Calculation of the Greeks Using Malliavin Calculus

Calculation of the Greeks using Malliavin Calculus

Christian Bayer University of Technology, Vienna

09/25/2006 – SCS Seminar 09/26/2006 – Mathematics Seminar

Christian Bayer University of Technology, Vienna Calculation of Greeks Outline: Part I

1 Hedging in Mathematical Finance The Model A Reminder on Mathematical Finance Hedging in the Model

2 Calculation of the Greeks Introduction Selected Results from Malliavin Calculus The Bismut-Elworthy-Li Formula

3 Summary

Christian Bayer University of Technology, Vienna Calculation of Greeks Outline: Part II

4 Universal Malliavin Weights Stochastic Taylor Expansion Step-m Nilpotent, Free Lie Groups m Iterated Stratonovich Integrals on Gd,0 Universal Malliavin Weights on Nilpotent, Free Lie Groups

5 Two Approaches to Approximate Universal Weights Calculating the Greeks by Cubature Formulas Approximation of the Heat Kernel

6 Summary

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging Calculation of the Greeks Summary

Part I

Greeks and Malliavin Calculus in Finance

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model The Model

1 d Let Bt = (Bt ,..., Bt ), t ∈ [0, T ], be a d-dimensional Brownian motion on the Wiener space (Ω, F, P). Ft denotes the ﬁltration generated by B and we assume that F = FT . We model a ﬁnancial market, in which n + 1 assets are traded, n ≤ d. 0 St , t ∈ [0, T ], is the “bank account” earning a risk free interest rate r > 0 (continuous compounding). 1 n St = (St ,..., St ) gives the risky assets (“stocks”). n n For bounded, measurable functions a : [0, T ] × R → R and n n×d σ : [0, T ] × R → R we set

0 0 dSt = rSt dt i i i Xd ij i j dS = a (t, St )S dt + σ (t, St )S dB , i = 1,..., n. t t j=1 t t

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Strategies

A strategy describes the amount of money invested in each available asset at any given time t ∈ [0, T ]. Definition n A (self-financing) strategy is a predictable, R -valued process π such that “all of the following integrals are well-defined”.

Remark The strategy is self-ﬁnancing, because we neither allow consumption nor external money entering the ﬁnancial market. Thus, strategies are determined by n coordinates. All positions are allowed to be negative (short selling).

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Strategies

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model The Wealth Process of a Portfolio

The wealth process X x,π with initial capital x associated to the x,π strategy π is deﬁned by X0 = x and its dynamics

n x,π n X πi X − P πi dX x,π = t dSi + t i=1 t dS0. t Si t S0 t i=1 t t

Deﬁnition A strategy is admissible if the corresponding wealth process is bounded from below by some ﬁxed real number.

The restriction to admissible strategy is economically sensible and disallows “doubling strategies”.

The wealth process X x,π with initial capital x associated to the x,π strategy π is deﬁned by X0 = x and its dynamics

n x,π n X πi X − P πi dX x,π = t dSi + t i=1 t dS0. t Si t S0 t i=1 t t

Deﬁnition A strategy is admissible if the corresponding wealth process is bounded from below by some ﬁxed real number.

The restriction to admissible strategy is economically sensible and disallows “doubling strategies”.

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model No Arbitrage

Deﬁnition An arbitrage opportunity is an admissible strategy π such that 0,π 0,π P XT ≥ 0 = 1 and P XT > 0 > 0.

Deﬁnition Let Q be a probability measure on (Ω, F) equivalent to P. Q is called equivalent (local) martingale measure if the discounted price −rt process Set = e St , t ∈ [0, T ], is a (local) martingale under Q.

Remark The Fundamental Theorem of Asset Pricing roughly says that the existence of (local) martingale measures is equivalent to the non-existence of arbitrage opportunities.

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model No Arbitrage – 2

Proposition d Assume there exists a predictable, R -valued process θ such that i Pd ij j (i) a (t, St ) − r = j=1 σ (t, St )θ (t), i = 1,..., n, R T 2 (ii) 0 kθ(t)k dt < ∞ a. s., R T 1 R T 2 (iii) E exp − 0 hθ(t),dBt i − 2 0 kθ(t)k dt = 1. Then our model is free of arbitrage and the probability measure Q R T 1 R T 2 with density ZT = exp − 0 hθ(t),dBt i − 2 0 kθ(t)k dt is a martingale measure.

