Malliavin Differentiability of CEV-Type Heston Model

4.3 Arbitrage ...... 18 4.4 Malliavin Differentiability of the CEV-Type Heston Model (Logarithmic Price) ...... 20 4.5 Malliavin Differentiability of the CEV-Type Heston Model (Actual Price) ...... 23 4.6 Delta and Rho ...... 25

5 Conclusion 27

1 Introduction

Malliavin calculus is the infinite-dimensional differential calculus on the Wiener space in order to give a probabilistic proof of Holmander’s¨ theorem. It has been developed as a tool in mathematical finance. In 1999, Founie´ et al. [1] gave a new method for more efficient computation of Greeks which represent sensitivities of the derivative price to changes in parameters of a model under consideration, by using the integration by parts formula related to Malliavin calculus. Following their works, more general and efficient application to computation of Greeks have been introduced by many authors (see [2], [3], [4]). They often considered this method for tractable models typified by the Black-Scholes model.

In the Black-Scholes model, an underlying asset St is assumed to follow the stochastic differential equation dSt = rSt dt + σSt dWt, where r and σ respectively imply the risk free interest rate and the volatility. The Black-Scholes model seems standard in business. The reason is that this model has the analytic solution for famous options, so it is fast to calculate prices of derivatives and risk parameters (Greeks) and easy to evaluate a lot of deals and the whole portfolios and to manage the risk. However, the Black-Scholes model has a defect that this model assumes that volatility is a constant. In the actual financial market, it is observed that volatility fluctuates. However, the Black-Scholes model does not suppose the prospective fluctuation of volatility, so when we use the model there is a problem that we would underestimate prices of options. Hence, more accurate models have been developed. One of the models is the stochastic volatility model. One of merits to consider this model is that even if prices of derivatives such as the European options are not given for any strike and maturity, we can grasp the volatility term structure. In particular, the Heston model, which is introduced in [5], is one of the most popular stochastic volatility models. This model assumes that the underlying asset St and the volatility νt follow the stochastic differential equations √ dSt = St(r dt + νt dBt), (1.1) 1 dνt = κ(µ − νt) dt + θνt 2 dWt, (1.2) where Bt and Wt denote correlated Brownian motions. In the equation (1.2), κ, µ and θ imply respectively the rate of mean reversion (percentage drift), the long-run mean (equilibrium level) and the volatility of volatility. This volatility model is called the Cox-Ingersoll-Ross model and more complicated than the Black-Scholes model. We have not got the analytic solution yet. However, even this model can not grasp fluctuation of volatility accurately. In 2006 (see [6]), Andersen

2 and Piterbarg generalized the Heston model. They extended the volatility process of (1.2) to

1 dν = κ(µ − ν ) dt + θν γ dW , γ ∈ , 1 . (1.3) t t t t 2

This model is called the constant elasticity of variance model (we will often shorten this model as the CEV 1 model). Naturally, in the case γ ∈ 2 , 1 , the volatility model (1.3) is more complicated than the volatility model (1.2). Here, consider the European call option and let φ is a payoff function. Then we can estimate the option −rT price by the following formula V (x) = E[ e φ(ST )]. However, the computation of Greeks is much ∂V (x) important in the risk-management. A Greek is given by ∂α where α is one of parameters needed to compute the price, such as the initial price, the risk free interest rate, the volatility and the maturity etc.. Most of ﬁnancial institutions have calculated Greeks by using ﬁnite-difference methods but there are some demerits such that the results depend on the approximation parameters. More than anything, the methods need the assumption that the payoff function φ is differentiable. However, in business they often consider the payoff functions such as φ(x) = (x − K)+ or φ(x) = 1{x≥K}. Here we need Malliavin calculus. In 1999 Founie´ et al. in [1] gave the new methods for Greeks. To come to the point, they calculated Greeks by the ∂V (x) −rT following formula ∂α = E[ e φ(ST )·(weight)]. We can calculate this even if φ is polynomial growth. Instead, we need the Malliavin differentiability of St.

The solution Xt satisfying the stochastic differential equation with Lipschitz continuous coefficients is known as Malliavin differentiable. Hence we can easily verify that the Black-Scholes model is Malliavin γ 1 differentiable. However the diffusion coefficient x , γ ∈ 2 , 1 is neither differentiable at x = 0 nor Lipschitz continuous and then we cannot find whether the CEV-type Heston model is Malliavin differentiable 1 or not. In [7], Alos and Ewald proved that the volatility process (1.2), that is the case where γ = 2 of (1.3), was Malliavin differentiable and gave the explicit expression for the derivative. However, in the case 1 γ ∈ 2 , 1 , we can not simply prove the Malliavin differentiability in the exact same way. 1 In this paper we concentrate on the case γ ∈ 2 , 1 , that is, we extend the results in [7] and give the explicit expression for the derivative. Moreover we consider the CEV-type Heston model and give the formula to compute Greeks.

2 Summary of Malliavin Calculus

We give the short introduction of Malliavin calculus on the Wiener space. For further details, refer to [8].

2.1 Malliavin Derivative

We consider a Brownian motion {W (t, ω)}t∈[0,T ] (in the sequel, we often denote W (t, ω) by Wt) on a complete filtered probability space (Ω, F, P; (Ft)) where (Ft) is the filtration generated by Wt, and the Hilbert space H := L2([0,T ]). When fixing ω, we can consider ω(t) := W (t, ω) ∈ C([0,T ]). Then the Z T Z T Itoˆ integral of h ∈ H is constructed as h(t) dW (t, ω) = h(t) dω(t) on C([0,T ]). We denote by 0 0

3 ∞ n n Cp (R ) the set of inﬁnitely continuously differentiable functions f : R → R such that f and all its partial derivatives have polynomial growth. Let S be the space of smooth random variables expressed as

F (ω) = f(W (h1),...,W (hn)), (2.1)

Z T ∞ n ∞ n where f ∈ Cp (R ) and W (h) := h(t) dWt where h1, . . . , hn ∈ H, n ≥ 1. We denote by C0 (R ) 0 the set of infinitely continuously differentiable functions f : Rn → R such that f has compact support. ∞ n n Moreover we denote by Cb (R ) the set of infinitely continuously differentiable functions f : R → R such that f and all of its partial derivatives are bounded. Denote by S0 and Sb respectively, the spaces of ∞ n ∞ n smooth random variables of the form (2.1) such that f ∈ C0 (R ) and f ∈ Cb (R ). We can find that p ∂ S0 ⊂ Sb ⊂ S and S0 is a linear subspace of and dense in L (Ω) for all p > 0. We use the notation ∂i = ∂xi in the sequal. We define the derivative operator D, so called the Malliavin derivative operator.

Deﬁnition 2.1 (Malliavin derivative). The Malliavin derivative DtF of a smooth random variable expressed as (2.1) is deﬁned as the H-valued random variable given by

n X DtF = ∂if(W (h1),...,W (hn))hi(t). (2.2) i=1

We sometimes omit to write the subscript t.

Since S is dense in Lp(Ω), we will deﬁne the Malliavin derivative of a general F ∈ Lp(Ω) by means of taking limits. We will now prove that the Malliavin derivative operator D : Lp(Ω) → Lp(Ω; H) is closable. Please refer to [8] for proves of the following results.

Lemma 2.1. We have E[ GhDF, hiH ] = −E[ F hDG, hiH ] + E[ F GW (h), for F,G ∈ S and h ∈ H.

Lemma 2.2. For any p ≥ 1, the Malliavin derivative operator D : Lp(Ω) → Lp(Ω; H) is closable.

For any p ≥ 1, we denote by D1,p the domain of D in Lp(Ω) and then it is the closure of S by the norm

p 1 ( " #) p 1 Z T 2 p p p p 2 kF k1,p = E[ |F | ] + E[ kDF kH ] = E[ |F | ] + E |DtF | dt . (2.3) 0

1,2 Note that D is a Hilbert space with the scalar product hF,Gi1,2 = E[ FG] + E[ hDF,DGiH ]. Moreover, the Malliavin derivative {DtF }t∈[0,T ] is regarded as a stochastic process deﬁned almost surely with the measure P × u where u is a Lebesgue measure in [0,T ]. Indeed, we can observe

Z T Z T 2 2 2 2 kDF kL2(Ω;H) = E (DtF ) dt = E[(DtF ) ] dt = kD·F kL2(Ω×[0,T ]). (2.4) 0 0

The following result will become a very important tool.

