Nonparametric Pricing and Hedging of Volatility Swaps in Stochastic Volatility Models
aa r X i v : . [ q -f i n . P R ] A p r Nonparametric Pricing and Hedging of Volatility Swaps inStochastic Volatility Models
Frido Rolloos ∗ March 28, 2020
Abstract
In this paper the zero vanna implied volatility approximation for the price of freshlyminted volatility swaps is generalised to seasoned volatility swaps. We also derive howvolatility swaps can be hedged using a strip of vanilla options with weights that are directlyrelated to trading intuition. Additionally, we derive first and second order hedges for volatil-ity swaps using only variance swaps. As dynamically trading variance swaps is in generalcheaper and operationally less cumbersome compared to dynamically rebalancing a continu-ous strip of options, our result makes the hedging of volatility swaps both practically feasibleand robust. Within the class of stochastic volatility models our pricing and hedging resultsare model-independent and can be implemented at almost no computational cost. ∗ [email protected] Assumptions and notations
We will work under the premise that the market implied volatility surface is generated bythe following general stochastic volatility (SV) model dS = σS [ ρ dW + ¯ ρ dZ ] (1.1) dσ = a ( σ , t ) dt + b ( σ , t ) dW (1.2)where ¯ ρ = p − ρ , dW and dZ are independent standard Brownian motions, and the func-tions a and b are deterministic functions of time and volatility. The results derived in thispaper are valid for any SV model satisfying (1.1) and (1.2), which includes among others theHeston model, the lognormal SABR model, and the 3 / C ( S , K ) = E t [( S ( T ) − K ) + ] (1.3)The option price C can always be expressed in terms of the Black-Merton-Scholes (BS) price C BS with an implied volatility parameter I : C ( S , K ) = C BS ( S , K , I ) (1.4)It is assumed that the implied volatility parameter I = I ( S , t , K , T , σ , ρ ) . The BS price is givenby the following well-known formula, C BS ( S , K , I ) = SN ( d + ) − KN ( d − ) (1.5)where N ( d ± ) are normal distribution functions, and d − = log ( S / K ) I √ τ − I √ τ , d + = d − + I √ τ (1.6)and τ = T − t .The generalised Hull-White formula also gives a relationship between options prices andBS prices C ( S , K ) = E (cid:2) C BS (cid:0) SM t , T ( ρ ) , K , σ t , T ( ρ ) (cid:1)(cid:3) (1.7)where M t , T ( ρ ) = exp (cid:26) − ρ ∫ Tt σ du + ρ ∫ Tt σ dW (cid:27) (1.8)and σ t , T ( ρ ) is σ t , T ( ρ ) = s τ ∫ Tt ¯ ρ σ du (1.9)This can be written as C BS ( S , K , I ) = E t (cid:2) C BS (cid:0) SM t , T ( ρ ) , K , σ t , T ( ρ ) (cid:1)(cid:3) (1.10) e will also need BS greeks for vanilla options, in particular the delta ( ∆ BS ) , vega ( ν BS ) ,vanna ( va BS ) , and volga ( vo BS ) : ∆ BS ( S , K , I ) = N ( d + ) (1.11) ν BS ( S , K , I ) = S √ τ N ′ ( d + ) (1.12) va BS ( S , K , I ) = − N ′ ( d + ) I d − (1.13) vo BS ( S , K , I ) = S √ τ N ′ ( d + ) I d + d − (1.14)Special notation will be given to the strike and corresponding implied volatility where d − = S / K − = I − √ τ (1.15) K − is called the zero vanna strike, and I − the zero vanna implied volatility.In Rolloos and Arslan [2017] it is proved that for general SV models the price of a volatil-ity swap at trade inception can be read directly from the