Consequently, the dynamics of the model under Q are given by

i i Xd ij i j dS = rS dt + σ (t, St )S dW , i = 1,..., n, t t j=1 t t where W denotes a Brownian motion under Q. Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Complete and Incomplete Markets

Deﬁnition

A contingent claim is an FT -measurable, Q-absolutely integrable random variable. A contingent claim Y is attainable or replicable if there is an admissible, self-ﬁnancing portfolio π and a number x x,π such that XT = Y a. s. π is called replicating portfolio for Y .

Deﬁnition A ﬁnancial market is complete, if every contingent claim is replicable. Otherwise, it is called incomplete.

Proposition

Our market is complete if and only if n = d and σ(t, St (ω)) is invertible for dt ⊗ P - a. e. (t, ω) ∈ [0, T ] × Ω.

Deﬁnition

Deﬁnition A ﬁnancial market is complete, if every contingent claim is replicable. Otherwise, it is called incomplete.

Proposition

Our market is complete if and only if n = d and σ(t, St (ω)) is invertible for dt ⊗ P - a. e. (t, ω) ∈ [0, T ] × Ω.

Remark Under realistic conditions, completeness is a very strong property. Many experts agree that realistic models are not complete.

Assumptions From now on, we assume that our model is arbitrage-free and complete.

Deﬁnition x ∈ R is an arbitrage-free price of a contingent claim Y if there is x,π an admissible strategy π such that Y = XT a. s.

Proposition In a complete, arbitrage-free model, any claim Y has a unique −rT arbitrage-free price x = EQ e Y , where Q is the unique e. m. m.

Remark In incomplete markets, there is, in general, no replicating portfolio, only super-replicating ones. Typically, there is an interval of arbitrage-free prices bounded by the super replication prices of the buyer and of the seller.

Deﬁnition x ∈ R is an arbitrage-free price of a contingent claim Y if there is x,π an admissible strategy π such that Y = XT a. s.

Proposition In a complete, arbitrage-free model, any claim Y has a unique −rT arbitrage-free price x = EQ e Y , where Q is the unique e. m. m.

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Black-Scholes PDE

For simplicity, we pass to the discounted model by setting 0 i −rt i Set = 1 and Set = e St , i = 1,..., n. For a European contingent claim of the form Y = f (ST ), let C(t, x) denote the price of Y at time t given St = x, −r(T −t) i. e. C(t, x) = EQ e f (ST ) St = x . By the Feynman-Kac formula, C satisﬁes the PDE ∂ C(t, x) + L C(t, x) = rC(t, x), t ∈ [0, T ], x ∈ n, ∂t t R with C(T , x) = f (x). Here, the inﬁnitesimal generator is

n n 2 X ∂g 1 X ij ∂ g L g(x) = rxi (x) + σσT (t, x)xi xj (x). t ∂xi 2 ∂xi ∂xj i=1 i,j=1

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Black-Scholes PDE

n n 2 X ∂g 1 X ij ∂ g L g(x) = rxi (x) + σσT (t, x)xi xj (x). t ∂xi 2 ∂xi ∂xj i=1 i,j=1

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Delta Hedging

Proposition (Delta Hedging) The replicating portfolio for the discounted, European claim −rT −rT i ∂C e Y = e f (ST ) is given by πt = ∂xi (t, St ), i = 1,..., n: Z T −rT D E e Y = C(0, S0) + ∇x C(t, St ),dSet . 0

Proof. −rt Apply Itˆo’sformula to the process e C(t, St ). Note that we have to use the risk-neutral dynamics. C(0,S0),π −rt In particular, ∀t ∈ [0, T ]: Xt = e C(t, St ).

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Delta Hedging

Proof. −rt Apply Itˆo’sformula to the process e C(t, St ). Note that we have to use the risk-neutral dynamics. C(0,S0),π −rt In particular, ∀t ∈ [0, T ]: Xt = e C(t, St ).

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Delta Hedging – 2

1 The derivative of the price with respect to the price of the underlying is called the Delta of a derivative, e. g. ∇x C(t, x).

2 We can construct a self-ﬁnancing portfolio as follows. t Y Se0 Sei 0 −1 C(0, S0) 0 D E ∂C t −1 C(t, St ) − ∇x C(t, St ),Set ∂xi (t, St ) T −1 Y 0

3 The term “Delta hedging” comes from the fact that this portfolio is Delta neutral, i. e. the Delta of the portfolio is 0. Note, however, that the portfolio requires continuous trading!

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model The Greeks

Deﬁnition The derivatives of the price of a contingent claim with respect to model parameters are called the Greeks.

