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Investors find it difficult to determine the movement of prices of stock due to volatility. Empirical evidence has shown that volatility is stochastic which contradicts the Black-Scholes framework of assuming it to be constant. In this paper, stochastic volatility is estimated theoretically in a model-free way without assuming its functional form. We show proof of an identity establishing an exact expression for the volatility in terms of the price process. This theoretical presentation for estimating stochastic volatility with the presence of a compensated Poisson jump is achieved by using Fourier Transform with Bohr’s convolution and quadratic variation. Our method establishes the addition of a compensated Poisson jump to a stochastic differential equation using Fourier Transforms around a small time window from the observation of a single market evolution.

Volatility measures uncertainty of returns which plays a major role in cash flows from selling assets at a precise future date. It is very essential in financial markets due to price fluctuations, prediction of stock prices, option pricing, portfolio management and hedging. Decision and policy makers depend on volatility to determine the bullish and bearish nature of the market to avoid loss. The varying nature of volatility makes it difficult to predict stock prices. The Black-Scholes framework assumes constant volatility but empirical evidence has proved otherwise leading researchers to explore more into modeling volatility of an asset. It is important to note that the original Black-Scholes framework did not include jumps into price processes. It is then of interest to explore to what extend the inclusion of jumps affects the dynamics of stock price volatility. Volatility can be estimated through parametric and nonparametric methods. When using the parametric methods, it is modeled using its functional form of observed variables in the market. These include discrete-time volatility models such as Autoregressive Conditional Heteroscedasticity(ARCH) models, where volatility relies on the past returns and other variables that are directly observed only [

Malliavin and Mancino [

All these and many researches have been done in volatility estimation using Fourier analysis without the consideration of the addition of a compensated Poisson jump. In this paper, we are motivated to propose a nonparametric estimation method based on Fourier analysis incorporated with Bohr’s convolu- tion and quadratic variation, applying it to continuous semi-martingale process with the addition of a compensated Poisson jump as an extension of the results of Malliavin and Mancino [

The volatility is reconstructed as a function of time. The volatility matrix Σ j , k ( t ) on the time window [ 0, t ] is computed by changing the origin of time, rescaling it to reduce the time window to [ 0,2 π ] and using Fourier transform with Bohr’s convolution and quadratic variation to estimate it. We prove a general identity relating the Fourier transform of the price process p ( t ) with a compensated Poisson jump to that of the instantaneous multivariate volatility Σ j , k ( t ) under the hypothesis that the volatility process is square integrable. We then derive an instantaneous volatility estimator from the identity based on a discrete, unevenly spaced asset prices. This is important when the derivation of stochastic volatility is performed along the time evolution in terms of contingent claim pricing-hedging [

We present some important mathematical preliminaries that are significant to this paper. These preliminaries will outline some definitions and theorems that will be used in this paper.

Definition 1. A stochastic process N is a counting process if there exists an increasing family of sequence of random variables

0 < T 1 ≤ T 2 ≤ ⋯ ∈ ℝ + ∪ { ∞ } : T j < T j + 1

where T j is finite and

N t = ∑ j = 1 ∞ I [ T j , ∞ ) ( t )

and

I [ T j , ∞ ) ( t ) = { 1 if t ≥ T j 0 if 0 ≤ t < T j [

Definition 2. Suppose ℝ is a metric space of a real line with metric θ ( x , y ) then a continuous real-valued function x ( t ) on ℝ is called an almost periodic function if, for every ε > 0 , there exists m = m ( ε ) > 0 such that every interval [ t 0 , t 0 + m ( ε ) ] contains at least one number τ for which θ [ x ( t ) , x ( t + τ ) ] < ε for ( − ∞ < t < ∞ ) . [

Definition 3. Let Φ , Ψ be functions on ℤ , where ℤ is an integer, then their Bohr convolution is:

( Φ ∗ B Ψ ) ( k ) : = lim n → ∞ 1 2 n + 1 ∑ s = − n n Φ ( s ) Ψ ( k − s ) .