market smile of European vanillaoptions, and is approximately equal to the zero vanna implied volatility. The approximationis accurate and valid for a large set of SV models, independent of the specific model param-eters. Moreover, it has been demonstrated in Alos et al. [2019] that for SV models driven byfractional noise, the zero vanna implied volatility approximation remains robust.Except for at trade inception, however, a volatility swap is ‘seasoned’, meaning that itwill have a realised volatility component. To be able to price a volatility swap throughoutits life we must therefore be able to calculate V , T ( t ) = E t s T ∫ T σ du (1.16)for all t ∈ [ , T ] . Furthermore, if we are not able to price seasoned volatility swaps we willnot know how to hedge volatility swaps as the change in the value of a volatility swap is thechange in the seasoned volatility swap price.To introduce notation which will later be used in our discussion on hedging of volatilityswaps, the price of a variance swap is given the notation V , T ( t ) = E t (cid:20) T ∫ t σ du + T ∫ Tt σ du (cid:21) (1.17)Variance swaps are clearly far easier to price and hedge compared to volatility swaps. Insome markets, trading variance swaps is in fact more liquid than trading the theoreticalreplicating portfolio for variance swaps, which is a continuous strip of options. From thatperspective it would be attractive to be able to hedge volatility swaps directly with varianceswaps, and if possible in an as model-independent manner as possible. The historical adjusted spot process
Let us introduce an auxiliary geometric Brownian motion H with constant volatility c , dH = cH dB (2.1)In addition, we require that the process H is independent of S and σ , which means that dBdW = dBdZ =
0. We shall later see that c is related to the historical realised volatility of S . Define the ‘historical adjusted’ spot process as S H = SH (2.2)The SDE for S H reads dS H = S H [ c dB + ρσ dW + ¯ ρσ dZ ] (2.3)Equation (2.3) can be rewritten in the following two equivalent ways: dS H = S H hp c + ¯ ρ σ dW ⊥ + ρσ dW i (2.4) dS H = S H (cid:2) c dB + σ dB ⊥ (cid:3) (2.5)where dW ⊥ = c dB + ¯ ρσ dZ p c + ¯ ρ σ , dB ⊥ = ρσ dW + ¯ ρσ dZσ (2.6)Integrating the SDE (2.4) gives S HT = e S HT M t , T ( ρ ) (2.7)with e S HT = S H exp (cid:26) − ∫ Tt (cid:2) c + ¯ ρ σ (cid:3) du + ∫ Tt p c + ¯ ρ σ dW ⊥ (cid:27) (2.8) M t , T ( ρ ) = exp (cid:26) − ∫ Tt ρ σ du + ∫ Tt ρσ dW (cid:27) (2.9)Similarly we can integrate (2.5) to obtain S HT = S T H T (2.10)with S T = S exp (cid:26) − ∫ Tt σ du + ∫ Tt σ dB ⊥ (cid:27) (2.11) H T = H exp (cid:26) − ∫ Tt c du + ∫ Tt c dB (cid:27) (2.12)In the remainder of the paper we will set the initial value of H to one, i.e. H = S H = S . The historical adjusted volatility smile
What we mean with the historical adjusted smile is the volatility smile I H of vanilla optionson the process S H . Although this process is not directly traded, we can express options on S H ( T ) in terms of options on S by making use of (2.10) and the independence of dB and dB ⊥ .To construct I H note that by conditioning we can write E t (cid:2) ( S HT − K ) + (cid:3) = E t [( S T H T − K ) + ] = E t [ H T C ( S , K / H T )] = ∫ ∞ hC ( S , K / h ) q ( h ) dh (3.1)where q ( h ) is the lognormal distribution because we