Diamonds: A quants best friend Jim Gatheral (joint work with Elisa - PowerPoint PPT Presentation

Decomposition Trees and forests Exponentiation Rough Heston Diamonds: A quant’s best friend Jim Gatheral (joint work with Elisa Al` os and Radoˇ s Radoiˇ ci´ c) Workshop on Finance, Insurance, Probability and Statistics King’s College London, September 10, 2018

Decomposition Trees and forests Exponentiation Rough Heston Outline of this talk The Itˆ o decomposition formula of Al` os Diamond and dot notation Stochasticity Trees and forests The Exponentiation Theorem Explicit computations in the rough Heston model Leverage swaps Stochasticity

Decomposition Trees and forests Exponentiation Rough Heston The Al` os Itˆ o decomposition formula Following Elisa Al` os in [Al` o12], let X t = log S t / K and consider the price process dX t = σ t dZ t − 1 2 σ 2 t dt . Now let H ( x , w ) be some function that solves the Black-Scholes equation. Specifically, − ∂ w H ( x , w ) + 1 2 ( ∂ xx − ∂ x ) H ( x , w ) = 0 which is of course the gamma-vega relationship. Note in particular that ∂ x and ∂ w commute when applied to a solution of the Black-Scholes equation.

Decomposition Trees and forests Exponentiation Rough Heston Now, define w t ( T ) as the integral of the expected future variance: �� T � � σ 2 � w t ( T ) := E s ds � F t . � t Notice that � t σ 2 w t ( T ) = M t − s ds , 0 �� T � � 0 σ 2 where the martingale M t := E s ds � F t . Then it follows that � dw t ( T ) = − σ 2 t dt + dM t .

Decomposition Trees and forests Exponentiation Rough Heston Applying Itˆ o’s Lemma to H t := H ( X t , w t ( T )), taking conditional expectations, simplifying using the Black-Scholes equation and integrating, we obtain Theorem (The Itˆ o Decomposition Formula of Al` os) �� T � � � E [ H T | F t ] = H t + E ∂ xw H s d � X , M � s � F t � t �� T � � +1 � ∂ ww H s d � M , M � s � F t . (1) 2 E � t Note in particular that (1) is an exact decomposition.

Decomposition Trees and forests Exponentiation Rough Heston Freezing derivatives Freezing the derivatives in the Al` os Itˆ o decomposition formula (1) gives us the approximation �� T � � � E [ H T | F t ] ≈ H t + E d � X , M � s � F t ∂ xw H t � t �� T � � +1 � d � M , M � s � F t ∂ ww H t 2 E � t H t + ( X ⋄ M ) t ( T ) · H t + 1 = 2 ( M ⋄ M ) t ( T ) · H t . Remark The essence of the Exponentiation Theorem is that we may express E [ H T | F t ] as an exact expansion consisting of infinitely many terms, with derivatives in each such term frozen.

Decomposition Trees and forests Exponentiation Rough Heston Diamond and dot notation Let A t and B t be semimartingales (here some combinations of X and M ). Then �� T � � � ( A ⋄ B ) t ( T ) = E d � A , B � s � F t . � t When ( A ⋄ B ) t ( T ) appears before some solution H t of the Black-Scholes equation, the dot · is to be understood as representing the action of ∂ x and ∂ w applied to H t . So for example �� T � � � ( X ⋄ M ) t ( T ) · H t = d � X , M � s � F t ∂ xw H t E � t and so on.

Decomposition Trees and forests Exponentiation Rough Heston Diamond functionals as covariances Diamond (or autocovariance) functionals are intimately related to conventional covariances. Lemma Let A and B be martingales in the same filtered probability space. Then ( A ⋄ B ) t ( T ) = E [ A T B T | F t ] − A t B t = cov [ A T , B T | F t ] . By finding the appropriate martingales, it is thus always possible to re-express autocovariance functionals in terms of covariances of terminal quantities. For example, it is easy to show that ( M ⋄ M ) t ( T ) = var [ � X � T | F t ].

Decomposition Trees and forests Exponentiation Rough Heston Autocovariance functionals vs covariances Covariances are typically easy to compute using simulation. Diamond functionals are expressible directly in terms of the formulation of a model in forward variance form.

Decomposition Trees and forests Exponentiation Rough Heston Conditional variance of X T Consider t + w t ( T ) (1 − X t ) + 1 F t = X 2 4 w t ( T ) 2 . F ( x , w ) satisfies the Black-Scholes equation and F T = X 2 T . ∂ x , w F = − 1 and ∂ w , w F = 1 2 . Plugging into the Decomposition Formula (1) gives �� T w t ( T ) + 1 � � 4 w t ( T ) 2 − E � F t X 2 � � � � = d � X , M � s � F t E T � t �� T � +1 � � d � M , M � s � F t 4 E � t w t ( T ) + 1 4 w t ( T ) 2 = − ( X ⋄ M ) t ( T ) + 1 4 ( M ⋄ M ) t ( T ) .

Decomposition Trees and forests Exponentiation Rough Heston Volatility stochasticity We can rewrite this as Lemma ζ t ( T ) := var [ X T |F t ] − w t ( T ) = − ( X ⋄ M ) t ( T ) + 1 4 ( M ⋄ M ) t ( T ) . Recall that in a stochastic volatility model, the variance of the terminal distribution of the log-underlying is not in general equal to the expected quadratic variation. In the Black-Scholes model of course ζ t ( T ) = 0. We call the difference ζ t ( T ) volatility stochasticity or just stochasticity .

