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Lecture 7: Quadratic Variation-based Payoffs Jim Gatheral, Merrill Lynch Case Studies in Financial Modeling Course Notes, Courant Institute of Mathematical Sciences, Fall Term, 2004 17 Spanning Generalized European Payoffs As usual, we assume that European options with all possible strikes and expirations are traded. In the spirit of the paper by Carr and Madan (1998), we now show that any twice-differentiable payoff at time T may be statically hedged using a portfolio of European options expiring at time T . From Breeden and Litzenberger (1978), we know that we may write the pdf of the stock price ST at time T as ¯ ¯ ¯ 2 2 ∂ C̃(St , K, t, T ) ¯ ∂ P̃ (St , K, t, T ) ¯¯ p(ST , T ; St , t) = = ¯ ¯ ¯ ¯ ∂K 2 ∂K 2 K=ST K=ST where C̃ and P̃ represent undiscounted call and put prices respectively. Then, the value of a claim with a generalized payoff g(ST ) at time T is given by Z ∞ E [ g(ST )| St ] = dK p(K, T ; St , t) g(K) 0 Z F Z ∞ ∂ 2 P̃ ∂ 2 C̃ = dK g(K) + dK g(K) ∂K 2 ∂K 2 0 F where F represents the time-T forward price of the stock. Integrating by parts twice and using the put-call parity relation C̃(K) − P̃ (K) = F − K gives ¯F ¯∞ Z F Z ∞ ¯ ¯ ∂ P̃ ∂ P̃ ∂ C̃ ∂ C̃ 0 ¯ ¯ 0 E [ g(ST )| St ] = g(K)¯ − dK g (K) + g(K)¯ − dK g (K) ¯ ¯ ∂K ∂K ∂K ∂K 0 F 0 F Z F Z ∞ ∂ P̃ 0 ∂ C̃ 0 = g(F ) − dK g (K) − dK g (K) ∂K ∂K 0 F Z F ¯F ¯ 0 = g(F ) − P̃ (K)g (K)¯ + dK P̃ (K) g 00 (K) 0 Z ∞ 0 ¯∞ ¯ dK C̃(K) g 00 (K) − C̃(K)g 0 (K)¯ + F F Z F Z ∞ 00 = g(F ) + dK P̃ (K) g (K) + dK C̃(K) g 00 (K) (70) 0 F 120 By letting t → T in equation (70), we see that any European-style twicedifferentiable payoff may be replicated using a portfolio of European options with strikes from 0 to ∞ with the weight of each option equal to the second derivative of the payoff at the strike price of the option. This portfolio of European options is a static hedge because the weight of an option with a particular strike depends only on the strike price and the form of the payoff function and not on time or the level of the stock price. Note further that equation (70) is completely model-independent. Example: European Options In fact, using Dirac delta-functions, we can extend the above result to payoffs which are not twice-differentiable. Consider for example the portfolio of options required to hedge a single call option with payoff (ST − L)+ . In this case g 00 (K) = δ(K − L) and equation (70) gives Z F Z ∞ £ ¤ + + E (ST − L) = (F − L) + dK P̃ (K) δ(K − L) + dK C̃(K) δ(K − L) 0 F ½ (F − L) + P̃ (L) if L < F = C̃(L) if L ≥ F = C̃(L) with the last step following from put-call parity as before. In other words, the replicating portfolio for a European option is just the option itself. Example: Amortizing Options A useful variation on the payoff of the standard European option is given by the amortizing option with strike L with payoff g(ST ) = (ST − L)+ ST Such options look particularly attractive when the volatility of the underlying stock is very high and the price of a standard European option is prohibitive . The payoff is effectively that of a European option whose notional amount declines as the option goes in-the-money. Then, ¾¯ ½ δ(ST − L) ¯¯ 2L 00 g (K) = − 3 θ(ST − L) + ¯ ST ST ST =K 121 Without loss of generality (but to make things easier), suppose L > F . Then substituting into equation (70) gives · ¸ Z ∞ (ST − L)+ E = dK C̃(K) g 00 (K) ST F Z ∞ C̃(L) dK = − 2L C̃(K) L K3 L and we see that an Amortizing call option struck at L is equivalent to a European call option struck at L minus an infinite strip of European call options with strikes from L to ∞. 17.1 The Log Contract Now consider a contract whose payoff at time T is log( SFT ). Then g 00 (K) = ¯ ¯ − ST1 2 ¯ and it follows from equation (70) that ST =K · µ ¶¸ Z F Z ∞ dK dK ST = − P̃ (K) − C̃(K) E log 2 F K K2 0 F Rewriting this equation in terms of the log-strike variable k ≡ log get the promising-looking expression · µ ¶¸ Z ∞ Z 0 ST dk c(k) E log dk p(k) − = − F 0 −∞ y ¡K ¢ F , we (71) y e ) e ) with c(y) ≡ C̃(F and p(y) ≡ P̃ (F representing option prices expressed F ey F ey in terms of percentage of the strike price. 