Remark 1 The Greeks are generally used to “hedge against risks”. 2 Some of the risks – like the “risk” of changing prices of the underlying (Delta hedging) – are inherent to the model. 3 Other risks are inherent to real-life restrictions: e. g., an investor implementing Delta hedging can only trade at discrete times. The corresponding risk can be countered by “Delta-Gamma-hedging”. 4 There are also risks concerning changes of model parameters.

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model The Greeks

Deﬁnition The derivatives of the price of a contingent claim with respect to model parameters are called the Greeks.

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model The Greeks – 2

Remark In reality, hedging is usually very expensive. Therefore, the Greeks are rather used to monitor the development of a portfolio.

A non-exhaustive list of Greeks: Delta: derivative w. r. t. the price of the underlying. Gamma: second derivative w. r. t. the price of the underlying. Vega: derivative w. r. t. the volatility. Rho: derivative w. r. t. the interest rate. Theta: derivative w. r. t. time.

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model The Greeks – 2

Remark In reality, hedging is usually very expensive. Therefore, the Greeks are rather used to monitor the development of a portfolio.

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging Introduction Calculation of the Greeks Selected Results from Malliavin Calculus Summary The Bismut-Elworthy-Li Formula Finite Diﬀerences

In very few models like the classical Black-Scholes model explicit formulas for the option prices exist. Of course, these formulas can be differentiated to get formulas for the Greeks. Numerical differentiation is one method for calculation of the Greeks. Let u(α) denote the dependence of the price u of a derivative on some parameter α and choose > 0 small enough. Use u(α + ) − u(α) du as an approximation of dα . Even more elaborate methods for numerical differentiation often do not give satisfactory results in this context.

α Given a family of random variables X , α ∈ R, having densities 1 p(α, x), x ∈ R, α ∈ R, C in α, and a bounded, measurable function f . d d Z E(f (X α)) = f (x)p(α, x)dx dα dα R Z ∂p (α, x) = f (x) ∂α p(α, x)dx p(α, x) R Z ∂ log(p(α, x)) = f (x) p(α, x)dx = Ef (X α)πα), ∂α R α ∂ α where π = ∂α log(p(α, X )) does not depend on f .

x n Let Xt , x ∈ R , t ∈ [0, T ], be the solution of the SDE

x x x t dXt = a(Xt )dt + σ(Xt )dB (1)

x n n n n×d 1 with X0 = x. Here, a : R → R and σ : R → R are C functions with linear growth. The ﬁrst variation process is the n × n-dimensional process given by

d x X i x i dJ0→t (x) = da(Xt ) · J0→t (x)dt + dσ (Xt ) · J0→t (x)dBt , (2) i=1

i and J0→0(x) = In, where σ denotes the ith column of σ.

1 If the coefficients of the SDE (1) are C 1+, then there is a x version of the solution Xt which is differentiable in x. In this case, the first variation is its Jacobian, i. e.

x J0→t (x) = dx Xt .

. 2 The first variation is almost surely invertible. In fact, it is not −1 difficult to find the SDE for J0→t (x) .

1 If the coefficients of the SDE (1) are C 1+, then there is a x version of the solution Xt which is differentiable in x. In this case, the first variation is its Jacobian, i. e.

x J0→t (x) = dx Xt .

. 2 The first variation is almost surely invertible. In fact, it is not −1 difficult to find the SDE for J0→t (x) .

Without getting into details, we present the formal set-up of Malliavin calculus. The Malliavin derivative is a closed operator D : D(D) ⊂ 2 2 d L (Ω, F, P) → L ([0, T ] × Ω, B([0, T ]) ⊗ F, dt ⊗ P; R ). We write Ds F (ω) = DF (s, ω), F ∈ D(D). The dual map δ is called Skorohod stochastic integral, 2 d i. e. δ : D(δ) ⊂ L ([0, T ] × Ω, B([0, T ]) ⊗ F, dt ⊗ P; R ) → L2(Ω, F, P) satisﬁes

Z T E(F δ(u)) = E hDs F ,us i d ds , (3) R 0 where F ∈ D(D) ⊂ L2(Ω) and u ∈ D(δ) ⊂ L2([0, T ] × Ω).

Remark Usually, calculation of Malliavin derivatives or Skorohod integrals is a hard problem. There are, however, some important special cases.