have taken c as constant: q ( h ) = hν √ π exp ( − (cid:18) ln h − µν (cid:19) ) , ν = c √ τ , µ = − c τ (3.2)The options prices C ( S , K / h ) under the integral are market prices of vanilla options on S andare available for all strikes. Thus C H ( S , K ) = ∫ ∞ hC ( S , K / h ) q ( h ) dh (3.3)where the C H left of the equality sign denotes the price of options on S H (recall we set H = C BS (cid:16) S , K , I H (cid:17) = ∫ ∞ hC BS ( S , K / h , I ) q ( h ) dh (3.4)As all quantities on the right hand side of (3.4) are known the numerical integration canbe carried out to back out I H . It is important to remember that for each h the impliedvolatility I on the right hand side of (3.4) is the implied volatility corresponding to the strike K / h and not the implied volatility corresponding to K .We have priced options on S HT making use of (2.10). However, options on S HT can also bepriced using (2.7) and conditioning on M t , T ( ρ ) . We shall see that equating the two naturallyleads us to the fair strike of seasoned volatility swaps. Indeed, E t (cid:2) ( S HT − K ) + (cid:3) = E t h ( e S HT M t , T ( ρ ) − K ) + i = E t h C BS (cid:16) SM t , T ( ρ ) , K , σ Ht , T ( ρ ) (cid:17)i (3.5)with σ Ht , T ( ρ ) = s τ ∫ Tt [ c + ¯ ρ σ ] du (3.6)Hence, C BS (cid:16) S , K , I H (cid:17) = E t h C BS (cid:16) SM t , T ( ρ ) , K , σ Ht , T ( ρ ) (cid:17)i (3.7) Volatility swaps pricing
Following the method introduced in Rolloos and Arslan [2017] the right hand side of (3.7)can be Taylor expanded around ρ = I H − where the BS vanna of a vanilla option on S HT is zero. The Taylor expansion of (3.7)around ρ = C BS ( S , K , I H ) ≈ E t (cid:2) C BS ( S , K , σ Ht , T ( )) (cid:3) + ρSE t (cid:20) ∆ BS ( S , K , σ Ht , T ( )) ∫ Tt σ dW (cid:21) (4.1)with σ Ht , T ( ) = s τ ∫ Tt [ c + σ ] du (4.2)Now, restricting to an option with zero vanna strike K − and zero vanna historical adjustedimplied volatility I H − ( t ) , and expanding around I H − ( t ) , we obtain C BS ( S , K − , I H − ) ≈ C BS ( S , K − , I H − ) + ν BS ( S , K − , I H − ) E t (cid:2) ( σ Ht , T ( ) − I H − ) (cid:3) + vo BS ( S , K − , I H − ) E t (cid:2) ( σ Ht , T ( ) − I H − ) (cid:3) + ρSva BS ( S , K − , I H − ) E t (cid:20) ( σ Ht , T ( ) − I H − ) ∫ Tt σ dW (cid:21) (4.3)Since ν BS ( S , K − , I H − ) = vo BS ( S , K − , I H − ) = C BS ( S , K − , I H − ) ≈ C BS ( S , K − , I H − ) + ν BS ( S , K − , I H − ) E t (cid:2) ( σ Ht , T ( ) − I H − ( t )) (cid:3) (4.5)This can only be the case if E t (cid:2) σ Ht , T ( ) (cid:3) = E t s τ ∫ Tt [ c + σ ] du ≈ I H − (4.6)Define the constant c as c = τ ∫ t σ du (4.7)and we obtain our desired result: V , T ( t ) = E t s T ∫ t σ du + T ∫ Tt σ du ≈ I H − r τT (4.8) he above equation contains as a special case the formula for freshly minted volatility swapsderived in Rolloos and Arslan [2017].We summarize the steps required to calculate the price of a seasoned volatility swapat time t with a realized volatility given by (4.7): Given a market smile which is assumedto be generated by a process of the type (1.1) - (1.2), introduce an adjusted spot processdefined by (2.1) - (2.2). Options on the adjusted spot process of maturity corresponding tothe maturity of the volatility swap can be priced with (3.4). Once options on the adjustedprocess have been