18 Variance and Volatility Swaps We now revert to our usual assumption of zero interest rates and dividends. In this case, F = S0 and applying Itô’s Lemma, path-by-path µ ¶ µ ¶ ST ST log = log F S0 Z T = d log (St ) 0 Z T Z T σSt 2 dSt − dt (72) = St 2 0 0 122 The second term on the r.h.s of equation (72) is immediately recognizable as half the total variance WT over the period {0, T }. The first term on the r.h.s represents the payoff of a hedging strategy which involves maintaining a constant dollar amount in stock (if the stock price increases, sell stock; if the stock price decreases, buy stock so as to maintain a constant dollar value of stock). Since the log payoff on the l.h.s can be hedged using a portfolio of European options as noted earlier, it follows that the total variance WT may be replicated in a completely model-independent way so long as the stock price process is a diffusion. In particular, volatility may be stochastic or deterministic and equation (72) still applies. Now taking the risk-neutral expectation of (72) and comparing with equation (71), we obtain ¾ ·Z T ¸ · µ ¶¸ ½Z 0 Z ∞ ST 2 dk c(k) E σSt dt = −2 E log =2 dk p(k) + F 0 0 −∞ (73) We see explicitly that the fair value of total variance is given by the value of an infinite strip of European options in a completely model-independent way so long as the underlying process is a diffusion. 18.1 Variance Swaps Although variance and volatility swaps are relatively recent innovations, there is already a significant literature describing these contracts and the practicalities of hedging them including articles by Chriss and Morokoff (1999) and Demeterfi, Derman, Kamal, and Zou (1999). In fact, a variance swap is not really a swap at all but a forward contract on the realized annualized variance. The payoff at time T is ( µ µ ¶¾2 ) ¶¾2 ½ N ½ 1 X Si 1 SN N ×A× log − log − N × Kvar N i=1 Si−1 N S0 where N is the notional amount of the swap, A is the annualization factor and Kvar is the strike price. Annualized variance may or may not be defined as mean-adjusted in practice so the corresponding drift term in the above payoff may or may not appear. From a theoretical perspective, the beauty of a variance swap is that it may be replicated perfectly assuming a diffusion process for the stock price 123 as shown in the previous section. From a practical perspective, market operators may express views on volatility using variance swaps without having to delta hedge. Variance swaps took off as a product in the aftermath of the LTCM meltdown in late 1998 when implied stock index volatility levels rose to unprecedented levels. Hedge funds took advantage of this by paying variance in swaps (selling the realized volatility at high implied levels). The key to their willingness to pay on a variance swap rather than sell options was that a variance swap is a pure play on realized volatility – no labor-intensive delta hedging or other path dependency is involved. Dealers were happy to buy vega at these high levels because they were structurally short vega (in the aggregate) through sales of guaranteed equity-linked investments to retail investors and were getting badly hurt by high implied volatility levels. 18.2 Variance Swaps in the Heston Model Recall that in the Heston model, instantaneous variance v follows the process: p dv(t) = −λ(v(t) − v̄)dt + η v(t) dZ It follows that the expectation of the total variance WT is given by ¸ ·Z T vt dt E [WT ] = E 0 Z T = v̂t dt 0 1 − e−λT = (v − v̄) + v̄T λ The expected annualized variance is given by 1 1 − e−λT E [WT ] = (v − v̄) + v̄ T λT We see that the expected variance in the Heston model depends only on v, v̄ and λ. It does not depend on the volatility of volatility η. Since the value of a variance swap depends only on the prices of European options, it follows that a variance swap would be priced identically by both Heston and our local volatility approximation to Heston. 