1 For the process X x solution of (1), the Malliavin derivative is given by

x −1 x Ds Xt = J0→t (x)J0→s (x) σ(Xs )1[0,t](s). (4)

2 For a predictable process u, the Skorohod integral coincides with the Itˆointegral. 1 m i 1 m 3 Let F = (F ,..., F ),F ∈ D(D), and ϕ ∈ C (R ), then ϕ(F ) ∈ D(D) and Dϕ(F ) = h∇ϕ(F ),DF i m . R

Remark Usually, calculation of Malliavin derivatives or Skorohod integrals is a hard problem. There are, however, some important special cases.

1 For the process X x solution of (1), the Malliavin derivative is given by

x −1 x Ds Xt = J0→t (x)J0→s (x) σ(Xs )1[0,t](s). (4)

Definition x For a given stochastic process Xt , t ∈ [0, T ], as in (1) and a fixed time t, a Malliavin weight is a (sufficiently regular) random variable π such that

x x ∇x E(f (Xt )) = E(f (Xt )π) (5)

n for all, say, bounded, measurable f : R → R.

Remark By the “logarithmic trick”, Malliavin weights exist for all hypo-elliptic diﬀusions. For simplicity, we concentrate on the Delta.

Theorem x d×n Assume that σ(Xt (ω)) has a right-inverse Rt (ω) ∈ R for i 2 dt ⊗ P a. e. (t, ω) such that Rt J0→t (x) ∈ L ([0, T ] × Ω), i i = 1,..., n, where J0→t (x) denotes the ith column of J0→t (x). n Then for every bounded, measurable f : R → R, Z T x x 1 ∇x E(f (XT )) = E f (XT ) Rt J0→t (x)dBt . (6) 0 T

Remark −1 x The assumption is satisﬁed (with Rs = σ (Xs )) if d = n, and σ is uniformly elliptic, i. e. ∃ > 0 s. t.

T T 2 n ξ σ(x) σ(x)ξ ≥ |ξ| , ∀x, ξ ∈ R .

x ∇x E(f (XT )) = Z T 1 x T −1 x E ∇f (XT ) J0→T (x) J0→t (x) σ(Xt )Rt J0→t (x) dt T 0 | {z } =In The chain rule for Malliavin derivatives implies

x x T x x T −1 x Dt f (XT ) = ∇f (XT ) Dt XT = ∇f (XT ) J0→T (x)J0→t (x) σ(Xt ) and we get Z T x x ∇x E(f (XT )) = E hDt f (XT ),Rt J0→t (x)/T i dt 0 x = E(f (XT )δ(t 7→ Rt J0→t (x)/T ))

Theorem Let a, σ depend on a real parameter α and assume that σ is uniformly elliptic (in particular, d = n). Then

∂ E(f (X x )) = E(f (X x )δ(t 7→ H(t, T , α))), (7) ∂α T T

x 1 x −1 −1 ∂XT with H(t, T , α) = T σ(Xt , α) J0→t (x)J0→T (x) ∂α .

Proof. Proceed similarly to the ﬁrst proof using

x T x x −1 −1 ∇f (XT ) = Dt f (XT )σ(Xt , α) J0→t (x)J0→T (x) .

Similar formulas are also possible in a hypo-elliptic set up. In general, the Malliavin weight is defined as Skorohod integral of some process. Note that – unlike the Itôintegral – the definition of the Skorohod integral is essentially non-constructive. Consequently, formulas for Malliavin weights especially in non-elliptic situations are often not directly usable for computational purposes.

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging Calculation of the Greeks Summary Summary

The Greeks, sensitivities of option prices with respect to model parameters, are not only important for computational issues, but also because of their rôlein hedging strategies. Calculation using finite differences is often inefficient. Malliavin weights provide an alternative, in many situations superior method. In general, they are only given in a non-constructive way. Extending the theory to more general models, e. g. models driven by jump-diffusions, is a popular research topic.

Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging Calculation of the Greeks Summary References

Jean-Michel Bismut: Large deviations and the Malliavin calculus, Birkäuser Verlag, 1984. Eric Fournié, Jean-Michel Lasry, JérômeLebuchoux, Pierre-Louis Lions, Nizar Touzi: Applications of Malliavin calculus to Monte Carlo methods in finance, Finance and Stochastics 3, 391–412, 1999. Ioannis Karatzas: Lectures on the mathematics of finance, CRM Monograph Series 8, AMS, 1996. David Nualart: The Malliavin calculus and related topics, Probability and Its Applications, Springer-Verlag, 1995.