4 1,2 2 2 Lemma 2.3. Suppose that a sequence {Fn : Fn ∈ D , supn E[ kDFnkH ] < ∞} converges to F in L (Ω). 1,2 2 Then F belongs to D and the sequence {DFn} converges to DF in the weak topology of L (Ω; H).

k F {Dk F, t ∈ [0,T ]} Ω × [0,T ]k Similarly, we define the -th Malliavin derivative of , t1,...,tk i , as a - measurable stochastic process defined P × uk-almost surely and the operator Dk is closable from S → Lp(Ω; Hk) for any p ≥ 1 and k ≥ 1. As with the Malliavin derivative D, from the closability of Dk, we can define the domain Dk,p of the operator Dk in Lp(Ω) as the completion of S with the norm

1 ( k ) p p X k p kF kk,p = E[|F | ] + E[kD F kH⊗i ] . (2.5) i=1 \ Moreover we deﬁne D1,∞ as D1,∞ := D1,p. We will now prove the chain rule and refer to the [8, p∈N Proposition 1.2.4] for details.

1,p n Lemma 2.4. For p ≥ 1, let F = (F1,...,Fn) ∈ D and ψ : R → R be a Lipschitz function with bounded partial derivatives, and then we have ψ(F ) ∈ D1,p and

n X Dtψ(F ) = ∂iψ(F )DtFi. (2.6) i=1

2.2 Skorohod Integral

For p, q > 1 satisﬁng 1/p + 1/q = 1, the adjoint D∗ of the operator D which is closable and has the domain on Lp(Ω) should be closable but with the domain contained in Lq(Ω). Focus on the case p = q = 2. We can deﬁne the divergence operator δ = D∗ so called the Scorohod integral which is the adjoint of the operator D such as

δ : L2(Ω; H) =∼ L2(Ω × [0,T ]) → L2(Ω). (2.7)

Deﬁnition 2.2 (Skorohod integral). Let u ∈ L2(Ω; H). If for all F ∈ D1,2, we can have

|E[ hDF, uiH ]| ≤ ckF kL2(Ω), (2.8) where c is some constant depending on u, then u is called to belong to the domain Dom(δ). Moreover if 2 u ∈ Dom(δ), then we have that δ(u) belongs to L (Ω) and the duality relation E[ F δ(u)] = E[ hDF, uiH ], for all F ∈ D1,2.

We can get the following results.

Lemma 2.5. Let F ∈ D1,2 and u ∈ Dom(δ) satisfy F u ∈ L2(Ω; H). And then we have that F u belongs to

Dom(δ) and δ(F u) = F δ(u) − hDF, uiH .

5 2 Lemma 2.6. Let u ∈ L (Ω; H) be an Ft-adapted stochastic process then u ∈ Dom(δ) and δ(u) = R T 0 ut dWt.

We give one of famous properties of δ. The following property implies the relationship between the Malliavin derivative and the Skorohod integral. Denote by D1,2(H) the class of processes u ∈ L2(Ω; H) =∼ (Ω × [0,T ]) such that u(t) ∈ D1,2 for almost all t and there exists a measurable version of the two variable hR T R T 2 i processes Dsut satisfying E 0 0 (Dsut) λ(ds) λ(dt) < ∞.

1,2 2 Lemma 2.7. Let u ∈ D (H) satisfy that Drut ∈ Dom(δ) and that δ(Drut) ∈ L (Ω; H). We have then that δ(u) belongs to D1,2 and

Dt(δ(u)) = u(t) + δ(Dtu). (2.9)

The following result is applied to calculate Greeks. For further details, refer to [8, Chapter 6].

1,2 Lemma 2.8. Let F,G ∈ D . Suppose that an random variable u(t, ·) ∈ H satisfy hDF, uiH 6= 0 a.s. and −1 Gu(hDF, uiH ) ∈ Dom(δ). For any continuously differentiable function f with bounded derivatives, we 0 −1 have E[ f (F )G] = E[ f(F )H(F,G)] where H(F,G) = δ(Gu(hDF, uiH ) ).

2.3 Malliavin Calculus for Stochastic Differential Equations

m Consider T > 0 and Ω = C0([0,T ]; R ). Let {Wt}t∈[0,T ] be the m-dimensional Brownian motion on ﬁl- tered probability space (Ω, F,P ; Ft) where P is the n-dimensional Wiener measure and F is the completion of the σ-ﬁeld of Ω with P . And then H = L2([0,T ]; Rm) is the underlying Hilbert space. We consider the solution {Xt}t∈[0,T ] of the following n-dimensional stochastic differential equation for all i = 1, . . . , n

m i i X i j i i dXt = b (Xt) dt + σj(Xt)dWt ,X0 = x , (2.10) j=1

n n n m where b : R → R and σj : R → R satisfy the following : there is a positive constant K < ∞ such that

|b(x) − b(y)| + |σ(x) − σ(y)| ≤ K|x − y|, for all x, y ∈ Rn, (2.11) |b(x)| + |σ(x)| ≤ K(1 + |x|), for all x ∈ Rn. (2.12)

i Here σj is the columns of the matrix σ = (σj). We can have the following result related to the uniqueness and refer to [8, Lemma 2.2.1] for the detail.

Theorem 2.1. There is a unique n-dimensional, continuous and Ft-adapted stochastic process {Xt}t∈[0,T ] h pi satisfying the stochastic differential equation (2.10) with E sup0≤t≤T |X(t)| < ∞, for all p ≥ 2.

i 1,∞ In the case the coefﬁcients are Lipschitz, the solution Xt belongs to D .

6 Theorem 2.2. Assume that coefﬁcients are Lipschitz continuous of the stochastic differential equation (2.10). i 1,∞ Then the solution Xt belongs to D for all t ∈ [0,T ] and i = 1, . . . , n and satisﬁes

h j i pi sup E sup |DrXs| < ∞. (2.13) 0≤r≤t r≤s≤T

j i Moreover the derivative DrXt satisﬁes the following

n m Z t n Z t j i i X X i j k l X i j k DrXt = σj(Xr) + ∂kσl (Xs)DrXs dWs + ∂kb (Xs)DrXs ds, (2.14) k=1 l=1 r k=1 r

j i j j for r ≤ t a.e., and DrXt = 0 for r > t a.e.. Here D denotes the Malliavin derivative for W .

Let Xt be the solution of the following stochastic differential equation

dXt = b(Xt) dt + σ(Xt) dWt,X0 = x, (2.15)

1,2 where Wt denotes a 1-dimensional Brownian motion. Assume that Xt ∈ D . We let Yt be the ﬁrst variation ∂Xt of Xt, that is, Yt = ∂x . We can easily have that Yt satisﬁes the folloing

0 0 dYt = b (Xt)Yt dt + σ (Xt)Yt dWt,Y0 = 1. (2.16)

Considering this as a stochastic differential equation for Yt, we can have the following solution

Z t Z t 0 1 0 2 0 Yt = exp (b (Xs) − (σ (Xs)) ) ds + σ (Xs) dWs . (2.17) 0 2 0

The following results will also be useful to calculate Greeks later.

−1 Lemma 2.9. Under the above conditions, we can have Yt = DsXtσ (Xs)Ys · 1{s≤t}. Z T Let {a(t)}t∈[0,T ] be a continuous function in H such that a(t) dt = 1. 0 R T −1 Lemma 2.10. Under the above conditions, we can have YT = 0 a(t)DtXT σ (Xt)Yt dt.

∂ Theorem 2.3. For any ψ : R → R of polynomial growth, we have ∂x E[ ψ(XT )] = E[ ψ(XT )π] where Z T −1 π = a(t)σ (Xt)Yt dWt. 0

For the more general case, the same result is proved as below. Let Xt denote the solution of the following n-dimensional stochastic differential equation just like as (2.10)

dXt = b(Xt) dt + σ(Xt) dWt,X0 = x, (2.18) where Wt denotes m-dimensional Brownian motion. For the sake of simpliﬁcation, we assume that n = m.