124 18.3 Dependence on Skew and Curvature We know that the implied volatility of an at-the-money forward option in the Heston model is lower than the square root of the expected variance (just think of the shape of the implied distribution of the final stock price in Heston). In practice, we start with a strip of European options of a given expiration and we would like to know how we should expect the price of a variance swap to relate to the at-the-money-forward implied volatility, the volatility skew and the volatility curvature (smile). It turns out that there is a very elegant exact expression for the fair value of variance; the proof is given in Appendix A. Define √ k σBS (k) T √ + z(k) = d2 = − 2 σBS (k) T Intuitively, z measures the log-moneyness of an option in implied standard deviations. Then, Z∞ 2 E[WT ] = dz N 0 (z) σBS (z)T (74) −∞ To see this formula is plausible, it is obviously correct when there is no volatility skew. Now consider the following simple parameterization of the BS implied variance skew: 2 σBS (z) = σ02 + α z + β z 2 Substituting into equation (74) and integrating, we obtain E[WT ] = σ02 T + βT We see that skew makes no contribution to this expression, only the curvature contributes. The intuition for this is simply that increasing the skew does not change the average level of volatility but increasing the curvature β increases the prices of puts and calls in equation (71) and always increases the fair value of variance. 18.4 The Effect of Jumps Suppose we have some continuous process and add a jump of size ² that occurs between times ti and ti+1 . Total variance is computed as a sum of 125 terms of the form µ µ ¶¶2 Si+1 log Si The jump will change this particular term in the sum to ¶¶2 µ µ Si+1 e² log Si The net impact of the jump on the total variance for this realization of the process is then given by µ ¶ Si+1 2 ² + 2² log Si which in expectation is just E[²2 ]. So, assuming that jumps are independent of the underlying continuous process, to get the effect of jumps on the expected total variance, we need only figure out the expected number of jumps in the interval (0, T ] and their average squared magnitude. In the Merton and SVJ models, jumps are generated by a Poisson process with parameter λ and the size of the jump is lognormally distributed with mean α and standard deviation δ. In this case, the expected number of jumps is just λT and the average squared magnitude is given by α2 + δ 2 . The expected total variance E[WT ] is then given by ¡ ¢ E[WT ] = E[WT ]c + λT α2 + δ 2 where E[WTc ] denotes the expected variance of the same process without jumps. This formula tells us in absolute terms how the value of a variance swap would change but not in terms of the European option hedge whose value obviously must also change when we add jumps to the underlying process. To see how the value of the variance swap relates to the value of the log-contract (the infinite strip of European options), we note that from the definition of the characteristic function ¯ · µ ¶¸ ¯ ∂ ST = −i φT (u)¯¯ E log F ∂u u=0 Also, recall that if jumps are independent of the continuous process as they are in both the Merton and SVJ models, the characteristic function may be written as the product of a continuous part and a jump part φT (u) = φcT (u) φjT (u) 126 Then ¯ ¯ · µ ¶¸ ¯ ¯ ST ∂ c ∂ j E log = −i φT (u)¯¯ −i φT (u)¯¯ F ∂u ∂u u=0 u=0 From equation (72), the first term on the r.h.s is just − 12 E[WTc ]. To get the second term, we use the explicit form of φjT (u) given by n ³ ´ ³ ´o 2 2 2 φjT (u) = exp −iuλT eα+δ /2 − 1 + λT eiuα−u δ /2 − 1 Then So ¯ ³ ´ ¯ ∂ j 2 = λT 1 + α − eα+δ /2 −i φT (u)¯¯ ∂u u=0 · µ ¶¸ ³ ´ ST 1 c α+δ 2 /2 E log = − E[WT ] + λT 1 + α − e F 2 and the difference in value between the log-contract hedge (2 log contracts) and the variance swap is given by · µ ¶¸ ³ ´ ¡ ¢ ST 2 = λT α2 + δ 2 + 2λT 1 + α − eα+δ /2 E[WT ] + 2E log F ¡ ¢ 1 = − λT α α2 + 3δ 2 + higher order terms 3 This shows that two log contracts is a good average hedge for a variance swap even with jumps. Putting λ = 0.6, α = −0.15 and δ = 0.1, we get an error of only 0.0016 on a one-year variance swap which corresponds to around 0.25 vol points if the volatility level is 30%. 