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Approximate Weights Summary

Part II

Approximation of Malliavin Weights

Christian Bayer University of Technology, Vienna Calculation of Greeks Stochastic Taylor Expansion Universal Malliavin Weights Nilpotent Lie Groups Approximate Weights Iterated Stratonovich Integrals on G m Summary d,0 Universal Malliavin Weights Rewriting the Equation

We rewrite equation (1) as

d d x x X x i X x i dXt = V0(Xt )dt + Vi (Xt ) ◦ dBt = Vi (Xt ) ◦ dBt , (8) i=1 i=0

i where ◦dBt denotes the Stratonovich stochastic integral and we 0 i use the notation “◦dBt = dt”. Vi (x) = σ (x), i = 1,..., d and 1 Pd V0(x) = a(x) − 2 i=1 dVi (x) · Vi (x). Remark n n We understand vector ﬁelds V : R → R as functions and as ∞ n diﬀerential operators acting on C (R ; R) by

∞ n n Vf (x) = df (x) · V (x), f ∈ C (R ; R), x ∈ R .

∞ The vector fields V0,..., Vd are smooth and C -bounded, i. e. their derivatives of order greater than 0 are bounded. The vector fields satisfy a uniform Hörmander condition, see [Kusuoka 2002] for details. Consequently, the corresponding diffusion X is hypo-elliptic.

We deﬁne a degree on the set of multi-indices S k A = {0,..., d} by deg((i1,..., i )) = k + #{j|ij = 0}, k∈N k i. e. 0s are counted twice. Theorem ∞ n Fix m ∈ N and f ∈ C (R ). Then

X Z f (X x ) = V ··· V f (x) ◦dBi1 ◦· · ·◦dBik t i1 ik t1 tk α=(i ,...,i )∈A 1 k 0

+ Rm(f , t, x),

p 2 (m+1)/2 with E(Rm(f , t, x) ) = O(t ).

Proof. By Itˆo’sformula, we get

d Z t x X x i f (Xt ) = f (x) + Vi f (Xs ) ◦ dBs . i=0 0

x Now we apply Itˆo’s formula to Vi f (Xs ), i = 0,..., d, and obtain

d Z t d Z s x X X x j i f (Xt ) = f (x) + Vi f (x) + Vj Vi f (Xu ) ◦ dBu ◦ dBs . i=0 0 j=0 0

Iterate this procedure and then apply Itˆo’slemma to get the order estimate for the rest term.

Remark 1 The stochastic Taylor expansion is the starting point for many applications in stochastic analysis and for several numerical methods, including Stochastic Taylor schemes and Cubature on Wiener space, a method based on Terry Lyon’s theory of rough paths. 2 Following the latter concept, we interpret the process α (Bt )α∈A, deg(α)≤m, t ∈ [0, T ], as the “probabilistic core” of the solution of the SDE. 3 Consequently, a better understanding of this process might yield methods to calculate the non-constructive Malliavin-weight formulas presented in the last section.

In order to study the stochastic process given by the iterated Stratonovich integrals up to order m, we embed it into an appropriate algebraic/geometric framework. We additionally assume that the Stratonovich drift vanishes, i. e. V0 = 0. This assumption simpliﬁes the notation and allows us to use the usual degree deg((i1,..., ik )) = k for k (i1,..., ik ) ∈ {1,..., d} . Furthermore, it will allow us to omit some subtle restrictions in the following.

We stress that the results remain essentially true for V0 6= 0.

m Let Ad,0 be the space of all non-commutative polynomials in e1,..., ed of degree less than or equal to m. m Deﬁne a non-commutative multiplication on Ad,0 by cutting m oﬀ all monomials of higher degree than m. Ad,0 becomes the free associative, non-commutative, step-m nilpotent real algebra with unit in d generators e1,..., ed . m Let W0 denote the linear span of the unit element 1 of Ad,0, i. e. W0 is the space of all polynomials of degree 0. We m identify W0 ' R and denote by x0 the projection of x ∈ Ad,0 on W0.

m m P∞ xi exp : Ad,0 → Ad,0 is deﬁned by exp(x) = 1 + i=1 i! . The logarithm is deﬁned for x0 6= 0 by

∞ i−1 X (−1) x − x0 i log(x) = log(x0) + i x0 i=1 m i−1 X (−1) x − x0 i = log(x0) + . i x0 i=1

m Ad,0 equipped with the commutator bracket [x, y] = xy − yx, m x, y ∈ Ad,0, is a Lie algebra.

m Let gd,0 denote the free, step-m nilpotent Lie algebra generated by {e1,..., ed }, i. e.

m gd,0 = {ei , [ei , ej ], [ei , [ej , ek ]],... | i, j, k = 1,..., d} .