priced the adjusted implied volatility can be backed out. The final step isto find the adjusted implied volatility where an option on the adjusted process has zero BSvanna. Equation (4.8) then gives the approximate value of the seasoned volatility swap.Note that at time t ′ = t + dt we will update the value of the ‘constant’ c to c ′ to take intoaccount of the new historical realised volatility, and set H ( t ′ ) = c is constant in this context is that at time t we treat it as if it will be constant untilmaturity date in order to be able get the historical volatility ‘under the square root sign’. InAppendix A we show that the pricing as described above gives the exact seasoned volatilityswap price in a BS setting. In order to find the hedge for a (seasoned) volatility swap suppose that at time t we havefound the historical adjusted zero vanna strike and implied volatility for a vanilla option onthe historical adjusted spot price. If we buy this ‘meta option’, which according to equation(3.4) is a strip of market traded vanilla options, the change in value of the option over dt is dC BS ( S , K − , I H − ) = ∫ ∞ h [ dC BS ( S , K − / h , I )] q ( h ) dh (5.1)where in the integrand I = I ( S , K − / h ) . Hence, E t (cid:2) dC BS ( S , K , I H ) (cid:3) = C BS ( S , K − , I H − ) satisfies the BS partial differential equation, and takinginto account that the vanna and volga contributions are zero, √ τν BS ( S , K − , I H − ) (cid:2) dC BS ( S , K − , I H − ) − ∆ BS ( S , K − , I H − ) dS (cid:3) = d ( I H − √ τ ) − σ I H − √ τ dτ (5.3)The risk-neutral drift of the zero vanna implied volatility at time t is thus E t (cid:2) d ( I H − √ τ ) (cid:3) = σ I H − √ τ dτ (5.4) s the volatility swap price is a martingale, and from equation (4.8) we see that at eachinstant it is (approximately) equal to the zero vanna implied volatility, the change in a fixedstrike implied volatility which is initially zero vanna cannot be equal to the change in thevolatility swap price: d (V , T ( t ) T ) , d ( I H − √ τ ) (5.5)This is because a fixed strike implied volatility which is zero vanna at time t will not be zerovanna at t + dt . Indeed, from equation (5.4) it is clear that an initially zero vanna impliedvolatility is not a martingale. The new zero vanna implied volatility at t + dt will correspondto a different strike.Let us introduce the total change in zero vanna implied volatility D ( I H − √ τ ) . This totalchange must satisfy d (V , T ( t ) T ) ≈ D ( I H − √ τ ) (5.6)The total change, which is the change from the zero vanna implied volatility at t to the zerovanna implied volatility at t + dt , can be written as: D ( I H − √ τ ) = d ( I H − √ τ ) + δ ( I H − √ τ ) + ∂ ( I H − √ τ ) ∂ log K d log K − + H . O . (5.7)The term d ( I H − √ τ ) is the usual change in fixed strike implied volatility. However, we knowthat to recalculate the volatility swap price the new realised volatility needs to be takeninto account. Since the infinitesimal change in realised volatility is σ dt , the change infixed strike implied volatility purely due to the realised volatility update will be of order dt : δ ( I H − √ τ ) = O ( dt ) . The term involving the slope of the adjusted implied volatility skew at theinitial zero vanna strike is the correction needed to find the new zero vanna