7 Z T −1 2+ Theorem 2.4. Suppose that the diffusion coefﬁcient σ is invertible and that E |σ (Xt)Yt| dt < 0 ji j 1,∞ ∞, for some > 0, where Y denotes the ﬁrst variation process, that is, Yt = ∂iXt . Let G ∈ D be a random variable which does not depend on the initial condition x. Then for all measurable function φ with polynomial growth we have ∂iE[ φ(XT )G] = E[ φ(XT )πi(G), where a(t) is an Ft-adapted process R T satisfying 0 a(t) dt = 1,

n X k −1 ki πi(G) = δ (Ga(t)(σ (Xt)Yt) ) k=1 n Z T Z T X −1 ki k k −1 ki = G a(t)(σ (Xt)Yt) dWt − Dt Ga(t)(σ (Xt)Yt) ds , (2.19) k=1 0 0

k k and δ denotes the adjoint to the Malliavin derivative with respect to a Brownian motion Wt .

The following theorem introduced in [9] is useful. From now on, we will now denote by ∂t the once derivative with respect to t, by ∂x the once derivative with respect to x and by ∂xx the second derivative with respect to x.

Theorem 2.5. Consider a stochastic process Xt satisfying the 1-dimensional stochastic differential equation

dXt = µ(t, Xt) dt + σ(t, Xt) dWt, (2.20)

1 2 where Wt denotes a Brownian motion and the coefﬁcients µ(t, x) ∈ C ([0,T ]×R) and σ(t, x) ∈ C ([0,T ]× R) satisfy the linear growth condition and the Lipschitz condition. Moreover, we assume that σ is positive and bounded away from 0, and that µ(t, 0) and σ(t, 0) are bounded for all t ∈ [0,T ]. Then Xt belongs to D1,2 and the derivative is given by

Z t µ∂xσ 1 ∂tσ DrXt = σ(t, Xt) exp ∂xµ − − (∂xxσ)σ − (s, Xs) ds , (2.21) r σ 2 σ for r ≤ t and DrXt = 0 for r > t.

Proof. We omit the proof. For further details, refer to [Theorem 2.1 [9]].

3 Mean-Reverting CEV Model

Following the construction in [7], we will now prove that the mean-reverting constant elasticity of variance model is Malliavin differentiable. The mean-reverting CEV model follows the stochastic differential equation

1 dν = κ(µ − ν ) dt + θν γ dW , γ ∈ , 1 , (3.1) t t t t 2

8 with ν0 = ν > 0 and where µ, κ and θ > 0. In [7], Alos and Ewald proved the Malliavin differentiability of 1 1 2 the case γ = 2 of (3.1). In the case, the function x is neither continuously differentiable in 0 nor Lipschitz continuous so they circumvented various problems by some transforming and approximating. However, in 1 the case γ ∈ 2 , 1 , there are more complex problems. Following [7], we will extend their results.

3.1 Existence and Uniqueness

We will now prove that the solution to (3.1) not only exists uniquely but is also positive a.s.

Lemma 3.1. There exists a unique strong solution to (3.1) which satisﬁes P ( νt ≥ 0, t ≥ 0 ) = 1. Moreover, let τ = inf{ t ≥ 0; νt = 0 or = ∞} with inf{∅} = ∞. Then we have P ( τ = ∞ ) = 1.

Proof. Instead of (3.1), consider the following

1 dv = κ(µ − v ) dt + θ|v |γ dW , γ ∈ , 1 . (3.2) t t t t 2

If we have concluded that the unique strong solution of (3.2) is positive a.s., then (3.2) coincides with (3.1). The existence of non-explosive weak solution for (3.2) follows from the continuity and the sub-linear growth condition of drift and diffusion coefﬁcients. Moreover, from [10, Proposition 5.3.20, Corollary 5.3.23], we have the pathwise uniqueness. From [10, Proposition 5.2.13], we can verify that the pathwise uniqueness holds for (3.2).

We will now prove that the second claim is true. Let τv = inf{ t ≥ 0; vt = 0 or = ∞} with inf{∅} =

∞. In order to use [10, Theorem 5.5.29], we verify that for a ﬁxed number c ∈ R, limx→0 p(x) = −∞ R x n R y κ(µ−z) o where p(x) is deﬁned as p(x) = c exp −2 c θ2z2γ dz dy. Since we have known that the solution vt of (3.2) does not explode at ∞, if we could prove that the above formula holds, we can claim that P (τv = ∞) = 1, that is, P (τ = ∞) = 1. We can assume without restriction that x < 1 and let c = 1. Then we have

Z y κ(µ − z) Z y κµ κ −2 2 2γ dz = −2 2 2γ − 2 2γ−1 dz 1 θ z 1 θ z θ z 2κµ 1 1 2κ 1 1 = 2 2γ−1 − 2 2γ−2 θ (2γ − 1) z y θ (2 − 2γ) z y 2κµ 1 2κ ≥ − 1 + . (3.3) θ2(2γ − 1) y2γ−1 θ2(2 − 2γ)

Letting w = y−1, we can calculate p(x). From the last inequality, there exists a constant C > 0 satisfying the following inequality and then we have as x → 0,

Z 1 2κµ 1 p(x) ≤ −C exp 2 2γ−1 dy x θ (2γ − 1) y Z 1 x 1 2κµ 2γ−1 = −C 2 exp 2 w dw → −∞. (3.4) 1 w θ (2γ − 1)

9 3.2 Lp-Integrability

Consider the stochastic differential equation

γ dνt = b(νt) dt + θνt dWt, (3.5)

1 with ν0 = x > 0, where b is such that b(0) > 0 and satisﬁes the Lipschitz condition, θ > 0 and γ ∈ 2 , 1 . The following lemma ensures the existence of its moments of any order.

p Lemma 3.2. Consider the solution of the (3.5). For any p ≥ 0, we have E supt∈[0,T ] νt < ∞ and −p E supt∈[0,T ] νt < ∞.

Proof. At ﬁrst we consider the positive moments. We deﬁne the stopping time τn = inf { 0 ≤ t ≤ T ; νt ≥ n } with inf{ ∅ } = ∞. By Ito’sˆ formula,

Z t∧τn Z t∧τn p p p−1 1 p−2 2 νt∧τn = x + pνs dνs + p(p − 1)νs (dνs) 0 2 0 Z t∧τn Z t∧τn 2 Z t∧τn p p−1 p−1+γ p(p − 1)θ p−2+2γ ≤ x + p νs b(νs) ds + pθ νs dWs + νs ds. (3.6) 0 0 2 0

From the Lipschitz condition of the drift function b(x), there exists a positive constant K which satisﬁes b(νs) ≤ Kνs + b(0). By the above inequality and Young’s inequality, we have

Z t∧τn 2 Z t∧τn p p p−1 p(p − 1)θ p−2+2γ E[ νt∧τn ] ≤x + pE νs b(νs) ds + E νs ds 0 2 0 Z t∧τn Z t∧τn p p p−1 ≤x + pKE νs ds + pE νs b(0) ds 0 0 2 Z t∧τn p(p − 1)θ p−2+2γ + E νs ds 2 0 hR t∧τn p i Z t∧τn E νs ds p p 0 ≤x + pKE νs ds + p p 0 p−1  hR t∧τn p i  b(0)p p(p − 1)θ2 E 0 νs ds 1 + p +  p + p  p 2 p−2+2γ 2−2γ Z t 0 p =C + C E[ νs∧τn ] ds. (3.7) 0

p 0 0 By Gronwall’s lemma, we can have E[ νt∧τn ] ≤ C exp{C t}, where both C and C do not depend on n. As n → ∞, we can obtain the result. Next we consider the negative moments. Deﬁne the stopping time as

10 1 τ = inf 0 ≤ t ≤ T ; ν ≤ , with inf{∅} = ∞. By Ito’sˆ formula, we have n t n

Z t∧τn Z t∧τn −p −p −(p+1) 1 −(p+2) 2 νt∧τn = x + (−p)νs dνs + p(p + 1)ν (dνs) 0 2 0 Z t∧τn b(ν ) Z t∧τn 1 p(p − 1)θ2 Z t∧τn 1 = x−p − p s ds − pθ dW + ds. p+1 (p+1)−γ s 2(1−γ)+p (3.8) 0 νs 0 νs 2 0 νs