We define the step-m nilpotent free Lie group as the m m m exponential image of gd,0, i. e. Gd,0 = exp(gd,0). m m Gd,0 is a Lie group and gd,0 is its Lie algebra, which can be seen by the Campbell-Baker-Hausdorff formula 1 1 exp(x) exp(y) = exp x+y+ [x, y]+ ([x, [x, y]]−[y, [y, x]])+··· . 2 12 m m m m Note that exp : gd,0 → Gd,0 and log : Gd,0 → gd,0 define a global chart of the Lie group.

2 The Heisenberg group is the group G2,0, a 3-dimensional 2 7 submanifold of A2,0 ' R . It allows the following representation as a matrix group:

2 n 1 a c o G2,0 = 0 1 b a, b, c ∈ . 0 0 1 R

2 The Lie algebra – as tangent space at I3 ∈ G2,0 – is

2 n 0 x z o g2,0 = 0 0 y x, y, z ∈ R . 0 0 0

0 1 0 0 0 0 An isomorphism is given by e1 = 0 0 0 , e2 = 0 0 1 . 0 0 0 0 0 0

Deﬁnition m y For y ∈ Ad,0 we deﬁne a stochastic process Yt , t ∈ [0, T ], by

y X α Y = y B eα , t α∈A, deg(α)≤m t

α where eα = ei1 ··· eik for α = (i1,..., ik ), e∅ = 1, and Bt denotes the corresponding iterated Stratonovich integral, i. e. Bα = R ◦dBi1 · · · ◦ dBik . t 0

Remark 1 Yt is the vector of the iterated Stratonovich integrals written in m the canonical basis of Ad,0.

Theorem m Deﬁne vector ﬁelds Di (x) = xei , x ∈ Ad,0, i = 1,..., d. y 1 Yt is solution of the SDE

y Xd y i y dY = Di (Y ) ◦ dB , Y = y. t i=1 t t 0 y 2 m m Given y ∈ Gd,0, we have Yt ∈ Gd,0 a. s. for all t ∈ [0, T ]. 3 By the Feynman-Kac formula,

t Xd E(Y y ) = y exp e2 . t 2 i=1 i

Theorem m Deﬁne vector ﬁelds Di (x) = xei , x ∈ Ad,0, i = 1,..., d. y 1 Yt is solution of the SDE

y Xd y i y dY = Di (Y ) ◦ dB , Y = y. t i=1 t t 0 y 2 m m Given y ∈ Gd,0, we have Yt ∈ Gd,0 a. s. for all t ∈ [0, T ]. 3 By the Feynman-Kac formula,

t Xd E(Y y ) = y exp e2 . t 2 i=1 i

Theorem m Deﬁne vector ﬁelds Di (x) = xei , x ∈ Ad,0, i = 1,..., d. y 1 Yt is solution of the SDE

y Xd y i y dY = Di (Y ) ◦ dB , Y = y. t i=1 t t 0 y 2 m m Given y ∈ Gd,0, we have Yt ∈ Gd,0 a. s. for all t ∈ [0, T ]. 3 By the Feynman-Kac formula,

t Xd E(Y y ) = y exp e2 . t 2 i=1 i

Remark 1 The formula (3) allows eﬃcient calculation of all moments of 1 m iterated Stratonovich integrals. Observe that E(Yt ) ∈/ Gd,0. 2 By Statement 2, we can carry out all relevant calculations for m iterated Stratonovich integrals in the vector space gd,0 by 1 passing to the process Zt = log(Yt ).

Example 1 2 For m = d = 2, we get Zt = Bt e1 + Bt e2 + At [e1, e2], where At denotes L´evy’sarea

Z t Z t 1 1 2 1 2 1 At = Bs ◦ dBs − Bs ◦ dBs . 2 0 2 0

Proposition m Fix w ∈ gd,0 and t ∈ [0, T ]. There is a non-adapted, Skorohod d integrable, R -valued process as , 0 ≤ s ≤ T such that for any m m bounded, measurable function f : Gd,0 → R and any y ∈ Gd,1

∂ y+w y m E(f (Yt )) = E(f (Yt )πd,0), ∂ =0

m where πd,0 = δ(a).

Proof. The proof is very similar to the proof for the existence of Malliavin weights in a general, hypo-elliptic setting. See [Teichmann 06].

Remark 1 The strategy a is universal in the sense that it does only depend on m, d, t and – in a linear way – w. 2 Yet again, the Malliavin weight is given in an essentially non-constructive way due to the Skorohod integration. Note that even calculation of the strategy a is diﬃcult. However, m approximation of the weight πd,0 turns out to be possible.