implied volatilitystrike after the usual change in implied volatility and change due to the realised volatilityupdate.Recall that the zero vanna strike and implied volatility at t satisfieslog ( S / K − ) = ( I H − √ τ ) (5.8)We can therefore relate the total change in zero vanna adjused implied volatility to therequired change in strike to maintain zero vanna: d log K − = d log S − I H − √ τ D ( I H − √ τ ) + ( D ( I H − √ τ )) (5.9)Substituting this into equation (5.7) and after some rearranging, gives (cid:18) + I H − √ τ ∂ ( I H − √ τ ) ∂ log K (cid:19) D ( I H − √ τ ) = d ( I H − √ τ ) + δ ( I H − √ τ ) + ∂ ( I H − √ τ ) ∂ log K d log S + H . O . (5.10)Using equation (5.6), it follows that E t (cid:2) D ( I H − √ τ ) (cid:3) ≈ E t (cid:2) d (V , T ( t ) T ) (cid:3) = nd hence, because δ ( I H − √ τ ) = O ( dt ) , it follows from (5.10) that δ ( I H − ( t )√ τ ) = E t (cid:2) δ ( I H − √ τ ) (cid:3) ≈ − E t (cid:2) d ( I H − √ τ ) (cid:3) − ∂ ( I H − √ τ ) ∂ log K E t [ d log S ] − H . O . ≈ − σ I H − √ τ dτ − ∂ ( I H − √ τ ) ∂ log K E t [ d log S ] − H . O . (5.12)Substituting this back into (5.10), (cid:18) + I H − √ τ ∂ ( I H − √ τ ) ∂ log K (cid:19) D ( I H − √ τ ) ≈ d ( I H − √ τ ) − σ I H − √ τ dτ + ∂ ( I H − √ τ ) ∂ log K dSS (5.13)because d log S = E t [ d log S ] + dS / S . The above can be simplified further by noting that forthe class of SV models considered, ∂ ( I H − √ τ ) ∂ log K = − ∂ ( I H − √ τ ) ∂ log S (5.14)and so, (cid:18) + I H − √ τ ∂ ( I H − √ τ ) ∂ log K (cid:19) D ( I H − √ τ ) ≈ d ( I H − √ τ ) − σ I H − √ τ dτ − ∂ ( I H − √ τ ) ∂ S dS (5.15)Comparing equation (5.15) with the BS PDE (5.3) we see that d (V , T ( t ) T ) ≈ D ( I H − √ τ ) ≈ N (cid:2) dC H ( S , K − ) − ∆ H ( S , K − ) dS (cid:3) (5.16)where ∆ H is the SV skew-adjusted delta ∆ H ( S , K − ) = ∆ BS ( S , K − , I H − ) + ν BS ( S , K − , I H − ) ∂ I H − ∂ S (5.17)and the notional N is N = (cid:18) ν BS ( S , K − , I H − )√ τ (cid:19) − (cid:18) + I H − √ τ ∂ ( I H − √ τ ) ∂ log K (cid:19) − (5.18)What the hedging formula (5.16) says is that the change in volatility swap price can beapproximately hedged in a self-financing manner by trading a delta-hedged zero vanna optionon the historical adjusted spot process, where the delta is the skew adjusted delta, and thenotional is a skew adjusted and vega-weighted notional. Note that this formula is identicalin form to the formula in [] for hedging forward starting volatility swaps. The reason thatthere is no delta hedge in the formula for forward starting volatility swaps is because theforward starting options used in the paper are insensitive to the spot price movements.As in Carr and Lee [2009], our hedge for the volatility swap involves continuous rebal-ancing of vanilla options of all strikes with appropriate weights (recall that a vanilla option n the adjusted spot process is a strip of market traded vanilla options). The differencebetween our approach and Carr and Lee [2009] is that the weights are directly related tointuitive trading concepts such as BS greeks and slope of the skew. Additionally, the weightsare smooth functions of strike and not highly oscillatory.Even though hedging with theoretical strip of options is, within our approach and frame-work, the most accurate hedge for the volatility swap, it is