Taking the expectation and using the Fubini’s theorem, we have " # " # Z t∧τn b(ν ) ds θ2 Z t∧τn ds E[ ν−p ] = x−p − pE s + p(p + 1)E t∧τn (p+1) 2(1−γ)+p 0 νs 2 0 νs " ! # Z t 1 Z t∧τn p(p + 1)θ2 pb(0) ≤ x−p + pK E ds + E − ds . (3.9) νp 2(1−γ)+p p+1 0 s∧τn 0 2νs νs

2 q(x) = p(p+1)θ − pb(0) x > 0 Here let 2x2(1−γ)+p xp+1 , then we can easily evaluate the boundedness for any

2(1−γ)+p p(2γ − 1)θ2 θ2 2γ−1 q(x) ≤ D := (2(1 − γ) + p) . (3.10) 2 2b(0)

E[ ν−p ] ≤ x−p + Dt + pK R t E[ ν−p ] ds Summarizing the calculation, we have t∧τn 0 s∧τn , and from Gronwall’s −p −p lemma we ﬁnally have E[ ν ] ≤ (x +Dt) exp{pKt}. Taking the limit n → ∞, then lim τn = ∞ a.s. t∧τn n→∞ −p −p so we have E[ νt ] ≤ (x + Dt) exp{pKt}. Hence we can deduce the result. remark 1. Since the CEV model satisﬁes the assumptions of Lemma 3.2, so the result holds for the CEV model.

3.3 Transformation and Approximation

1−γ We consider the process transformed as σt := νt . By Ito’sˆ formula, we have

γ 2 − 1−γ γθ 1 dσt = (1 − γ) κµσt − − κσt dt + (1 − γ)θ dWt, (3.11) 2 σt

1−γ with σ0 = ν > 0. If σt is the solution of the stochastic differential equation(3.11), then we can prove 1 1−γ that σt = νt is also the solution of the stochastic differential equation(3.1) satisfying the initial condition 1 1−γ σ0 = ν0. By this transformation, we can replace (3.1) by (3.11) with the constant volatility term. In order γ 1 − 1−γ to use Theorem 2.5, we must approximate x and x by the Lipschitz continuous functions, respectively.

11 For all > 0, deﬁne the continuously differentiable functions Φ and Ψ as

 − γ x 1−γ ( for x ≥ ), Φ(x) = 1 γ (3.12) γ − 1−γ 1 − 1−γ − 1−γ x + 1−γ ( for x < ),   1 ( for x ≥ ), Ψ(x) = x (3.13) 1 2 − 2 x + ( for x < ).

1 0 γ − 1−γ 0 1 For the functions Φ and Ψ, we can easily verify that for all x ∈ R, |Φ (x)| ≤ 1−γ and |Ψ (x)| ≤ 2 1 γ − 1−γ 1 and then we have that for all x, y ∈ R, |Φ(x) − Φ(y)| ≤ 1−γ |x − y| and |Ψ(x) − Ψ(y)| ≤ 2 |x − y|. γ − 1−γ 1 Moreover, note that for all x ∈ R+, Φ(x) ≤ x and Ψ(x) ≤ x . Deﬁne our approximations σt as the stochastic process following the stochastic differential equation

γθ2 dσ = (1 − γ) κµΦ(σ) − Ψ(σ) − κσ dt + (1 − γ)θ dW , (3.14) t t 2 t t t

with σ0 = σ0 for all > 0. The coefﬁcients of the equation (3.14) are Lipschitz continuous because we can have for all x, y ∈ R,

2 2 γθ γθ κµΦ(x) − Ψ(x) − κx − κµΦ(y) − Ψ(y) − κy 2 2 γθ2 ≤κµ|Φ(x) − Φ(y)| + |Ψ(x) − Ψ(y)| + κ|x − y| 2 2 κµγ − 1 γθ 1 ≤ 1−γ + + κ |x − y|. (3.15) 1 − γ 2 2

2 We will prove that σt converges to σt in L (Ω). First we prove that σt converges to σt pointwise.

Lemma 3.3. The sequence σt converges to σt a.s., for all t ∈ [0,T ].

Proof. Deﬁne for all > 0 the stopping time as τ := inf{ 0 ≤ t ≤ T ; σt ≤ } with { ∅ } = ∞. By the deﬁnition of Φ, Ψ, and τ , we have

1 2 By Gronwall’s lemma, σt = σt for t < τ and by Lemma 3.1 and the fact that τ ≤ τ for 1 ≥ 2, we have lim τ = ∞ a.s. so lim σt = σt for all t ∈ [0,T ]. →0 →0 Next we prove that there exist square integrable processes ut and wt with ut ≤ σt ≤ wt for all t ∈ [0,T ].

12 Actually, we will see that wt is σt. Before starting with the proof, we prove the following inequality.

γ 1 − 1−γ b Lemma 3.4. For γ ∈ 2 , 1 and a, b > 0, let f(x) = ax − x . We have, for x ∈ R+,

− γ aγ 2γ−1 1 − γ f(x) ≥ − . (3.17) b(1 − γ) a(2γ − 1)

Proof. By differentiating f(x), we can easily have the result.

γθ2 Consider a = κµ and b = 2 in the above inequality, then we can have the below result.

Lemma 3.5. Let ut be the solution of the following stochastic differential equation

dut = (1 − γ)(C − κut) dt + θ(1 − γ) dWt, (3.18)

− γ 2κµ 2γ−1 1−γ with u0 = σ0, where C = − θ2(1−γ) κµ(2γ−1) . Then ut ≤ σt ≤ σt a.s. for all t ∈ [0,T ].

γθ2 Proof. From the deﬁnitions of Φ and Ψ, κµΦ(x)− 2 Ψ(x) ≥ C for all x ∈ R+, that is, the drift coefﬁcient of ut is smaller than one of σt . By Yamada-Watanabe’s comparison lemma (see [10, Proposition 5.2.18]) and Lemma 3.1, we have ut ≤ σt a.s.. We prove the second inequality. In order to use Yamada-Watanabe’s comparison lemma, we must prove that, 2 2 γθ2 − γ γθ2 − γ γθ γθ for x ∈ R , κµΦ(x) − Ψ(x) ≤ κµx 1−γ − . Let g(x) := κµx 1−γ − − κµΦ(x) + Ψ(x). + 2 2x 2x 2 γ 2 1 2 γ 2 − 1−γ γθ κµγ − 1−γ γθ κµ − 1−γ γθ We can easily verify g(x) = κµx − 2x + 1−γ − 22 x − 1−γ + , for x < and g(x) = 0 for x ≥ . For all x < , we have

2 2 κµγ − 1 γθ κµγ − 1 γθ g0(x) = − x 1−γ + + 1−γ − , (3.19) 1 − γ 2x2 1 − γ 22 − 2−γ κµγ 2γ−1 g00(x) = x 1−γ − γθ2x 1−γ . (3.20) (1 − γ)2

Then there is a constant η > 0 with g00(η) < 0 for all x < η and g00(η) > 0 for all x > η. For < η, g0(x) is 0 decreasing for all x < . Then g() = 0 and g () = 0 imply for all x < , g(x) > 0, that is, for x ∈ R+

2 2 γθ − γ γθ κµΦ(x) − Ψ(x) ≤ κµx 1−γ − . (3.21) 2 2x

By Yamada-Watanabe’s comparison lemma, we have σt ≤ σt a.s.

2 Theorem 3.1. For all t ∈ [0,T ], the sequence σt converges to σt in L (Ω).

2 Proof. From Lemma 3.5, we have |σt − σt| ≤ |ut − σt| ≤ |ut| + |σt|. Lemma 3.2 implies |σt| ∈ L (Ω). 2 Moreover, the Ornstein-Uhlenbeck process ut ∈ L (Ω). By the dominated convergence theorem we can have the convergence.

13 3.4 Malliavin Differentiability

We will prove the Malliavin differentiability of both σt and νt. To do this, we consider our approximation sequence σt . The approximating stochastic differential equation (3.14) of σt satisﬁes the assumption of Theorem 2.5, so we can prove the Malliavin differentiability of σt .

