Some Practical Issues in FX and Equity Derivatives


 Amos Roberts
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1 Some Practical Issues in FX and Equity Derivatives
2 Phenomenology of the Volatility Surface The volatility matrix is the map of the implied volatilities quoted by the market for options of different strikes and different maturities. Implied volatility is the parameter σ needed to calculate the B&S formula. In practice the matrix can be built according to two main rules: Sticky Delta: the matrix of implied volatilities is mapped, for each expiry, with respect to the of the option; this rule is usually adopted in the over the counter markets (e.g.: fx options). Options are priced depending on their. That has subtle implications for the running of a book of options. Sticky Strike: the matrix of implied volatilities is mapped, for each expiry, with respect to the strike prices; this the rule is usually adopted in official markets (e.g.: equity options and futures options). Implied volatilities remain constant for each strike, even if the underlying asset price changes.
3 Phenomenology of the Volatility Surface Sticky Delta Rule: For each strike K and expiry T the implied volatility of any option is dependent on the level of its and consequently on the movements of the underlying asset (and on the elapsing of time).
4 Phenomenology of the Volatility Surface Sticky Strike Rule:For each strike K and expiry T the implied volatility of any option is independent of the movements of the underlying asset.
5 Sticky Strike Arbitrage Consider the P&L of a perfectly hedged portfolio in a short period dt: P&L = dπ + f = dc ds + f where f is the financing cost of the position. Let s assume the underlying asset s evolution is commanded by a Brownian motion: ds = µsdt + σ t SdZ and apply Ito s lemma (note that σ t is the actual and not the implied volatility) dc = (Θ σ2 ts 2 Γ + µ )dt + σ t SdZ
6 Sticky Strike Arbitrage Since f is: f = ( r d (t)c + r d (t) S r f (t) S)dt = ( r d (t) S + Df d r d (t)kφ(d 2 ) + r d (t) S r f (t) S)dt we get Θdt + f = 1 2 σ2 KS 2 Γdt and hence (recalling Θ definition): P&L = 1 2 S2 Γ[σ 2 t σ 2 K]dt
7 Sticky Strike Arbitrage Now, let s assume we have two call option C 1 (K 1, σ 1 ) and C 2 (K 2, σ 2 ) with σ 1 > σ 2. We build a portfolio such as: long Γ 1 options C 2 and short Γ 2 options C 1 The P&L in the interval dt is. which is a positive amount. P&L = Γ 1Γ 2 2 S2 [σ 1 σ 2 ]dt Thus Sticky Strike rule imply an arbitrage if it is realized in the market
8 Sticky Delta Arbitrage The Sticky Delta rule imply arbitrage as well, if it is realized in the market, though in less evident way. One should consider the shape of the surface in terms of slope and convexity and then build complex portfolios Generally speaking, with relatively symmetric matrices (such as in the Eur/Usd FX market), a position short butterfly grants a positive Γ and positive Θ portfolio. A short butterfly = long an ATM straddle and short a symmetric strangle (e.g.: short a 25 Call and a 25 Put), in an amount such that the total Vega of the portfolio is nil
9 Conclusions on the Sticky Strike and Sticky Delta Rules Matrices with different implied volatility for different levels of strike show that the constant volatility assumption of the B&S model is not realistic. The two Sticky Strike and Sticky Delta rules imply arbitrage should they actually be realized: so they both cannot be considered as a feasible model of the evolution of the volatility surface. They can be considered just as two convention for quoting volatility surfaces and they are respectively chosen according to their suitability to different markets.
10 Phenomenology of the Volatility Surface The evolution of the volatility surface can be decomposed in three main movements, for each expiry: Parallel Shift Convexity Increase/Decrease Slope Increase/Decrease. To represent these movements in terms of market instruments, one can consider: The ATM straddle volatility as an indicator of the level The Vega Weighted Butterfly as an indicator of the convexity The Risk Reversal as an indicator of the slope.