not the cheapest way to hedge. Forthis reason, the next section will discuss an approach to hedge volatility swaps using varianceswaps only, which for some indices is cheaper to trade than a portfolio consisting of optionsof all possible strikes. The price to pay for hedging with relatively cheaper instruments ispotential loss of accuracy (increased hedging error). Two approximations for hedging volatility swaps with variance swaps will be given: a firstorder and second order approximation. Both will be based on Gatheral’s formula for varianceswaps. We recall that for a freshly minted variance swap at time t it reads V t , T ( t ) τ = ∫ ∞−∞ N ′ ( d − ) ∂ d − ∂ log K I τ d log K (5.19)This formula can be generalised to seasoned volatility swaps. Note that the price of aseasoned volatility swap is given by V , T ( t ) T = − E t (cid:2) log S HT / S H (cid:3) (5.20)with c given by (4.7). A log-contract on the adjusted spot process S H can be synthesised usingoptions on the adjusted spot process C BS ( S , K , I H ) . Hence, Gatheral’s formula for seasonedvolatility swaps reads V , T ( t ) T = ∫ ∞−∞ N ′ ( d − ) ∂ d − ∂ log K ( I H √ τ ) d log K (5.21)which can be rewritten in the slightly more convenient form V , T ( t ) T = ∫ ∞−∞ N ′ ( d − )( I H √ τ ) dd − (5.22) The first order approximation is found by expanding I H √ τ around d − =
0. This leads to ∫ ∞−∞ N ′ ( d − )( I H √ τ ) dd − ≈ ( I H − √ τ ) ∫ ∞−∞ N ′ ( d − ) dd − + ∂ ( I H − √ τ ) ∂ d − ∫ ∞−∞ d − N ′ ( d − ) dd − += ( I H − √ τ ) (5.23) ence, the lowest order approximation is V , T ( t ) T ≈ ( I H − √ τ ) ≈ V , T ( t ) T (5.24)From which it follows that d V , T ( t ) ≈ I H − √ τ dV , T ( t ) (5.25)By dynamically trading variance swaps with notional 1 /( I H − √ τ ) we can hedge a volatilityswap. Note however that this first order approximation ignores the convexity correction. A more accurate approximation can be derived by expanding up to order two. That is, V , T ( t ) T ≈ ( I H √ τ ) + ∂ ( I H − √ τ ) ∂ d − ∫ ∞−∞ d − N ′ ( d − ) dd − = ( I H − √ τ ) + ∂ ( I H − √ τ ) ∂ d − (5.26)This formula gives an intuitive interpretation of the convexity correction, namely as theconvexity of the historical adjusted implied variance at the zero vanna strike: V , T ( t ) T − V , T ( t ) T ≈ V , T ( t ) T − ( I H − √ τ ) ≈ ∂ ( I H − √ τ ) ∂ d − (5.27)To find the hedge ratio, note that dV , T ( t ) T − D (( I H − √ τ ) ) = dV , T ( t ) T − I H − √ τ D ( I H − √ τ ) − ( D ( I H − √ τ )) ≈ D (cid:18) ∂ ( I H − √ τ ) ∂ d − (cid:19) ≈ ∂ ( I H − √ τ ) ∂ d − D ( I H − √ τ ) + D (· · · ) (5.28)As E ( dV , T ( t ) T ) = E ( D ( I H − √ τ )) ≈ dV , T ( t ) T and D ( I H − √ τ ) willapproximately cancel out. We are therefore left with dV , T ( t ) T − I H − √ τ D ( I H − √ τ ) ≈ ∂ ( I H − √ τ ) ∂ d − D ( I H − √ τ ) (5.29)Using (5.6) we arrive at d V , T ( t ) ≈ I H − √ τ + ∂ ( I H − √ τ ) ∂ d − dV , T ( t ) (5.30) eferences E. Alos, F. Rolloos, and K. Shiraya. On the difference between the volatility swap strike andthe zero vanna implied volatility, 2019.P. Carr and R. Lee. Robust replication of volatility derivatives, 2009. Working Paper, NewYork University.D. B. Owen. A table of normal integrals.
Communications in Statistics - Simulation andComputation , 1980.F. Rolloos and M. Arslan. Taylor-made volatility swaps.