1,2 Lemma 3.6. σt belongs to D and we have

Z t 2 0 γθ 0 Drσt = (1 − γ)θ exp (1 − γ) κµΦ (σs) − Ψ (σs) − κ ds , (3.22) r 2

for r ≤ t, and Drσt = 0 for r > t.

Proof. By Theorem 2.5, we have the result.

We will now prove the Malliavin differentiability of σt. To start with, we prove some useful lemmas.

γ 1 − 1−γ b Lemma 3.7. For γ ∈ ( 2 , 1) and a, b > 0, let f(x) = ax − x , then for x ∈ R+ we have

− 1 aγ 2γ−1 aγ(2γ − 1) f 0(x) ≤ . (3.23) 2b(1 − γ)2 2(1 − γ)2

Proof. By differentiating f 0(x) we can easily have the result.

γθ2 By Lemma 3.7, considering the case where a = κµ and b = 2 , we have for x ≥

− 1 γθ2 κµ 2γ−1 κµγ(2γ − 1) κµΦ0(x) − Ψ0(x) ≤ := ξ. (3.24) 2 θ2(1 − γ)2 2(1 − γ)2

We have for x < ,

2 2 γθ − 1 γθ − 2γ−1 κµγ κµΦ0(x) − Ψ0(x) = 1−γ − 1−γ + , (3.25) 2 2 1 − γ

0 γθ2 0 so there exists a constant 0 > 0 such that for all < 0, κµΦ (x) − 2 Ψ (x) < 0. Hence, for < 0, we 0 γθ2 0 have κµΦ (x) − 2 Ψ (x) ≤ ξ, for all x ∈ R+. Note that ξ is independent of . By this inequality, we have the following result.

Lemma 3.8. We have for all t ∈ [0,T ] and < 0,

|Drσt | ≤ (1 − γ)θ exp{(1 − γ)(ξ − κ)(t − r)}. (3.26)

Proof. When r > t, Drσt = 0 so the result follows. Moreover when r ≤ t, putting above results together, we obtain the result.

Putting the scenarios together, we can prove the following.

14 1,2 Theorem 3.2. σt belongs to D and we have

Z t 1 2 κµγ − 1−γ γθ Drσt = (1 − γ)θ exp (1 − γ) − σs + 2 − κ ds , (3.27) r 1 − γ 2σs for r ≤ t, and Drσt = 0 for r > t.

2 1,2 Proof. We have proved that σt → σt in L (Ω) and σt ∈ D . Moreover, by Lemma 3.8, we have 2 sup E kDσt k < ∞. Here σt converges to σt also pointwise, we can conclude that Drσt converges Z t κµγ − 1 γθ2 : 1−γ to G = (1 − γ)θ exp (1 − γ) − σs + 2 − κ ds . Using the bounded convergence r 1 − γ 2σs 2 theorem, we can have that Drσt converges to G in L (Ω; H). Hence by Lemma 2.4, we can conclude that 1,2 σt ∈ D and Drσt = G.

Moreover we can prove the following Malliavin differentiability in more detail.

1,p 1,∞ Theorem 3.3. For all p ≥ 1, σt belongs to D , that is, σt belongs to D .

p Proof. We only have to prove that kσtk1,p < ∞. We have

" p # Z T 2 p p 2 kσtk1,p =E[ σt ] + E (Drσt) dr 0 (1−γ)p =E[ νt ]  p  2 2 Z T Z t κµγ − 1 γθ2 + E (1 − γ)θ exp (1 − γ) − σ 1−γ + − κ ds dr  s 2  0 r 1 − γ 2σs " p # Z T 2 (1−γ)p 2 2 ≤E[ νt ] + E (1 − γ) θ exp{2(1 − γ)(ξ − κ)(t − r)}dr 0 p (1 − γ)θ2(1 − exp{−2(1 − γ)(ξ − κ)T }) 2 =E[ ν(1−γ)p] + exp{p(1 − γ)(C − κ)t}. (3.28) t 2(ξ − κ)

p Hence we can conclude that kσtk1,p < ∞.

By the chain rule, we can conclude that νt is also Malliavin differentiable.

1,∞ Theorem 3.4. For all p ≥ 1, νt belongs to D and the Malliavin derivative is given by

Z t κµγ γθ2 D ν = θνγ exp (1 − γ) − + − κ ds , r t t 2(1−γ) (3.29) r (1 − γ)νs 2νs for r ≤ t, and Drνt = 0 for r > t.

15 Proof. Consider only the case where r ≤ t. Similarly, we can easily prove the case where r > t. We have 1 1−γ 1,∞ shown that νt = σt and σt ∈ D . By Lemma 2.5, we have

1 Z t κµγ γθ2 D ν = D σ 1−γ = θνγ exp (1 − γ) − + − κ ds . r t r t t 2(1−γ) (3.30) r (1 − γ)νs 2νs

p 1,∞ For all p ≥ 1, using Young’s inequality and the fact νt ∈ L (Ω) and σt ∈ D , we can prove that νt belongs to D1,∞. Indeed, we have

" p # " p # Z T 2 p pγ Z T 2 p p 2 p 1 1−γ 2 kνtk1,p =E[ νt ] + E (Drνt) dr = E[ νt ] + E σt (Drσt) dr 0 1 − γ 0

p " 2pγ Z T p# p 1 1 1−γ 1 2 ≤E[ νt ] + E σt + (Drσt) dr 1 − γ 2 2 0 p " Z T p# p 1 1 2pγ 1 2 =E[ νt ] + E νt + (Drσt) dr 1 − γ 2 2 0 <∞. (3.31)

4 CEV-Type Heston Model and Greeks

We will now consider the CEV-type Heston model and Greeks. Fournie´ et al. introduced new numerical methods for calculating Greeks using Malliavin calculus for the first time in 1999 (see [1]). We call this methods Malliavin Monte-Carlo methods. They focused on models with Lipschitz continuous coefficients, and then a lot of researchers have considered Malliavin Monte-Carlo methods to compute Greeks. However, lately, there is need to focus on models with non-Lipschitz coefficients such as stochastic volatility models. In 2008, Alos and Ewald proved that the Cox-Ingersoll-Ross model was Malliavin differentiable (see [7]). We apply Malliavin calculus for calculating Greeks of the CEV-type Heston model which is one of the important in business but mathematically complex models. Basically, we consider the European option but we can easily extend this result to other options.

4.1 Greeks

We introduce the concept of Greeks. For example, consider a European option with payoff function φ depending on the final value of the underlying asset ST where St denotes a stochastic process expressing −rT the asset and T denotes the maturity of the option. The price V is given by V = E[ e φ(ST )] where r is the risk-free rate. We can estimate this by Monte-Carlo simulations. Greeks are derivatives of the option price V with respect to the parameters of the model. Greeks are the useful measure for the portfolio risk management by traders in financial institutions. Most of financial institutions estimate Greeks by finite difference methods. However, there are some demerits. For examples, the numerical results depend on the

16 approximation parameters and, in the case where φ is not differentiable, this methods do not work well. In [1], Founie´ et al. gave the new methods to circumvent these problems. The idea is that we calculate Greeks by multiplying the weight, so-called Malliavin weight, as following

∂V (x) = E[ e−rT φ(S )·(weight)]. (4.1) ∂α T

This methods are much useful since we do not require the differentiability of the payoff function φ. Instead, there is need to assume that the underlying assert St is Malliavin differentiable. From Theorem 2.2, we ﬁnd that the solution of the stochastic differential equation with Lipschitz continuous coefﬁcients are Malliavin differentiable. However, if a model under consideration becomes more complex just like the CEV-type Heston model, we could not apply this Malliavin methods. Through Section 4, we consider the Malliavin differentiability of the CEV-type Heston model in order to give formulas for Greeks, in particular, Delta and Rho. Here, Delta ∆ and Rho % respectively measure the sensitivity of the option price with respect to the initial price and the risk-free rate. In particular, ∆ is one of the most important Greeks which also describes the replicating portfolio.