11 Phenomenology of the Volatility Surface Parallel
12 Phenomenology of the Volatility Surface Convexity
13 Phenomenology of the Volatility Surface Slope
14 Phenomenology of the Volatility Surface Composite
15 Model Risk in Hedging Derivatives Exposures Every time we choose a model to price and to hedge a derivative, we make more or less realistic assumptions about the risk factors we want to take into consideration. In practice, markets never behave in the way predicted by the model, so that the risk one incurs in by using a misspecified model is very high. In order to minimize the model risk, one can analyze which is the hedging error arising from the nonrealistic modeling of the factors included in the model; subsequently, one tries to minimize this error. In what follows we will analyze which is the hedging error produced by the false assumption of a constant volatility (as in the B&S model). We also analyze which is the error produced by the more realistic assumption of a changing implied volatility, in case we are not able to correctly describe its evolution.
16 Model Risk: Hedging by B&S (constant volatility) We have shown above that the P&L in short interval dt of a perfectly hedged (under B&S s assumptions) portfolio is: P&L = 1 2 S2 Γ[σ 2 t σ 2 K]dt We make a profit if the realized volatility σ t higher than the implied volatility σ K, and the magnitude of this profit is directly linked to the level of the Γ (which is always positive in the case of a plain vanilla call option). If we integrate over the entire option s life, we obtain the total P&L resulting from running a hedged book at constant implied volatility: P&L = T S2 t Γ(S t, σ K, t)[σ 2 t σ 2 K]dt
17 Model Risk: Hedging by B&S (constant volatility) From the formula above we can infer that: Continuous hedging of a single option revalued at a constant volatility generates a P&L directly proportional to the Γ of the option; In general the P&L of a long position in the option, continuously rehedged, is positive if the realized volatility is, on average during the option s life, higher than the constant implied volatility; it is negative in the opposite case; The previous statement is not always true since the total P&L is dependent on the path followed by the underlying: if periods of low realized volatility are experienced when the Γ is high, whereas periods of high realized volatility are experienced when the Γ is negligible, then the total P&L is negative, though the realized volatility can be higher than implied volatility for periods longer then those when it is lower.
18 Model Risk: Hedging by a Floating Implied Volatility In practice, every day the trader s book is marked to the market, so to have a revaluation as near as possible to the true current value of the assets and other derivatives. That means that the book is revaluated at current market conditions regarding the price of the underlying asset and the implied volatility (we drop for the moment the fact that also the interest rates are updated to the current level). We would like to explore now the impact on the hedging performance when the implied volatility is floating and continuously updated to the market levels.
19 Model Risk: Hedging by a Floating Implied Volatility Under the real probability measure P, the underlying asset s price evolves according to the following SDE: ds = µsdt + σ t dz 1 The implied volatility σ K is now considered as a new stochastic factor and model its evolution, under the real probability measure P, as: dσ K = αdt + ν t dz 2 where dz 1 and dz 2 are two correlated Brownian motion. Under the equivalent martingale measure Q: ds = (r d r f )Sdt + σ t dw 1 dˆσ K = ˆαdt + ν t dw 2 where dw 1 and dw 2 are again two correlated Brownian motion with correlation parameter ρ.
20 Model Risk: Hedging by a Floating Implied Volatility Under the real probability measure P, the option s price evolves as: ( C dc = t C 2 S 2σ2 ts 2 + C C µs + α S σ K C 2 σ 2 K ν 2 t + 2 C σ K S ρσ tsν t ) dt + C S σ tsdz 1 + C σ K ν t dz 2 or, writing the partial derivatives as Greeks: dc = ( Θ Γσ2 ts 2 + µs + Vα Wν2 t + Xρσ t Sν t ) dt + σt SdZ 1 + Vν t dz 2 where V is the Vega, X is the vanna and the W is the volga.