18.5 Volatility Swaps Realized volatility ΣT is the square root of realized variance VT and we know that the expectation of the square root of a random variable is less than√(or equal to) the square root of the expectation. The difference between VT and ΣT is known as the convexity adjustment. Figure 1 shows how the payoff of a variance swap compares with the payoff of a volatility swap. Intuitively, the magnitude of the convexity adjustment must depend on the volatility of realized volatility. Note that volatility does not have to be 127 Figure 1: Payoff of a variance swap (dashed line) and volatility swap (solid line) as a function of realized volatility. Both swaps are stuck at 30% volatility. ΣT Payoff 0.1 0.05 0.15 0.2 0.25 0.3 0.35 0.4 ST -0.05 -0.1 -0.15 -0.2 stochastic for realized volatility to be volatile; realized volatility ΣT varies according to the path of the stock price even in a local volatility model. In general, there is no replicating portfolio for a volatility swap and the magnitude of the convexity adjustment is highly model-dependent. As a consequence, market makers’ prices for volatility swaps are both wide (in terms of bid-offer) and widely dispersed. As in the case of live-out options, price takers such as hedge funds may occasionally have the luxury of being able to cross the bid-offer – that is, buy on one dealer’s offer and sell on the other dealer’s bid. Assuming no jumps however (Matytsin (1999) discusses the impact of jumps), as we will see in Section 18.7, the convexity adjustment is model independent. We will now compute it for the Heston model. 18.6 Convexity Adjustment in the Heston Model To compute the expectation of volatility in the Heston model we use the following trick: £ ¤ Z ∞ hp i 1 − E e−ψWT 1 dψ (75) E WT = √ ψ 3/2 2 π 0 128 From Cox, Ingersoll, and Ross (1985), the Laplace transform of the total RT variance WT = 0 vt dt is given by £ ¤ E e−ψWT = A e−ψvB where ½ 2φ e(φ+λ)T /2 A = (φ + λ)(eφT − 1) + 2φ 2 (eφT − 1) B = (φ + λ)(eφT − 1) + 2φ ¾2λv̄/η2 p with φ = λ2 + 2ψη 2 . With some tedious algebra, we may verify that ¯ ∂ £ −ψWT ¤¯¯ E [WT ] = − E e ¯ ∂ψ ψ=0 = 1 − e−λT (v − v̄) + v̄T λ as we found earlier in Section 18.2. Computing the integral in equation (75) numerically using the usual parameters from Homework 2 (v = 0.04, v̄ = 0.04, λ = 10.0, η = 1.0), we get the graph of the convexity adjustment as a function of time to expiration shown in Figure 2. Using Bakshi, Cao and Chen parameters (v = 0.04, v̄ = 0.04, λ = 1.15, η = 0.39), we get the graph of the convexity adjustment as a function of time to expiration shown in Figure 3. To get intuition for what is going on here, compute the limit of the variance of VT as T → ∞ with v = v̄ using £ ¤ var [WT ] = E WT 2 − {E [WT ]}2 ( ¯ ¯ )2 ∂ 2 £ −ψWT ¤¯¯ ∂ £ −ψWT ¤¯¯ − = E e E e ¯ ¯ ∂ψ 2 ∂ψ ψ=0 ψ=0 = v̄T η2 + O(T 0 ) λ2 Then, as T → ∞, the q standard deviation of annualized variance has the v̄ η leading order behavior . The convexity adjustment should be of the T λ 129 Figure 2: Annualized Heston convexity adjustment as a function of T with parameters from Homework 2. Convexity Adjustment 0.01 0.008 0.006 0.004 0.002 1 2 3 4 5 T Figure 3: Annualized Heston convexity adjustment as a function of T with Bakshi, Cao and Chen parameters. Convexity Adjustment 0.012 0.01 0.008 0.006 0.004 0.002 1 2 130 3 4 5 T order of the standard deviation of annualized volatility over the life of the contract. From the last result, we expect this to scale as λη . Comparing Bakshi, Cao and Chen (BCC) parameters with Homework 2 parameters, we deduce that the convexity adjustment should be roughly 3.39 times greater for BCC parameters and that’s what we see in the graphs. 