4.2 CEV-Type Heston Model

In [5], Heston supposed that the stock price St follows the stochastic differential equation √ dSt = St(r dt + νt dBt), (4.2) √ where Bt, r and νt respectively mean a Brownian motion, the risk-free rate and the volatility. Moreover

Heston assumed that the volatility process νt becomes a mean-reverting stochastic process of the form √ dνt = κ(µ − νt) dt + θ νt dWt, (4.3) where Wt, µ, κ and θ respetively mean a Brownian motion, the long-run mean, the rate of mean reversion and the volatility of volatility. This model is called the Cox-Ingersoll-Ross model. Here Bt and Wt are two correlated Brownian motions with

dBt dWt = ρ dt, ρ ∈ (−1, 1), (4.4) where ρ is the correlation coefficient between two Brownian motions. Moreover we assume that the dynamics following stochastic differential equations (4.1), (4.2), and (4.3) are satisfied under the risk neutral measure. However even the Heston model can not grasp the fluctuation of the volatility accurately. In [6],

17 Andersen and Piterbarg extended the Heston model to the model of which dynamics follow √ dSt =St(r dt + νt dBt), (4.5) 1 dν =κ(µ − ν ) dt + θν γ dW , γ ∈ , 1 , (4.6) t t t t 2

dBt dWt =ρ dt, ρ ∈ (−1, 1), (4.7) with the initial conditions S0 = x and ν0 = ν. We call this model the CEV-type Heston model. For the equation (4.5) with γ = 1, the Malliavin differentiability obviously follows by Theorem 2.2. In the case 1 γ = 2 , Alos and Ewald proved the Malliavin differentiability in [7]. In Section 3, we have proved the 1 1 Malliavin differentiability in the case γ ∈ ( 2 , 1). Fron now on, we concentrate on γ ∈ 2 , 1 . In order to give the formulas for the CEV-type Heston model, we will now prove the Malliavin differentiability of the model. Before considering the Malliavin differentiability, we now prove that there is a following Brownian motion Wˆ t which will become useful later.

p 2 Lemma 4.1. There exists a Brownian motion Wˆ t independent of Wt with Bt = ρWt + 1 − ρ Wˆ t.

1 ρ Proof. From the deﬁnition of Wˆ t, we have Wˆ t = √ Bt − √ Wt. At ﬁrst we prove that Wˆ t is 1−ρ2 1−ρ2 independent of Wt. Since we easily have E[ WtWˆ t] = 0, so Wˆ t is independent of Wt. Using Leby’sˆ theorem, we conclude Wˆ t is a Brownian motion. We can easily verify that Wˆ t is also martingale. Consider ˆ ˆ the quadratic variation hW it of Wt. Then we have * + 1 ρ hWˆ i = B − W = t. (4.8) t p1 − ρ2 p1 − ρ2 t

Hence by the Levy’sˆ theorem, Wˆ t is a Brownian motion.

Instead of the dynamics (4.5), (4.6) and (4.7), replacing Bt by Wˆ t, then we can consider the following

√ p 2 dSt =St(r dt + νt(ρ dWt + 1 − ρ dWˆ t)), (4.9) γ dνt =κ(µ − νt) dt + θνt dWt, (4.10) where Wt and Wˆ t are independent. Note that we assume that St and νt follow the dynamics (4.7) and (4.8) under the risk neutral measure.

4.3 Arbitrage

Under the real measure, the CEV-type Heston model follows the following dynamics

√ p 2 dSt =St(u dt + νt(ρ dWt + 1 − ρ dWˆ t)), (4.11) γ dνt =κ(µ − νt) dt + θνt dWt, (4.12)

18 where Wt and Wˆ t are independent. Here u denotes the expected return of St. In business, u is assumed to equal to the risk free rate. In order to do this, we will change the real measure P to the measure Q called the risk-neutral measure. We consider the arbitrage but this problem is complicated, since the volatility is not tractable. However, we obtain the following theorem.

Theorem 4.1. The CEV-type Heston model following (4.9) and (4.10) is free of arbitrage and there is a risk-neutral measure Q

√ p 2 dSt =St(r dt + νt(ρ dWt + 1 − ρ dWˆ t)), (4.13) γ dνt =κ(µ − νt) dt + θνt dWt. (4.14)

√ p 2 1 √ 2 Proof. We consider the interval [0,T ]. First we solve the equation u − r = νt 1 − ρ zt + νtρzt . In order to solve this, we put z2 = 0. From Lemma 3.1, ν is positive a.s. so we have z1 = √ u−r . Here t t t 2√ 1−ρ νt 1 1 zt is obviously progressively measurable. Moreover, we can easily see that zt is locally bounded and in Z t 2 1 1 ˆ L (Ω). Let (M)t := exp Mt − 2 hMit where Mt = − zs dWs. It is well-known that if we can prove 0 that (M)t is a martingale, then the market is free of arbitrage and under the risk neutral measure Q with p Q(A) = E [ (M)T · 1A ],A ∈ FT . Note that Wˆ t is replaced by W¯ t which is a Brownian motion under 1 Q. Here we must prove that for all t ≥ 0, E[(M)t] = 1. Fix t ≥ 0 and let τn = inf{s ≥ 0; |zt | ≥ n} with Z t 1 τn 1 2 inf{∅} = ∞. Here zτn∧· is bounded, so we have hM it = (zs∧τn ) ds is bounded. From Novikov’s 0 τn criteria, we have that (M )t is a uniformly integrable martingale for any t ≥ 0. Moreover, from the 1 continuity of zt and Lemma 3.1, τn increases to inﬁnity. Since (M)t is positive a.s., (M)t · 1{t≤τn} converges to (M)t as n → ∞, and then by using the monotone convergence theorem

E[(M)t] = lim E[(M)t · 1{t≤τ }]. (4.15) n→∞ n

n τn∧t n dQ Here we have (M)t · 1{t≤τn} = (M )T · 1{t≤τn}, so letting Q be the measure satisfying dP = t∧τn (M )T , and then we have

t∧τn n E[(M)t] = lim E[(M)t] · 1{t≤τ } = lim E[(M )T ] · 1{t≤τ } = lim Q (t ≤ τn). (4.16) n→∞ n n→∞ n n→∞

n n We must prove lim Q (t ≤ τn) = 1. First we prove Q (t ≤ τn) = P (t ≤ τn). From Girsanov’s theorem, n→∞ Z t ˆ 1 n the processes Wt + zs · 1[0,τn∧t](s) ds and Wt are Ft-Brownian motions under the measure Q . Note 0 n that Wt is an Ft-adapted Brownian motion under Q for all n. We have known that under the measure P , νt follows the equation

γ dνt =κ(µ − νt) dt + θνt dWt. (4.17)

n Integrals under P and Q are the same, so νt also satisﬁes the above stochastic differential equation under

19 n n Q . From Lemma 3.1, the solution νt is unique. Hence the distribution of νt under the measure Q must be the same as the distribution of νt under the measure P , and then we can conclude that the distribution τn is n n the same under P and Q , that is, Q (t ≤ τn) = P (t ≤ τn). Since τn tends to ∞ a.s., lim P (t ≤ τn) = 1. n→∞ Hence we can conclude E[(M)t] = 1 and (M)t is a martingale. Then the market is free of arbitrage.

This theorem implies that the dynamics for the volatility process is preserved, and the drift term of the underlying asset is changed from u to r. In the sequel, we will consider the CEV-type Heston model under the risk-neutral measure denoted by P not by Q.

4.4 Malliavin Differentiability of the CEV-Type Heston Model (Logarithmic Price)

From now on, we denote by D and Dˆ two Malliavin derivatives with respect to Wt and Wˆ t, respectively. We now consider the logarithmic price Xt := log St. First, we will prove that Xt is Malliavin differentiable. By Ito’sˆ formula, we have ν √ dX = r − t dt + ν dB , (4.18) t 2 t t √ with X0 = log x. Here νt is neither differentiable at νt = 0 in 0 nor Lipschitz continuous. Hence we will now approximate this stochastic differential equation by one with Lipschitz continuous coefﬁcients and prove the Malliavin differentiability of Xt. Let

 x  e − e + (x < ),  1 φ (x) = x ( ≤ x < ), (4.19)  1  −x+ 1 1 −e + + 1 ( ≤ x).