21 Model Risk: Hedging by a Floating Implied Volatility Under equivalent martingale measure Q the dynamics of the call option is: dĉ = Θ Γσ2 ts 2 + (r d r f )S + V ˆα Wν2 t + Xρσ t Sν t = r d Ĉ Let s build a portfolio of: long one call option and short quantity of the underlying; it s P&L over a small period dt is: dπ = dc ds + f where f is the cost born to finance the position: f = ( r d (t)c + r d (t) S r f (t) S)dt
22 Model Risk: Hedging by a Floating Implied Volatility After a few substitutions we get dπ = ( Θ+ 1 2 Γσ2 ts 2 + (r d r f )S+Vα+ 1 2 Wν2 t +Xρσ t Sν t ) dt r d Cdt+Vν t dz 2 Adding and subtracting V ˆα and by means of previous equations we have dπ = Vν t dz 2 + (Vα V ˆα)dt = V(dσ k ˆαdt) By integrating over the option s life: P&L = T 0 dπ = T 0 V(dσ k ˆαdt)
23 Model Risk: Hedging by a Floating Implied Volatility From the formula above we can infer that: Continuous hedging of a single option revalued at a running implied volatility generates a P&L proportional to the Vega of the option. In general the P&L of a long position in the option, continuously rehedged, is positive if the realized variation in implied volatility is, on average during the option s life, higher than its expected (riskneutral) variation, it is negative in the opposite case. The previous statement is not always true since the total P&L is dependent on the path followed by the underlying: if periods of low realized variations of the actual implied volatility are experienced when the Vega is high, whereas periods of high realized variations of the actual implied volatility are experienced when the Vega is negligible, then the total P&L is negative, though the realized variations could be on average grater then expected implied volatility s variation.
24 Hedging Volatility Risk in a B&S World In practice, the a trader s book is frequently updated in terms of the underlying asset price and implied volatility. If the book is revalued and hedged as in a B&S world, then we know from the previous analysis that we have to minimize the model risk by minimizing the Vega exposure. Then the book will be hedged against the movements of the underlying asset; it will be Vegahedged against the change in the implied volatility. Vegahedging must be considered in a very extended meaning: the portfolio must remain Vegahedged even after movements in the implied volatility and/or the underlying asset.
25 Hedging Volatility Risk in a B&S World So, hedging a book in a B&S world implies setting to zero the following Greeks: V X (Vanna or DvegaDspot) W (Volga or DvegaDvol) The exposure is (usually) easily set to zero by trading in the underlying asset s cash market. The volatilityrelated Greeks are set to zero by trading (combinations of) other options.
26 Hedging Volatility Risk in a B&S World Tools to cancel Vega exposures are: ATM straddle: this structure has a strong Vega exposure, low Volga exposure and nil Vanna. Risk Reversal 25 : no Vega and Volga exposures, strong Vanna exposure. Vega Weighted Butterfly 25 : no Vega and Vanna exposures, strong Volga exposures. By combining the three structures above, traders make their book Vegahedged, and the keep this hedging stable to implied volatility ad underlying asset movements.
27 Hedging Volatility Risk in a B&S World Exotic option (e.g.: barriers and One Touch) often show more sensitivity to the Volga and to the Vanna than to the Vega. As an example we consider an Up&Out Eur Call Usd Put option: Spot Ref.: Expiry: 6M Strike: Barrier Up&Out:
28 Hedging Volatility Risk in a B&S World Value spot time to ( 1.05
29 Hedging Volatility Risk in a B&S World Vega spot time to 1.05
30 Hedging Volatility Risk in a B&S World Volga spot time to 1.05
31 Hedging Volatility Risk in a B&S World Vanna spot time t 1.05
32 Hedging Volatility Risk in a B&S World Another example We consider a Down&Out Eur Put Usd Call option: Spot Ref.: Expiry: 3M Strike: Barrier Down&Out:
33 Hedging Volatility Risk in a B&S World Value spot time to (d 1.10
34 Hedging Volatility Risk in a B&S World Vega spot time to ( 1.10
35 Hedging Volatility Risk in a B&S World Volga spot time to ( 1.10
36 Hedging Volatility Risk in a B&S World Vanna spot time to 1.10
37 Hedging Volatility Risk in a B&S World Given the Vega, Vanna and Volga of an option, we calculate the equivalent position in terms of three basic options (ATM, 25 Call and 25 Put): these quantities can be easily converted in amounts of the hedging instruments we have shown above. The U&O Eur Call Usd Put and the D&O Eur Put Usd Call have a volatility exposure as presented in the following table: 25 put 25 call ATM put Up&Out call 79,008,643 54,195, ,556,533 Down&Out put 400,852, ,348, ,163,095 Table 1: Quantities of plain vanilla options to hedge the barrier options according to B&S model.