18.7 Replication of Generic Quadratic Variation-Based Claims in a Diffusion Context Ground-breaking work by Peter Carr and Roger Lee (Carr and Lee 2003) extends the well-known replication results for variance-swaps presented in Section 17 to arbitrary functions of realized variance assuming a diffusion process for the underlying and zero correlation between moves in the underlying and moves in volatility. We proceed to sketch out their results. From equation (70), we know that generalized payoffs may be spanned according to Z F Z ∞ 00 E [ g(ST )] = g(F ) + dK P̃ (K, T ) g (K) + dK C̃(K, T ) g 00 (K) (76) 0 F where C̃ and P̃ represent undiscounted call and put prices respectively and F represents the time-T forward price of the stock. With the substitution g(ST ) = STp , we obtain the replicating portfolio for a power payoff: Z F Z ∞ p p p−2 E [ ST ] = F +p (p−1) dK P̃ (K, T ) K +p (p−1) dK C̃(K, T ) K p−2 0 F Define the log-strike k ≡ log(K/F ). Then ½ Z Z 0 p pk p 1 + p (p − 1) dk p̂(k) e + p (p − 1) E [ ST ] = F −∞ ¾ ∞ dk ĉ(k) e 0 pk (77) where p̂(k) and ĉ(k) denote the prices of puts and calls respectively in terms of percentage of strike. If there is zero correlation between moves in the underlying and volatility moves, put-call symmetry holds, so that p̂(k) = e−k ĉ(−k). We can then rewrite equation (77) as ½ ¾ Z ∞ p p k/2 E [ ST ] = F 1 + 2 p (p − 1) dk e ĉ(k) cosh (p − 1/2) k 0 131 18.7.1 The Laplace transform of quadratic variation The main contribution of Carr and Lee (2003) under their assumptions of diffusion and zero correlation between spot moves and volatility moves is to have derived an expression for the Laplace transform (moment generating function) of the quadratic variation WT ≡ hxiT in terms of the fair value of the power payoff just derived in equation (77). Since we know the replicating strip for the power payoff, it follows that we know the replicating strip for quadratic variation. Explicitly, Carr and Lee show that £ ¤ £ ¤ E eλhxiT = E ep(λ)xT p where p(λ) = 1/2 ± 1/4 + 2 λ. From equation (77), we then have that ½Z 0 Z £ λhxi ¤ p(λ) k T dk p̂(k) e + E e = 1 + p(λ) (p(λ) − 1) ¾ ∞ dk ĉ(k) e 0 −∞ p(λ) k (78) or alternatively £ λhxiT E e ¤ Z ∞ = 1 + 2 p(λ) (p(λ) − 1) dk ek/2 ĉ(k) cosh (p(λ) − 1/2) k 0 Noting that p(λ) (p(λ) − 1) = 2 λ, we may simplify further to obtain Z ∞ £ λhxi ¤ T dk ek/2 ĉ(k) cosh (p(λ) − 1/2) k E e = 1 + 4λ (79) 0 The fair value of an exponential quadratic-variation (realized volatility) payoff follows immediately from equation (79). By taking derivatives, the fair value of any positive integral power of quadratic variation also follows. As a check, consider the fair value of the first power of quadratic variation – 132 expected realized total variance. We have ¯ ∂ £ λhxiT ¤¯¯ E e E [ hxiT ] = ¯ ∂λ ½ ¾¯ Zλ=0 ∞ ¯ ∂ k/2 = 1 + 4λ dk e ĉ(k) cosh (p(λ) − 1/2) k ¯¯ ∂λ 0 λ=0 Z ∞ = 4 dk ek/2 ĉ(k) cosh (p(0) − 1/2) k Z0 ∞ = 4 dk ek/2 ĉ(k) cosh k/2 Z0 ∞ © ª = 2 dk ĉ(k) 1 + ek ½0Z ∞ ¾ Z 0 = 2 dk ĉ(k) + dk p̂(k) 0 −∞ which agrees exactly with our earlier result (73). In principle, since we have an explicit expression for the moment generating function (mgf ) of realized variance, we know its entire terminal (time T ) pseudo-probability distribution and we may compute the fair value of any European-style claim on realized variance – knowing only market prices of European options! 18.7.2 Replicating volatility It turns out√that by far the dominant contribution to the fair value of realized volatility ( WT ) is the at-the-money forward European option. Indeed we have p √ E[ WT ] ≈ 2 π ĉ(0) This should be no surprise as we are already very familiar with the extremely accurate approximation √ σBS T ĉ(0) ≈ √ 2π for at-the-money forward European options. We see however that the volatility replication strategy differs fundamentally from the variance replication strategy. In the variance case, we trade a strip of options at inception and thereafter daily rebalancing in the underlying only and that only to maintain a constant dollar amount of the 133 underlying