Decomposition Trees and forests Exponentiation Rough Heston Model calibration Once again, stochasticity is given by ζ t ( T ) = − ( X ⋄ M ) t ( T ) + 1 4 ( M ⋄ M ) t ( T ) . The LHS may be estimated from the volatility surface using the spanning formula. ζ t ( T ) is a tradable asset for each T . We get a matching condition for each expiry T i , i ∈ { 1 , .. n } . The RHS may typically be computed in a given model as a function of model parameters. If so, we would be able to calibrate such a model directly to tradable assets with no need for any expansion.

Decomposition Trees and forests Exponentiation Rough Heston ζ t ( T ) directly from the smile Let √ − k ± σ BS ( k , T ) T d ± ( k ) = √ 2 σ BS ( k , T ) T and following Fukasawa, denote the inverse functions by g ± ( z ) = d − 1 ± ( z ). Further define √ σ − ( z ) = σ BS ( g − ( z ) , T ) T .

Decomposition Trees and forests Exponentiation Rough Heston In terms of the implied volatility smile, it is a well-known corollary of Matytsin’s characteristic function representation in [Mat00], that � dz N ′ ( z ) σ 2 σ 2 . w t ( T ) = − ( z ) =: ¯ Similarly, we can show that ζ t ( T ) = 1 � σ 2 � 2 dz + 2 � N ′ ( z ) σ 2 N ′ ( z ) z σ 3 � − ( z ) − ¯ − ( z ) dz . 4 3 We may thus in principle use stochasticity to calibrate any given model. In practice, we need a good parameterization of the implied volatility surface (see VolaDynamics later). Whether or not market implied stochasticity is robust to the interpolation and extrapolation method is still to be explored.

Decomposition Trees and forests Exponentiation Rough Heston Forward variance models Following [BG12], consider the model dS t √ v t � � 1 − ρ 2 dW ⊥ � = ρ dW t + t S t d ξ t ( u ) = λ ( t , u , ξ t ) dW t . (2) where v t = σ 2 t denotes instantaneous variance and the ξ t ( u ) = E [ v u | F t ] , u ∈ [ t , T ] are forward variances. To expand such a model, we scale the volatility of volatility function λ ( · ) so that λ �→ ǫ λ . Setting ǫ = 1 at the end then gives the required expansion.

Decomposition Trees and forests Exponentiation Rough Heston The Bergomi-Guyon expansion According to equation (13) of [BG12], in diamond notation, the conditional expectation of a solution of the Black-Scholes equation satisfies E [ H T | F t ] 1 + ǫ ( X ⋄ M ) t + ǫ 2 � = 2 ( M ⋄ M ) t + ǫ 2 � 2 [( X ⋄ M ) t ] 2 + ǫ 2 ( X ⋄ ( X ⋄ M )) t + O ( ǫ 3 ) · H t

Decomposition Trees and forests Exponentiation Rough Heston We notice that ǫ ( X ⋄ M ) t + ǫ 2 � E [ H T | F t ] = exp 2 ( M ⋄ M ) t � + ǫ 2 ( X ⋄ ( X ⋄ M )) t + O ( ǫ 3 ) · H t , the exponential of a sum of “connected diagrams”. Motivated by exponentiation results in physics, we are tempted to see if something like this holds to all orders.

Decomposition Trees and forests Exponentiation Rough Heston Trees Terms such as ( X ⋄ M ), ( M ⋄ M ) and X ⋄ ( X ⋄ M ) are naturally indexed by trees, each of whose leaves corresponds to either X or M . We end up with diamond trees reminiscent of Feynman diagrams, with analogous rules.

Decomposition Trees and forests Exponentiation Rough Heston Forests Definition Let ❋ 0 = M . Then the higher order forests ❋ k are defined recursively as follows: k − 2 ❋ k = 1 � ✶ i + j = k − 2 ❋ i ⋄ ❋ j + X ⋄ ❋ k − 1 . 2 i , j =0

Decomposition Trees and forests Exponentiation Rough Heston The first few forests Applying this definition to compute the first few terms, we obtain = ❋ 0 M = X ⋄ ❋ 0 = ( X ⋄ M ) ❋ 1 1 2( ❋ 0 ⋄ ❋ 0 ) + X ⋄ ❋ 1 = 1 = 2( M ⋄ M ) + X ⋄ ( X ⋄ M ) ❋ 2 ❋ 3 = ( ❋ 0 ⋄ ❋ 1 ) + X ⋄ ❋ 2 M ⋄ ( X ⋄ M ) + 1 = 2 X ⋄ ( M ⋄ M ) + X ⋄ ( X ⋄ ( X ⋄ M ))

Decomposition Trees and forests Exponentiation Rough Heston The first forest ❋ 1 = X ⋄ M ♦ X M

Decomposition Trees and forests Exponentiation Rough Heston The second forest ❋ 2 ❋ 2 = 1 2( M ⋄ M ) + X ⋄ ( X ⋄ M ) ♦ ♦ ♦ X M M X M

Decomposition Trees and forests Exponentiation Rough Heston The third forest ❋ 3 ❋ 3 = M ⋄ ( X ⋄ M ) + 1 2 X ⋄ ( M ⋄ M ) + X ⋄ ( X ⋄ ( X ⋄ M )) ♦ ♦ ♦ ♦ X ♦ ♦ ♦ M X X X M M M X M