38 Hedging Volatility Risk in a Stoch Vol World Managing the volatility risk on the B&S s assumption is inconsistent and incomplete. All the volatility related Greeks are zeroed, but the model assumes that the impled volatility is constant, so they should not be hedged. The book is revalued with one implied volatility (typically the ATM), whereas on the market a whole volatility surface is quoted and it changes over time (the three movements for any expiry have been analyzed before). The pricing of exotic options is not consistent with a volatility surface.
39 Hedging Volatility Risk in a Stoch Vol World Need for a model to capture smile effects Nonlognormal models (e.g.: CEV) Localvolatility models (e.g.: Dupire) Stochastc Volatility models (e.g.: Heston, SABR) Lognormal Mixture models (e.g.: Brigo & Mercurio; Brigo Mercurio & Rapisarda)
40 Hedging Volatility Risk in a Stoch Vol World Example: we try to hedge the volatility risk by the Brigo, Mercurio & Rapisarda (2004) model. The hedging procedure is based on the concept of sensitivity bucketing and reflects what a trader is willing to do in practice. This is possible thanks to the model capability of exactly reproducing the fundamental volatility quotes (at least for the three basic instruments). One shifts such a volatility by a fixed amount σ, say ten basis points. One then fit the model to the tilted surface and calculate the price of the exotic, π NEW, corresponding to the newly calibrated parameters. Denoting by π INI the initial price of the exotic, its sensitivity to the given implied volatility is thus calculated as: π NEW π INI σ
41 Hedging Volatility Risk in a Stoch Vol World In practice, it can be more meaningful to hedge the typical movements of the market implied volatility curves. To this end, we start from the three basic data for each maturity (the ATM and the two 25 call and put volatilities), and calculate the exotic s sensitivities to: i) a parallel shift of the three volatilities; ii) a change in the difference between the two 25 wings; iii) an increase of the two wings with fixed ATM volatility. This is actually equivalent to calculating the sensitivities with respect to the basic market quotes. In this way we capture the effect of a parallel, a twist and a convexity movement of the volatility surface. Once these sensitivities are calculated, it is straightforward to hedge the related exposure via plain vanilla options, namely the ATM calls, 25 calls and 25 puts for each expiry.
42 Hedging Volatility Risk in a Stoch Vol World We use the following volatility surface and interest rate data σ ATM σ RR σ V WB P d (0, T) P f (0, T) 1W 13.50% 0.00% 0.19% W 11.80% 0.00% 0.19% M 11.95% 0.05% 0.19% M 11.55% 0.15% 0.21% M 11.50% 0.15% 0.21% M 11.30% 0.20% 0.23% M 11.23% 0.23% 0.23% Y 11.20% 0.25% 0.24% Y 11.10% 0.20% 0.25% Table 2: Market data for EUR/USD as of 31 st March 2004.
43 Hedging Volatility Risk in a Stoch Vol World The hedging quantities calculated according to UVUR model with the scenario approach are shown below. The expiry of the hedging plain vanilla options is once again the same as the corresponding barrier options. It is noteworthy that both the sign and order of magnitude of the hedging options is the similar to those of the BS model we calculated before. 25 put 25 call ATM put Up&Out call 76,409,972 42,089, ,796,515 Down&Out put 338,476, ,078, ,195,436 Table 3: Quantities of plain vanilla options to hedge the barrier options according to UVUR model with the scenario approach.