in the hedge portfolio. In contrast, in the volatility case we have to continuously maintain a position in the at-the-money option: each day, we sell the entire position (which is no longer at-the-money) and buy a new one. Unlike the variance strategy, this strategy is clearly not practical – the option bid-offer would kill the hedger. Similar comments apply to the hedging in practice of any generic claim using this strategy: it involves daily rebalancing in options. Nevertheless, Carr and Lee’s result is powerful because it shows that the fair value of any claim on quadratic variation (under their assumptions) depends only on the prices of European options and is otherwise completely model-independent. We can suppose that a way will be found to extend their results further to the case of non-zero correlation between volatility moves and underlying moves although not to the general non-diffusive case (see Friz and Gatheral (2004) for example). Despite this, given our observations in Section 18.4 on the effect of jumps in practice, it seems likely that any such extension will be valuable in practice as well as in theory. 19 The VIX futures contract Earlier this year, the CBOE listed futures on the VIX - an implied volatility index. Originally, the VIX computation was designed to mimic the implied volatility of an at-the-money 1 month option on the OEX index. It did this by averaging volatilities from 8 options (puts and calls from the closest to ATM strikes in the nearest and next to nearest months). To facilitate trading in the VIX, the definition of the index was changed. Here is an excerpt from the FAQ section on the CBOE website: “How is VIX being changed? Three important changes are being made to update and improve VIX: 1. The New VIX is calculated using a wide range of strike prices in order to incorporate information from the volatility skew. The original VIX used only at-the-money options. 2. The New VIX uses a newly developed formula to derive expected volatility directly from the prices of a weighted strip of options. The original VIX extracted implied volatility from an option-pricing model. 3. The New VIX uses options on the S&P 500 Index, which is the primary U.S. stock market benchmark. The original VIX was based on S&P 100 Index (OEX) option prices. Why is the CBOE making changes to the VIX? CBOE is changing VIX to provide a more precise and robust measure of expected market 134 volatility and to create a viable underlying index for tradable volatility products. The New VIX calculation reflects the way financial theorists, risk managers and volatility traders think about - and trade - volatility. As such, the New VIX calculation more closely conforms to industry practice than the original VIX methodology. It is simpler, yet it yields a more robust measure of expected volatility. The New VIX is more robust because it pools information from option prices over a wide range of strike prices thereby capturing the whole volatility skew, rather than just the volatility implied by at-the-money options. The New VIX is simpler because it uses a formula that derives the market expectation of volatility directly from index option prices rather than an algorithm that involves backing implied volatilities out of an option-pricing model. The changes also increase the practical appeal of VIX. As noted previously, the New VIX is calculated using options on the S&P 500 index, the widely recognized benchmark for U.S. equities, and the reference point for the performance of many stock funds, with over $800 billion in indexed assets. In addition, the S&P 500 is the domestic index most often used in over-the-counter volatility trading. This powerful calculation supplies a script for replicating the New VIX with a static portfolio of S&P 500 options. This critical fact lays the foundation for tradable products based on the New VIX, critical because it facilitates hedging and arbitrage of VIX derivatives. CBOE has announced plans to list VIX futures and