Theorem n Fix x, v ∈ R , t ∈ [0, T ] and m ≥ 1 such that v can be written as X v = wα[Vi1 , [Vi2 , [··· , Vik ] ··· ](x) α∈A\{∅}, deg(α)≤m−1

P m for some wα ∈ R. Deﬁne w = wα[ei1 , [ei2 , [··· , eik ] ··· ] ∈ gd,0 m and let πd,0 denote the corresponding universal Malliavin weight. ∞ n Then for any C -bounded function f : R → R we have

∂ x+v x m (m+1)/2 E(f (Xt )) = E(f (Xt )πd,0) + O(t ). ∂ =0 We note that the constant in the leading order term in the error estimate depends only on the ﬁrst derivative of f .

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Cubature Formulas Approximate Weights Heat Kernel Summary Cubature

Definition n Given a finite Borel measure µ on R with finite moments of order up to m ∈ N.A cubature formula of degree m is a collection of n weights λ1, . . . , λk > 0 and points x1,..., xk ∈ supp(µ) ⊂ R such n that the following equality holds for any polynomial f on R of degree less or equal m:

k Z X f (x)µ(dx) = λi f (xi ). n R i=1

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Cubature Formulas Approximate Weights Heat Kernel Summary Chakalov’s Theorem

Theorem n For any Borel measure µ on R with ﬁnite moments of order up to n m there is a cubature formula with size k ≤ dim Polm(R ), the n space of polynomials on R of order up to m.

Remark Chakalov’s Theorem is non-constructive, and construction of eﬃcient cubature formulas remains a non-trivial problem in higher dimensions. The bound on the size is a consequence of Caratheodory’s Theorem.

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Cubature Formulas Approximate Weights Heat Kernel Summary Cubature Formulas for the Universal Weight

Theorem m m Fix t ∈ [0, T ] and w ∈ gd,0. There are points x1,..., xr ∈ Gd,0 and weights ρ1, . . . , ρr 6= 0 such that

d r t X X E(Y 1πm ) = w exp e2 = ρ x . t d,0 2 i j j i=1 j=1

m Furthermore, we may choose r ≤ 2 dim Ad,0 + 2.

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Cubature Formulas Approximate Weights Heat Kernel Summary Proof of Cubature for the Universal Weight

Proof. The ﬁrst equality is an easy consequence of

y t Xd 2 m E(Yt ) = y exp ei , y ∈ Ad,0. 2 i=1 Now deﬁne two positive measures on the Wiener space by dQ+ m dQ− m dP = (πd,0)+ and dP = (πd,0)−. By absolute continuity of Q± 1 m w. r. t. P, Yt ∈ Gd,0 Q±-a. s.. Chakalov’s Theorem applied to the 1 laws of Yt under Q+ and Q− yields

1 m 1 1 Xr EP (Y π ) = EQ (Y ) − EQ (Y ) = ρj xj . t d,0 + t − t j=1

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Cubature Formulas Approximate Weights Heat Kernel Summary A Reformulation of Cubature

A theorem from sub-Riemannian geometry (Chow’s Theorem) m implies that any x ∈ Gd,0 can be joined to 1 by the the solution of the ODE d d X X x˙t = Di (xt )ω ˙i (t) = xt ei ω˙i (t) i=1 i=1 for some paths of bounded variation ωi : [0, T ] → R, i = 1,..., d, i. e. x0 = 1 and xt = x. We denote the solution to the above ODE for some path ω of 1 bounded variation by Ys (ω), s ∈ [0, T ]. d Consequently, there are ωj ∈ Cbv ([0, T ]; R ), j = 1,..., r, s. t. r 1 m X 1 E(Yt πd,0) = ρj Yt (ωj ). j=1

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Cubature Formulas Approximate Weights Heat Kernel Summary Malliavin Weights by Cubature

Theorem n Fix x, v ∈ R , t ∈ [0, T ] and m ≥ 1 such that v can be written as X v = wα[Vi , [Vi , [··· , Vi ] ··· ](x) α∈A\{∅}, deg(α)≤m−1 1 2 k

P m for some wα ∈ R. Deﬁne w = wα[ei1 , [ei2 , [··· , eik ] ··· ] ∈ gd,0 and let ρj , ωj , j = 1,..., r, denote the cubature formula for the m corresponding weight πd,0. Then

∂ x+v Xr x (m+1)/2 E(f (X )) = ρj f (X (ωj )) + O(t ), t j=1 t ∂ =0

x dXt (ωj ) Pd x i x where dt = i=1 Vi (Xt (ωj ))ω ˙ j (t) with X0 (ωj ) = x.