44 Analogies between B&S and Stoch Vol Hedging The sign and magnitude of the hedging quantities show that some analogies exist between the exposures of an option priced by a B&S model and a model which consider the smile effects. The B&S Vega can be though of as the equivalent of the sensitivity of the option price to a parallel shift of the volatility surface. The B&S Volga is the equivalent of the sensitivity to a change in the convexity of the volatility surface, i.e.: an upward or downward movement of the wings with respect to the ATM level. The B&S Vanna is the equivalent of the sensitivity to a change in the slope of the volatility surface, i.e.: a twist of the wings with respect to the ATM level, considered as a pivot point.
45 More Risks Other risks, related to plain vanilla and exotic options, have to be managed P (Rho) and Φ exposure, i.e.: the sensitivity of the option price to the domestic interest rate and the foreign interest rate, or dividend yield in case of equity option. Risks related to some exotics, e.g.: gap at the breach of the barrier. Correlation risk: many exotic options (especially in the equity market) have as underlying basket of stocks, or the payoff is contingent on the future evolution of a given number of stocks. In these cases, correlation between the single assets become a main risk.
46 Correlation Risk As an example of correlation risk, we discuss three different option types with the following payout structures An atthemoney (ATM) call option on an equally weighted basket of n stocks. An option on the maximum performance of n assets An option on the minimum performance of n assets. Payouts are defined relative to S i, i = 1,.., n i.e.: the asset price at the expiry T = 0.
47 Correlation Risk The risk management of multiasset options implies the canceling of the first and second order spot and volatility sensitivities, though in this case we have to deal with matrices of sensitivities : The vector C S i The Γ matrix C S i S j The Vega vector C σ i The Volga matrix The Vanna matrix C σ i σ j C σ i S j Firstorder correlation risk (correlation Vega) can be calculated as a triangular matrix C ρ ij with i < j.
48 Correlation Risk Single stocks and plain vanilla options on single stocks hedge only the the Vega and the diagonal elements of the Γ, Volga and Vanna matrix. The remaining risks, i.e. the nondiagonal elements (cross Γ, cross Volga and cross Vanna) and the correlation Vega, can be hedged only by other multiasset options. It can be shown that in a B&S world the following relationship holds: C ρ ij = S i S j σ i σ j T C S i S j So that by hedging all the cross Γ exposure one hedges also the correlation Vega exposures.
49 Correlation Risk The correlation risk affects the price of an options in two ways, depending also on the kind of payoff of the structure: It impacts on the volatility of the entire basket of underlying stocks. It impacts on the dispersion of the single stocks within the basket. We make some intuitive considerations on these two effects with respect to the three kind of exotic options we listed above.
50 Correlation Risk Basket options: The value of the option is affected only by the basket volatility. The dispersion of individual assets does not influence the option price, because the payout only depends on the sum of the asset prices at maturity. Higher correlations increase basket volatility and thus the option price. Hence basket options are long in correlation.
51 Options on the Maximum: Correlation Risk Increasing correlations imply higher volatility of the basket Increasing dispersion of the single stocks increases the probability of any stock to reach a very high value at maturity. This effect grows with declining correlations. So for max options, the two effects operate in opposite directions. From moderate to high correlation, option prices decrease with increasing correlation: hence, the dispersion effect is stronger than the basket volatility effect and the max option is short in correlation. It should be stressed that this is the initial exposure when the S i (0) are fixed. Situations can occur where the max option is both short and long in correlation depending on the specific levels of correlation and spot prices.
52 Options on the Minimum: Correlation Risk The min option is affected by both effects, but both take the same direction in this case. The dispersion effect increases option prices as correlations become higher, since this minimizes the probability that any asset reaches a very low level at maturity and maximizes the value of the min option Higher correlation implies also a higher volatility of the basket and this increase the option value. Since both effects operate in the same direction, the correlation sensitivity is positive and especially high for this option type.
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