options in Q4 2003, pending regulatory approval. These will be the first of an entire family of volatility products.” This explanation is intriguing. To see what the CBOE means, consider the VIX definition (converted to our notation) as specified in the CBOE white paper (CBOE 2003). · ¸2 2 X ∆Ki 1 F V IX = Qi (Ki ) − −1 T i Ki2 T K0 2 where Qi is the price of the out-of-the-money option with strike Ki and K0 is the highest strike below the forward price F . We recognize this formula as a simple discretization of equation (73) and makes clear the reason why the CBOE implies that the new index permits replication of volatility. 135 19.1 How did they perform? Volume on VIX futures (strictly speaking VXB futures where the VXB is defined to be 10 times the VIX) has been negligible. For example, at the time of writing, the Jan-05 VXB future had been trading an average of 155 contracts per day. At the current price of 143.90, that corresponds to roughly $2.25 notional per day. In contrast, Dec04 SPX futures trade around 50,000 contracts per day which corresponds to around $15bn notional per day. As a practical example of how the replicating strip (73) is associated with expected variance, settlement is on the Wednesday before the third Friday based on the opening prices of options expiring in the following month. 19.2 A subtlety VXB futures settle based on the square root of the value of the replicating strip (i.e. on volatility rather than variance) so there must be a convexity adjustment. However, this isn’t exactly the same as the convexity adjustment p p E[WT ] − E[ WT ] that we computed earlier. As of the valuation date, the convexity adjustment relevant to the VXB futures contract is given by ·q ¸ q E[Wt,T ] − E Et [Wt,T ] where t is the settlement date, T is the expiration date of options in the strip and the expectation is computed as of some valuation date prior to t. As before, we can easily compute E[Wt,T ] from strips of options expiring hp at times i t and T respectively but we need a model to compute E Et [Wt,T ] . One obvious question is where the market prices this convexity adjustment. As of December 8th, 2004 the spot VXB was at 132.30. Computing the expected forward variances E[Wt,T ] from market prices of options we obtain the empirical convexity adjustments shown in Table 1. The next question is what the VXB futures are worth theoretically. 136 Table 1: Empirical VXB convexity adjustments as of 08-Dec-2004. Expiry Spot Jan-05 Feb-05 19.3 VXB Price Fwd Variance Convexity adjustment 132.30 132.30 0.00 144.20 149.09 4.89 153.30 159.63 6.33 VXB convexity adjustment in the Heston Model To compute the VXB convexity adjustment in the Heston model, we first note that expected one-month expected total variance Et [Wt,T ] given the instantaneous variance vt is f (vt ) = Et [Wt,T ] = (vt − v̄) with τ = T − t. Then ·q ¸ Z E Et [Wt,T ] = 1 − e−λ τ + v̄ τ λ ∞ q(vt |v0 ) p f (vt ) dv (80) 0 where q(v|v0 ) is the pdf of instantaneous variance v given the initial variance v0 . Computing the integral in equation (80) numerically using Heston parameters from a fit to December 8, 2004 data, we get the graph of the convexity adjustment as a function of time to expiration shown in Figure 4. We see that for the two futures maturities listed in Table 1, the Heston convexity adjustments were respectively around 7.4 and 8.6 points - not so different from the empirically-observed numbers. 19.4 Futures on realized variance The CBOE also lists futures on 3-month realized variance with symbol VT. These futures really hardly trade at all. Their payoff is given by N 1 X 2 Realized Variance = 252 r N −1 i i where N is the number of observation days in the 3-month trading period. The VT future becomes decreasingly volatile as settings are made; settlement 137 Figure 4: Annualized Heston VXB convexity adjustment as a function of t with Heston parameters from 08-Dec-2004 SPX fit. VXB Adjustment 14 12 10 8 6 4 2 0.25 0.5 0.75 1 1.25 1.5 t is effectively based on average realized daily variance. It would be nice to compare the price of the VT future with theory but as an example of its illiquidity, there wasn’t a single trade on December 8, 2004. 