Here we can easily verify that φ(x) is bounded and continuously differentiable. Moreover we can 1 1 verify that both (φ(x)) 1−γ and (φ(x)) 2(1−γ) are Lipschitz continuous. In Section 3, we have used the stochastic process σ with Lipschitz continuous coefﬁcients, in stead of νt. We will now prove the Malliavin differentiability of the two stochastic processes σ and the following approximation process X of X with

Lipschitz coefﬁcients. Naturally, instead of Xt, we consider the following stochastic differential equation

1 1 1 dX = r − (φ(σ)) 1−γ dt + (φ(σ)) 2(1−γ) dB , (4.20) t 2 t t t

with X0 = log x.

2 Lemma 4.2. We have Xt → Xt in L (Ω).

Proof. From the inquality (a + b)2 ≤ 2(a2 + b2), we have

t 2 t 1 2 1 Z 1 1 Z 1 2 1−γ 1−γ 2(1−γ) 2(1−γ) |Xt − Xt| ≤ σs − (φ (σs)) ds + 2 σs − (φ (σs)) dBs . (4.21) 2 0 0

20 We have using Cauchy-Schwarz’s inequality and Ito’sˆ isometry,

2 E |Xt − Xt|

2 " t 1 2 # " t 1 2 # t Z 1 Z 1 1−γ 1−γ 2(1−γ) 2(1−γ) ≤ E σs − (φ (σs)) ds + 2E σs − (φ (σs)) ds . (4.22) 2 0 0

2 2 2 For the second term, since both σt and σt are positive a.s. and for a, b > 0, (a − b) ≤ |a − b |, we have

2 " t 1 2 # t 1 t Z 1 Z 1 2 1−γ 1−γ 1−γ 1−γ E |Xt − Xt| ≤ E σs − (φ (σs)) ds + 2E σs − (φ (σs)) ds . (4.23) 2 0 0

By the scenarios in Subsection 3.3 and Subsection 3.4, we have that for almost all ω ∈ Ω there exists a 1 positive constant 0(ω) such that for all < 0(ω), < σt (ω) = σt(ω) < . For such , let

1 τ = inf t ≥ 0; σ = or σ = , (4.24) t t

1 1 1−γ with inf{∅} = ∞, then we have |σ − σt| = 0 for t ≤ τ. Hence we can have (φ (σ )) 1−γ − σ = t t t 1 1 1−γ 0 a.s., for t ≤ τ. And then we can have τ → ∞ as → 0, (φ(σ)) 1−γ − σ = 0 a.s. for all t ≥ 0. t t 1 1 1 1−γ p Since φ (x) ≤ x and σ ≤ σt, (φ (σ )) 1−γ − σ ≤ 2 |σt| 1−γ . Here σt is L -integrable for all p ≥ 1 so t t t 1 1 1−γ we can conclude that for all p ≥ 1, (φ(σ)) 1−γ − σ = 0 in Lp(Ω). We have from Fubini’s theorem, t t 2 |Xt − Xt| → 0 in L (Ω).

The following theorem implies that Xt is Malliavin differentiable.

1,2 Theorem 4.2. Xt belongs to D and the Malliavin derivatives are given by

Z t γ 1 Z t 2γ−1 1 1−γ 2(1−γ) 1 2(1−γ) DuXt = − σt Dsσs ds + ρσu + σs Duσs dBs, (4.25) 2(1 − γ) u 2(1 − γ) u 1 p 2 DˆuXt = 1 − ρ σu 2(1−γ) , (4.26) for u ≤ t, and DuXt = DˆuXt = 0 for u > t.

Proof. Since the coefﬁcients of stochastic differential equations for σt and Xt are Lipschitz continuous, we 1,∞ can use Theorem 2.2. At ﬁrst, we can conclude that σt ∈ D and the derivatives are given by

Z t 2 0 γθ 0 Duσt =(1 − γ)θ exp (1 − γ) (κµΦ (σs) − Ψ (σs) − κ) ds , (4.27) u 2 ˆ Duσt =0, (4.28)

21 ˆ for u ≤ t and Duσt = Duσt = 0 for u > t. 1,∞ Moreover we can also conclude that Xt ∈ D and the derivatives are given by the following

t t Z 1 1 1 Z 1 1−γ 2(1−γ) 2(1−γ) DuXt = Du r − (φ (σs)) ds + ρ(φ (σu)) + Du(φ (σs)) dBs u 2 u t 1 Z γ 1 1−γ 2(1−γ) = − (φ (σs)) Duσs ds + ρ(φ (σu)) 2(1 − γ) u 1 Z t 2γ−1 2(1−γ) + (φ (σs)) DuσsdBs, (4.29) 2(1 − γ) u 1 ˆ p 2 2(1−γ) DuXt = 1 − ρ (φ (σu)) , (4.30)

ˆ for u ≤ t, and DuXt = DuXt = 0 for u > t. ˆ We only consider the case u ≤ t. First we consider the Malliavin derivative DuXt . By Lemma 4.2 and 1 1 1 2(1−γ) 2(1−γ) 2 2 2(1−γ) the proof, we have (φ (σt )) → σt in L (Ω; H) and Xt →Xt in L (Ω). Moreover, (φ (x)) 1 p 2 is bounded, so we can use Lemma 2.4. Hence we can conclude DˆuXt = 1 − ρ σu 2(1−γ) . We consider the Malliavin derivative DuXt. For the ﬁrst term, we need prove

Z t γ Z t γ 1−γ 1−γ 2 (φ (σs)) Duσs ds → σt Duσs ds in L (Ω; H). (4.31) u u

Here we have that

Z t γ Z t γ 1−γ 1−γ (φ (σs)) Duσs ds − σt Duσt ds u u Z t γ γ 1−γ 1−γ = (φ (σs)) Duσs − σt Duσs ds u Z t γ γ 1 1 1−γ 1−γ 1−γ 1−γ = (φ (σs)) Duσs − σt Duσs + σt Duσs − σs Duσt ds u Z t γ γ Z t γ γ 1−γ 1−γ 1−γ 1−γ ≤ (φ (σs)) Duσs − σt Duσs ds + σt Duσs − σt Duσs ds u u Z t γ γ γ 1−γ 1−γ 1−γ ≤ (φ (σs)) − σt ds sup |Duσs| + (t − u) sup |Duσs − Duσs| sup σs . (4.32) u s∈[u,t] s∈[u,t] s∈[u,t]

This converges to 0 in L2(Ω) by the proof of Lemma 4.2, Lemma 3.8, Theorem 3.2, and Lemma 3.1. γ 1 R t 1−γ R t 1−γ 2 Hence we can conclude u(φ (σs)) Duσs ds → u σt Duσs ds in L (Ω; H). For the second term, γ 1 ˆ 2(1−γ) 2(1−γ) 2 as well as the case for DuXt, we can prove that ρ(φ (σu)) → ρσt in L (Ω; H). For the third 2γ−1 2γ−1 R t 2(1−γ) R t 2(1−γ) 2 term, we will prove u(φ (σs)) Duσs dBs → u σs Duσs dBs in L (Ω; H). We have from Ito’sˆ

22 isometry,

2 Z t 2γ−1 Z t 2γ−1 2(1−γ) 2(1−γ) (φ (σs)) Duσs dBs − σs Duσs dBs u u

Z t 2γ−1 2γ−1 2γ−1 2γ−1 2 2(1−γ) 2(1−γ) 2(1−γ) 2(1−γ) = (φ (σs)) Duσs − σs Duσs + σs Duσs − σs Duσs ds u

Z t 2γ−1 2γ−1 2 Z t 2γ−1 2γ−1 2 2(1−γ) 2(1−γ) 2(1−γ) 2(1−γ) ≤2 (φ (σs)) Duσs − σs Duσs ds + 2 σs Duσs − σs Duσs ds u u Z t 2γ−1 2γ−1 Z t 2γ−1 1−γ 1−γ 2 2 1−γ ≤2 (φ (σs)) − σs ds sup |Duσs| + 2 |Duσs − Duσs| ds sup σs . (4.33) u s∈[u,t] u s∈[u,t]

This converges to 0 in L2(Ω) as well as the ﬁrst term, so we can conclude that

Z t 2γ−1 Z t 2γ−1 2(1−γ) 2(1−γ) 2 (φ (σs)) Duσs dBs → σs Duσs dBs in L (Ω; H). (4.34) u u

By Lemma 2.4, we have

Z t 1 1 Z t 2γ−1 1 1−γ 2(1−γ) 1 2(1−γ) DuXt = − σt Duσs ds + ρσt + σs Duσs dBs. (4.35) 2(1 − γ) u 2(1 − γ) u

remark 2. For σt, as well as Theorem 4.1, we can more easily prove

Z t 1 2 κµγ − 1−γ γθ Duσt =(1 − γ)θ exp (1 − γ) − σs + 2 − κ ds , (4.36) u 1 − γ 2σs

Dˆuσt =0, (4.37) for u ≤ t, and Duσt = Dˆuσt = 0 for u > t.