Wilmott , 2017. The Black-Merton-Scholes case
It is clear that equation (3.4) is the key to pricing (seasoned) volatility swaps. In generalit cannot be solved analytically and although approximations are possible, we would like tolimit the number of approximations made in the pricing. Numerical integration is there-fore preferable to analytical approximations when evaluating (3.4). In this section we willdemonstrate that when we limit ourselves to a Black-Scholes world with deterministic termstructure an analytical solution can be obtained and we recover the exact price of a seasonedvolatility swap.In what follows we will need to evaluate a double integral involving the normal probabilitydensity function and its cumulative distribution function. Let ϕ ( x ) = √ π exp (cid:26) − x (cid:27) (A.1) Φ ( x ) = ∫ x −∞ ϕ ( y ) dy (A.2)Then, for constants a and b , it can be shown that ∫ ∞−∞ Φ ( a + bx ) ϕ ( x ) dx = Φ (cid:18) a √ + b (cid:19) (A.3)For the above and other Gaussian integrals we refer the reader to Owen [1980].In a Black-Scholes world with deterministic term structure of volatility, the impliedvolatilities I and those of the adjusted process I H will not depend on the strike of the option.Instead of trying to solve (3.4) directly, we make the observation that the zero vanna impliedvolatility is equivalently characterized by the point where the price of a binary option isone-half (in the presence of a skew this is not the case anymore). Thus, for the option onthe adjusted process we must have − (cid:18) ∂ C ( S , K , I H ) ∂ K (cid:19) I H = I H − ( t ) , K = K − =
12 (A.4)Now we can differentiate the right-hand side of (3.4) and find the strike K and correspondingimplied volatility I such that the integral is equal to one-half: − ∂∂ K ∫ ∞ hC BS ( S , K / h , I ) q ( h ) dh =
12 (A.5)Introduce a new variable y = ln h , then − ∂∂ K ∫ ∞ hC BS ( S , K / h , I ) q ( h ) dh = ∫ ∞−∞ Φ ( a + by ) ν √ π exp (cid:26) − (cid:16) y − µν (cid:17) (cid:27) dy (A.6)where a = ln S / K − I τI √ τ , b = I √ τ (A.7) efine now the variable y ′ = ( y − µ )/ ν and we obtain ∫ ∞−∞ Φ ( a + by ) ν √ π exp (cid:26) − (cid:16) y − µν (cid:17) (cid:27) dy = ∫ ∞−∞ Φ ( a + µb + νby ′ ) ϕ ( y ′ ) dy ′ = Φ (cid:18) a + µb √ + ν b (cid:19) (A.8)where we have used (A.3). This immediately gives the strike where the vanna of the optionon the adjusted process S H is zero: Φ (cid:18) a + µb √ + ν b (cid:19) = ⇐⇒ a = − µb (A.9)Since µ = − c τ , the above condition givesln S / K − − I τ = c τ (A.10)which means that the zero vanna strike for the option on the adjusted process isln S / K − = (cid:0) c + I (cid:1) τ (A.11)The zero vanna adjusted implied volatility I H − satisfiesln S / K − = ( I H − ) τ (A.12)and so, I H − = √ c + I (A.13)In the Black-Scholes model with deterministic volatility term structure σ , c = τ ∫ t σ du , I = τ ∫ Tt σ du (A.14)and so (4.8) gives us V , T ( t ) ≈ I H − r τT = s T ∫ t σ du + T ∫ Tt σ du (A.15)but of course the approximate result is in fact exact in the Black-Scholes case.We could have written down equation (A.15) without carrying out the previous cal-culations since in a Black-Scholes world there is no uncertainty about realized volatility.Nevertheless, the fact that we arrive at the result using our method which is applicable togeneral stochastic volatility models, shows that equation (4.8) is correct.(A.15)but of course the approximate result is in fact exact in the Black-Scholes case.We could have written down equation (A.15) without carrying out the previous cal-culations since in a Black-Scholes world there is no uncertainty about realized volatility.Nevertheless, the fact that we arrive at the result using our method which is applicable togeneral stochastic volatility models, shows that equation (4.8) is correct.