19.5 Spin-off benefits Although these futures contracts weren’t very successful by usual futures standards, they did stimulate the OTC market to very substantially tighten spreads. Now SPX variance swaps can be traded from 6 months to 5 years with bid-offer spreads of as little as 0.5 vol points. Single-stock variance swaps are also now actively quoted; for the top 50 US single stock names, bid-offer spreads are around 2.5 vol points. 20 Epilog I hope that this series of lectures has given students an insight into how financial mathematics is used in the derivatives industry. It should be apparent that modeling is an art in the true sense of the word – not a science, although when it comes to implementing the chosen solution approach, science becomes necessary. We have seen several examples of claims which may be priced differently under different modeling assumptions even though the 138 models generate identical prices for European options. The importance of lateral thinking outside the framework of a given model cannot be overemphasized. For example, what is the impact of jumps? of stochastic volatility? of skew? and so on. Finally, intuition together with the ability to express this intuition clearly is ultimately what counts; without intuition and the ability to communicate it clearly to non-mathematicians, financial mathematics becomes no more than a theoretical exercise divorced from the real business of making money. 139 References Breeden, D., and R. Litzenberger, 1978, Prices of state-contingent claims implicit in option prices, Journal of Business 51, 621–651. Carr, Peter, and Roger Lee, 2003, The robust replication of volatility derivatives, http://www.math.nyu.edu/research/carrp/papers/pdf/Voltradingoh1.pdf. Carr, Peter, and Dilip Madan, 1998, Towards a theory of volatility trading, in Robert A. Jarrow, ed.: Volatility: New Estimation Techniques for Pricing Derivatives . chap. 29, pp. 417–427 (Risk Books: London). CBOE, 2003, VIX: CBOE volatility index, http://www.cboe.com/micro/vix/vixwhite.pdf. Chriss, Neil, and William Morokoff, 1999, Market risk for variance swaps, Risk 12, 55–59. Cox, John C., Jonathan E. Ingersoll, and Steven A. Ross, 1985, A theory of the term structure of interest rates, Econometrica 53, 385–407. Demeterfi, Kresimir, Emanuel Derman, Michael Kamal, and Joseph Zou, 1999, A guide to volatility and variance swaps, Journal of Derivatives 6, 9–32. Friz, Peter, and Jim Gatheral, 2004, Valuing volatility derivatives as an inverse problem, Discussion paper Courant Institute of Mathematical Sciences, NYU. Matytsin, Andrew, 1999, Modeling volatility and volatility derivatives, Columbia Practitioners Conference on the Mathematics of Finance. 140 A Proof of Equation (74)1 As usual, the undiscounted European call option price is given by ½ µ µ √ ¶ √ ¶¾ k w k w k C = F N −√ + − e N −√ − 2 2 w w where k := ln(K/F ) is the log-strike. Differentiating wrt the strike K we obtain µ µ √ ¶ √ ¶ √ k 1 ∂C k ∂C w w ∂ w 0 + N −√ − = = −N − √ − ∂K K ∂k 2 2 ∂k w w Then with the notation √ √ k w k w d1 = − √ + ; d2 = − √ − 2 2 w w and differentiating again wrt K, we obtain ½ µ µ √ ¶ √ ¶ √ ¾ ∂ 2C 1 ∂ k w k w ∂ w 0 = −N − √ − + N −√ − 2 ∂K K ∂k 2 2 ∂k w w ½ µ √ ¶ √ ¾ 0 2 N (d2 ) ∂d2 ∂ w ∂ w = − 1 + d2 + K ∂k ∂k ∂k 2 à ( ! µ ¶ √ √ √ ) 2 0 2 ∂ w N (d2 ) 1 2k ∂ w ∂ w √ d1 d2 = −√ +1 + K ∂k ∂k 2 w w ∂k From Section 18, the fair value of a variance swap under diffusion may be obtained by valuing a contract that pays 2 ln (ST /F ) at maturity T . Bruteforce calculation leads to · ¸ µ ¶ 2 Z ∞ ST K ∂ C 2E ln = 2 dK ln F F ∂K 2 µ ½ √ ¶ √ ¾ Z0 ∞ ∂d2 ∂ w ∂2 w 0 = 2 dk k N (d2 ) − 1 + d2 + ∂k ∂k ∂k 2 −∞ ½ √ ¾ Z ∞ ∂d2 ∂ w − = 2 dk N 0 (d2 ) −k ∂k ∂k −∞ Z ∞ ∂d2 = dk N 0 (d2 ) w ∂k −∞ which recovers equation (74) as required. 1 This particularly neat proof is due to Chiyan Luo. 141