4.5 Malliavin Differentiability of the CEV-Type Heston Model (Actual Price)

From now on, we will concentrate on the underlying asset St and the volatility νt.

In Subsection 4.4, we proved the Malliavin differentiability of the logarithmic price Xt and the transformed volatility σt. Here we can prove that both of the underlying asset St and the volatility νt are Malliavin differentiabile by the chain rule.

23 1,2 Theorem 4.3. St and νt belong to D and we have

Z t Z t 1 1 √ 1 − 2 Dt0 St =St − Dt0 νs ds + ρ νt0 + νs Dt0 νs dBs , (4.38) 2 t0 2 t0 √ ˆ p 2 Dt0 St =St 1 − ρ νt0 , (4.39) Z t 2 γ κµγ −1 γθ −2(1−γ) Dt0 νt =θνt exp (1 − γ) − νs + νs − κ ds , (4.40) t0 1 − γ 2 ˆ Dt0 νt =0, (4.41)

0 ˆ ˆ 0 for t ≤ t, and Dt0 St = Dt0 St = Dt0 νt = Dt0 νt = 0 for t > t.

Proof. First we consider the Malliavin derivative for νt. By Lemma 2.5, we have

1 γ 1−γ 1 1−γ 1 γ D 0 ν =D 0 σ = σ D 0 σ = ν D 0 σ , (4.42) t t t t 1 − γ t t t 1 − γ t t t 1 γ Dˆ 0 ν = ν Dˆ 0 σ . (4.43) t t 1 − γ t t t

We have by Theorem 4.2

Z t 1 2 1 γ κµγ − 1−γ γθ 0 Dt νt = νt (1 − γ)θ exp (1 − γ) − σs + 2 − κ ds 1 − γ t0 1 − γ 2σs Z t 2 γ κµγ −1 γθ −2(1−γ) =θνt exp (1 − γ) − νs + νs − κ ds , (4.44) t0 1 − γ 2 ˆ Dt0 νt =0, (4.45)

0 ˆ 0 for t ≤ t, and Dt0 νt = Dt0 νt = 0 for t > t. Next, we consider the Malliavin derivative for St. By Lemma 2.5, we have

Xt Xt Dt0 St =Dt0 e = e Dt0 Xt = StDt0 Xt, (4.46) ˆ ˆ Dt0 St =StDt0 Xt. (4.47)

Hence by Theorem 4.2, we have

Z t γ 1 Z t 2γ−1 1 1−γ 2(1−γ) 1 2(1−γ) Dt0 St =St − σt Dt0 σs ds + ρσt0 + σs Dt0 σs dBs 2(1 − γ) t0 2(1 − γ) t0 Z t Z t 1 1 √ 1 − 2 =St − Dt0 νs ds + ρ νt0 + νs Dt0 νs dBs , (4.48) 2 t0 2 t0 1 √ ˆ p 2 2(1−γ) p 2 Dt0 St =St 1 − ρ σt0 = St 1 − ρ νt0 , (4.49)

0 ˆ 0 for t ≤ t and Dt0 St = Dt0 St = 0 for t > t.

24 4.6 Delta and Rho

Using Theorem 2.4 and Theorem 4.4, we can calculate Greeks of St. We now consider the following stochastic differential equations

√ p 2√ dSt =rSt dt + ρ νtSt dWt + 1 − ρ νtStdWˆ t, (4.50) γ dνt =κ(µ − νt) dt + θνt dWt. (4.51)

Rewrite the stochastic differential equations (4.15) and (4.16) as the integral form, and then we have

! ! t ! t p 2√ √ ! ! St x Z rSs Z 1 − γ νsSs ρ νsSs dWˆ s = + ds + γ . νt ν0 0 κ(µ − νs) 0 0 θνs dWs (4.52)

We now give the formula for Delta of this model.

Theorem 4.4. Consider the CEV-type Heston model following the dynamics (4.15) and (4.16). We have for any funtion with polynomial growth φ : R → R

" Z T # −rT 1 ∆S = E e φ(ST ) dWˆ t . (4.53) p 2√ 0 xT 1 − ρ νt

p 2√ √ ! 1 − γ νsSs ρ νsSs Proof. Let Υ be the diffusion matrix Υ = γ , then we can have the in- 0 θνs   √ 1 −√ ρ 2√ 2 γ verse Υ−1 = 1−γ νsSs 1−γ νs . We can have from the Ito’sˆ formula  1  0 γ θνs

Z t Z t Z t νs √ p 2 √ St = x exp r − ds + ρ νs dWs + 1 − ρ νsdWˆ s . (4.54) 0 2 0 0

! ! St ! St ∂ St x Hence we can directly calculate the ﬁrst variation process Zt of as Zt := ∂x = . νt νt 0 Then we can have

 1  √ √ 0 1−ρ2 ν S 1 −1 T T −1 T St t t √ 0 (Υ Zt) =Z (Υ ) = 0 = 2√ . (4.55) t x  ρ 1  x 1−ρ νt −√ γ 2 γ θν 1−ρ νt t

By Lemma 3.2, we have √ 1 ∈ L4(Ω × [0,T ]). As with Theorem 4.3, let dW˙ be the column with 2√ t x 1−ρ νt

25 ! dWˆ t the form . Since St and νt are Malliavin differentiable we have from Theorem 2.4 dWt

Z T −rT 1 −1 T ∆S =E[ e φ(ST ) (Υ Zt) dW˙ t] T 0 " Z T ˆ !# −rT 1 1 dWt =E e φ(S ) √ √ 0 T x 1−ρ2 ν T 0 t dWt " Z T # −rT 1 =E e φ(ST ) dWˆ t . (4.56) p 2√ 0 xT 1 − ρ νt

Moreover we can calculate a Greek, Rho %.

Theorem 4.5. Consider the CEV-type Heston model following the dynamics (4.15) and (4.16). Then for any φ : R → R of polynomial growth, we have

" Z T !# −rT 1 % = E e φ(ST ) dWˆ t − T . (4.57) p 2√ 0 1 − ρ νt

Proof. By the deﬁnition of %, we have

∂ ∂S % = E[ e−rT φ(S )] = E (−T )e−rT φ(S ) + e−rT φ0(S ) T . (4.58) ∂r T T T ∂r

! ! S TS ∂ST ∂ST ∂ T T and ∂r as ∂r = ∂r = = xT ZT . Here we have νT 0

Z T Z T Z T ∂ ∂ νs √ p 2 √ ST = x exp r − ds + ρ νs dWs + 1 − ρ νsdWˆ s ∂r ∂r 0 2 0 0

=T · ST . (4.59)

By the above formula, we have

−rT −rT 0 % =E (−T )e φ(ST ) + e φ (ST )xT ZT " Z T !# −rT 1 =E e φ(ST ) dWˆ t − T . (4.60) p 2√ 0 1 − ρ νt

26 5 Conclusion

From Section 3 and 4, it is proved by using unique transformation and approximation that we can apply Malliavin calculus to the CEV model and the CEV-type Heston model both which have non-Lipschitz coefficients in their processes. Then we can provide the formulas to calculate important Greeks as Delta and Rho of these models and contribute to finance, in particular for traders in financial institutions to measure market risks and hedge their portfolios in terms of Delta Hedge. In future it will be required how to calculate the Vega, one of the most important Greeks, for general stochastic volatility models including the CEV-type Heston model. Vega is the sensitivity for volatility but it is difficult to measure Vega for the stochastic volatility models since the volatility is also stochastic process. After the financial crisis, the necessity to grasp the behavior of volatility is increasing. We believe that we can calculate the vega of some important stochastic volatility models such as the Heston model or the CEV-type Heston model by using our results in Section 3 and 4.

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