the exchange rate (dollar up) leads to a rise in the implied volatility of an at-ten-money put and a decline in the implied volatility of an at-ten-money call along the volatility smile. In our Monte Carlo computation of VaR, we combine correlated shocks to spot and implied volatility. To understand the net effect of the smile on VaR, we need to analyze the effect on portfolio value of the four possible combinations of positive and negative shocks to these two factors: Spot up, vol up This is the worst combination for the hedged short at-the-money put. The rise in the dollar s value creates a bad gamma trading loss (as will be the case for a drop in spot). The dollar appreciation also drives the volatility of the put higher along the smile, as it becomes, say, a 40- rather than 50-delta option. Thus volatility rises both as a result of the shock to volatility and, in addition, as a result of the change in moneyness. Spot up, vol down The rise in the dollar s value creates a bad gamma trading loss and the dollar appreciation also drives the volatility of the put higher along the smile. In this case, the rise in vol along the smile is offset by its decline due to the vol shock. Spot down, vol up The fall in the dollar s value also creates a bad gamma trading loss. The vol falls along the volatility smile as the put becomes, say, a 60-delta option, offsetting the rise due to the vol shock. Spot down, vol down This is the best combination for the hedged short at-the-money put. The vol falls both along the smile and because of the vol shock. The result is gain offset only by the bad gamma trading loss. A similar analysis can be applied to hedged long put, and long and short call portfolios. The long put portfolio behaves precisely as does the short put portfolio, but with signs reversed. The hedged call portfolios fare exactly the same as hedged put portfolios. To understand the net effect of the smile on VaR, we need also to consider the correlation between the at-themoney implied volatility and the spot rate. Because of the pronounced negative correlation between spot and vol, there is a preponderance of scenarios in which spot and vol move in opposite directions, which leads to 23

26 offsetting effects of the smile and the vol shock. As a result, the smile has little effect on VaR, although it does have an effect on the distribution of returns. As seen in Figure 10, the distribution of gains and losses when the smile is taken into account is less disperse, skewed and kurtotic than when the smile is ignored. The smile thus appears to be oddly self-insuring. In general, one can expect a bearish volatility smile, that is, one skewed to lower strikes, when there is a negative correlation between implied volatility and the underlying price. Figure 10: Distribution of gains with and without a smile. Hedged short dollar-yen put freq with smile with no smile gain Table 2 shows the results of Monte Carlo simulations of hedged portfolios containing at-the-money dollar-yen puts using the fixed smile technique. The results for each portfolio are compared with the results for the same portfolio in the absence of a smile. 19 The smile risk for an at-the-money option portfolio is greatest when the skew runs counter to the correlation of spot and implied volatility. To illustrate, Table 2 shows the VaR of a short put in a market environment in which the risk reversal price is rather than vols. This would be an unusual case: Risk reversal prices would not be likely to reach without a positive correlation between spot and implied volatility 19 All the simulations in this paper are carried out using the same realization of 10,000 normally distributed shocks. 24

27 Table 2: VaR of hedged short dollar-yen put option portfolio with fixed smile 0.05 quantile 0.95 quantile mean median Short put with smile Short put without smile Short put with dollar-bullish smile quantile VaR ( 1) being observed or becoming likely going forward. If it were to happen, the effect would be to increase the VaR significantly, and generally to increase the dispersion and skewness of the return distribution. Example 3.2 (How to get murdered by the smile) As an example of how large an impact the smile can have on VaR, consider a risk reversal. A risk reversal is vega neutral: its delta is identically equal to 0.50 at the time it is initiated, since it combines a short and long position with deltas of 25 percent, while the vega is extremely close to zero at initiation. Figure 11 shows how the vegas of an at-the-money forward and a risk reversal vary as the spot rate varies. 20 As is the case for any portfolio of options, the delta hedge of a risk reversal must be adjusted as the spot exchange rate changes. Similarly, as the spot exchange rate fluctuates, it moves closer to the exercise price of one of the options and further from the exercise price of the other, so the risk reversal is no longer vega neutral. In other words, both the delta and the vega must be actively managed as the spot rate changes. However, at the time of initiation, in the absence of a volatility smile, the return profile of the risk reversal is symmetrical. In the presence of a smile, the return distribution is no longer symmetrical. In fact, the distribution of returns on the position can be heavily skewed. Consider a short position in a yen-bullish risk reversal consisting of a long position in a 25-delta dollar put and a 25-delta dollar call in a market in which the volatility smile is as shown in Figure 7. Dealers might have such position because of customer demand for protection against a weaker dollar. 20 The vegas are adjusted for the smile. 25