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Cubature Formulas Approximate Weights Heat Kernel Summary Remarks

The above formula is some kind of finite difference formula! Instead of solving the PDE problem for different starting points, we solve the SDE problem for different trajectories ω of the Brownian motion. m The same procedure without πd,0 gives a method for x Pl x calculation of E(f (Xt )) ≈ j=1 λj f (Xt (˜ωj )). This method – introduced by T. Lyons and N. Victoir – is known as “Cubature on Wiener space”. For actual computations, it is necessary to iterate the procedure along a partition 0 = t0 < t1 < ··· < tk = t of [0, t]. This is possible using cubature on Wiener space, m−1 yielding, at least in theory, a method of order 2 for approximation of Greeks.

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Cubature Formulas Approximate Weights Heat Kernel Summary m The Heat Kernel on Gd,0

Deﬁnition m 1 The density pt (x), x ∈ Gd,0, of the process Yt with respect to the Haar measure on the Lie group is called heat kernel.

Remark

1 The name comes from the fact that pt is the fundamental solution of the heat equation with respect to the sub-Laplacian 1 Pd 2 1 L = 2 i=1 Di , the inﬁnitesimal generator of Yt . 1 2 We can equivalently study the density of Zt = log(Yt ) with m respect to the Lebesgue measure on gd,0. 3 In the setting without drift, the heat kernel always exists as a Schwartz function. With non-vanishing drift, we need to m factor out the direction in gd,1 corresponding to t.

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Cubature Formulas Approximate Weights Heat Kernel Summary Approximation of the Heat Kernel

We want to ﬁnd polynomial approximations of the heat kernel 1 pt or the Malliavin weight d log pt ◦ Yt . In the following we m concentrate on the former problem and tacitly switch to gd,0 m using the same notation pt for the density of Zt on gd,0. Choose a suitable (Gaussian) measure Qt with density rt on m gd,0 and let hα(t, ·) denote the corresponding family of orthonormal (Hermite) polynomials. We need to calculate the integral Z Z pt (z) hα(t, z)Qt (dz) = pt (z)hα(t, z)dz = E(hα(t, Zt )). m rt (z) m gd,0 gd,0

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Cubature Formulas Approximate Weights Heat Kernel Summary Approximation of the Heat Kernel – 2

This procedure will give us an approximation

pt (z) X = aαh (t, z) r (z) t α t α 2 m α valid in L (gd,0, Qt ), where at = E(hα(t, Zt )). Note that hα(t, Zt ) is a (time dependent) polynomial in the iterated Stratonovich integrals of order up to m. Calculation α of at is enabled by t Xd E(Y˜ 1) = exp e2 , t m˜ 2 i=1 i ˜ m˜ where Y denotes the process of iterated integrals in Ad,0 with m˜ > m large enough. The use of Qt instead of dz is preferable because polynomials are not dz-square integrable.

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Cubature Formulas Approximate Weights Heat Kernel Summary Applications of Approximate Heat Kernels

Approximate heat kernels can be used to make higher order Taylor schemes for approximation of SDEs feasible. In the context of Mallivin weights, they provide approximations to the universal Malliavin weights deﬁned before. Finally, the subject of heat kernels on Lie groups is a well-established subject of mathematical research in its own right.

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Approximate Weights Summary Summary

By using Stochastic Taylor expansion, we constructed universal Malliavin weights, which yield approximations to the Greeks for a very general class of (d-dimensional) problems. Cubature formulas for Greeks – in combination with cubature on Wiener space – yield high-order methods for the calculation of the Greeks, even in situations, where no direct formula for the Greeks is possible. There are still many open problems regarding actual usability of this theory for computations.

Christian Bayer University of Technology, Vienna Calculation of Greeks Universal Malliavin Weights Approximate Weights Summary References

Shigeo Kusuoka: Appropximation of expectation of diﬀusion processes based on Lie algebra and Malliavin calculus, Advances in mathematical economics 6, 69-83, 2004. Terry Lyons: Diﬀerential equations driven by rough signals, Rev. Mat. Iberoam. 14, No.2, 215-310, 1998. Terry Lyons and Nicolas Victoir: Cubature on Wiener space, Proceedings of the Royal Society London A 460, 169-198, 2004. Josef Teichmann: Calculating the Greeks by Cubature formulas, Proceedings of the Royal Society London A 462, 647-670, 2006.

Christian Bayer University of Technology, Vienna Calculation of Greeks