28 Figure 11: Vega of a risk reversal Κ call strike 25 call strike 5 forward ATM forward risk reversal S t Table 3: VaR of hedged yen-bullish risk reversal with fixed smile 0.05 quantile 0.95 quantile mean median Risk reversal with smile Risk reversal without smile quantile VaR ( 1) Because a risk reversal is vega neutral and has low gamma risk, the VaR of a delta-hedged risk reversal is very low in the absence of the smile. In the presence of a smile, however, the VaR increases dramatically. In the case at hand VaR more than quadruples, as seen in Table Random volatility smile There is a more complicated approach to incorporating smile effects on VaR, but as we shall see, it does not add much information and is probably not worth implementing. In this approach, the smile is not fixed, but rather the 25-, 50-, and 75-delta implied volatilities are permitted to vary in correlated fashion. For exchange-traded options, one can permit the at-the-money implied volatility and the vols corresponding to 26

29 one or two strikes on each side of the current futures price vary together. The procedure is mechanically analogous to time bucketing along a maturity spectrum. To obtain an implied volatility for the shocked value of the underlying price, we interpolate between the shocked smile points using the interpolation scheme described above. The effect is shown in Figure 12, which shows a decline in the at-the-money forward volatility accompanied by a slight decline in skewness and curvature, which are more likely to be associated with a decline in the at-the-money volatility. Figure 12: Perturbation of spot and vol under the fixed smile approach Ν 0.19 initial perturbed X Example 3.3 (Example 2.3 continued) We first calculate correlated shocks to spot and the 25-, 50-, and 75-delta implied volatilities. The vector of standard deviations is (0.0098, , , ) and the correlation matrix is

30 Table 4: VaR of hedged short dollar-yen put option portfolio with random smile 0.05 quantile 0.95 quantile mean median Short put with random smile quantile VaR ( 1) The 25-, 50-, and 75-delta implied volatilities are highly correlated with one another. The 75-delta call vol has a markedly higher negative correlation with the spot rate, that is, when the dollar weakens the skewness against the dollar becomes more pronounced and the risk reversal price becomes more negative. This may amplify some of the effects of the smile on VaR outlined in the previous section. In the case of the short at-the-money put portfolio, the effects are small, as seen in Table 4. The tendency of the returns to become more symmetrical is somewhat more pronounced. Both the term structure and the shape of the volatility smile generally change slowly over time compared with the level of implied volatility. However, when the levels of spot and implied volatility change abruptly, it is often accompanied by a sharp change in the slope of the term structure and the curvature of the smile. We are unlikely to pick up these relationships with our lognormality assumption regarding implied volatilities, and it is probably best to stress test for their effects. A table of changes in term spreads and measures of smile curvature accompanying large changes in cash prices and implied volatility can provide scenarios for such stress tests. Figure 13 shows how abruptly the smile can change shape. 3.4 The term structure of volatility and vega risk One effect of the term structure of volatility is that as an option ages, its implied volatility changes along the term structure. Except for very short-term options (less than one month), the VaR horizon of one day is typically short compared with the spread between the vols of options with different maturities. The shortening of the option maturity by one day, or even ten days, is therefore unlikely to change the implied volatility materially along the term structure. For example, if the spread between the 1-month implied volatility and 28

31 Figure 13: Dollar-yen 1-month 25-delta risk reversals and strangles. Source: DataMetrics vols strangle risk reversal 4 Jan 96 Jan 97 Jan 98 Jan 99 the 3-month implied volatility is a relatively high 1.20 vols, the aging of a 3-month option will change its volatility by only 15 basis points (0.0015) per day. This is typically small compared with the vol of vol, which in Example 1.1 was over 5 percent. For now, we will therefore ignore changes in the remaining term of the option in our analysis. The mechanics of the aging of options in moving their implied volatilities along the prevailing term structure are analogous to the mechanics of cash price shocks moving implied volatilities of options along the smile. The difference is that the effect of aging is deterministic rather than random. Just as risk managers must address changes in the shape of the volatility smile as well changes along the smile due to variations in cash prices, they must take into account changes in the shape of the term structure of volatility. Term structure risks can be addressed by dividing the maturity spectrum into buckets. For each underlying asset type, options in each maturity bucket represent have a vega risk which is correlated with that of other maturity buckets, with the cash asset prices, and with other factors in the portfolio. In practice, the availability of implied volatility data may limit the fineness of the maturity grid. Example 3.4 (The term structure of dollar-yen volatilities) Returning to the example of dollar-yen, the 29