Definition 4. The map I m : x → p from P ( R n ) to P ( M ) is an Itô map where P ( M ) is the set of continuous maps from [ 0,1 ] to M and P ( R n ) is the set of continuous maps x , from [ 0,1 ] to R n such that x ( 0 ) = 0 and stochastic moving frame r ( τ ) = ( p ( τ ) , e ( τ ) ) such that p ( 0 ) = m ∈ M and e ( 0 ) is the identity [

Definition 5. Let M be an n-dimensional Riemannian Manifold, T ( M ) be a tangent space and T * ( M ) be the dual space of T ( M ) then a vector field Z along p is a map from [ 0,1 ] to T ( M ) such that for τ ∈ [ 0,1 ] , Z p ( τ ) ∈ T p ( τ ) ( M ) . If Z is an adapted vector field with respect to ω being an adapted differential one form along p , then e ( τ ) − 1 Z p ( τ ) with respect to e ( τ ) ω p ( τ ) is an adapted vector on ℝ n [

Definition 6. Suppose T m * ( M ) is the dual space of T * ( M ) = ∪ m ∈ M T * ( M )

then a differential form ω of degree one along p is a map from [ 0,1 ] to T * ( M ) such that for any τ ∈ [ 0,1 ] , ω p ( τ ) ∈ T p ( τ ) * ( M ) . If ω is an adapted differential one form along p , then e ( τ ) ω p ( τ ) is in a linear form [

Theorem 1. (Ito energy identity). Let ω be an adapted differential form and Z be an adapted vector field along p ; then it implies that we have

E [ ∫ 0 1 ω ( s ) Z ( s ) d s ] = E [ ∫ 0 1 ω ( s ) d p ( s ) ⋅ ∫ 0 1 Z ( s ) d p ( s ) ] (1)

Proof: see [

Theorem 2. (Ito Formula for a complex variable case). If μ ( V i , V j ) = V i V j , where V i , V j are martingales and we apply complex Itô formula to it, then we get,

d ( V i V j ) = V i d V j + V j d V i + d V i d V j (2)

Proof: see [

Theorem 3. (Burkholder-Davis-Gundy inequality) For every 1 ≤ p < ∞ there exist positive constants k p , K p such that, for all local martingales X with X 0 = 0 and stopping times τ , the following inequality holds.

k p E [ [ X ] τ p / 2 ] ≤ E [ ( X τ * ) p ] ≤ K p E [ [ X ] τ p / 2 ] .

If p = 2 , then we have a special case, then it implies that this inequality holds in this case with constants k 2 = 1 , K 2 = 4 .

E [ [ X ] τ ] ≤ E [ ( X τ * ) 2 ] ≤ 4 E [ [ X ] τ ] .

See proof [

Remark 1. The compensated Poisson process M t = N t − λ t , t ∈ ℝ + is a martingale with respect to its own filtration F t [

Definition 7. Let V t be a real-valued stochastic process defined on the probability space ( Ω , F , ℙ ) and with time t that ranges over non-negative real numbers then the pth variation is defined as,

V t = lim ‖ Π ‖ → 0 ∑ k = 1 n | V t k − V t k − 1 | p

where Π ∈ [ 0, t ] and ‖ Π ‖ is the norm of the partition 0 = t 0 < t 1 < t 2 < ⋯ < t n = t such that we have ‖ Π ‖ = max { ( t i − t i − 1 ) , ∀ i = 1 , ⋯ , n } if the above sum converges [

This is true under certain conditions for example, p = 1 defines the first variation or total variation process, for p = 2 , the pth variation equals the quadratic variation if the sum converges. Also, it is a bounded variation if and only if for p = 1 , V t < + ∞ .

For a generalize Itô processes,

X t = X 0 + ∫ 0 t σ s d B s + ∫ 0 t μ s d s ,

where B is a standard Brownian motion, its quadratic variation is given by

[ X ] t = ∫ 0 t σ s 2 d s . [

The quadratic variation of a compensated Poisson process M t = N t − λ t is

[ M ] t = ∑ s ≤ t Δ M s 2 = N t [

Fourier Transform is a nonparametric method which is the representation of frequency domain, and the mathematical operation that links the frequency domain representation to a function of time. A time varying data can be transformed from one domain into a different domain (frequency domain) and that is the main idea behind Fourier [

Fourier transform constructs the volatility as a function for it’s iteration and computation of the cross-correlation between price and volatility. Fourier Transform takes into consideration all the observations and avoids inconsistency in data since it is based on integration. Let p ( t ) be the log-price of assets which is a continuous semi-martingale on a fixed time window, then

d p ( t ) = α ( t , B ) d t + σ ( t , B ) d B ( t ) + d M ( t ) ,

where M ( t ) = N ( t ) − λ t and N ( t ) is a Poisson process with intensity λ , α is the drift, σ is the volatility, time is t and the standard Brownian motion B . B ( t ) and M ( t ) are independent.

We solve for the price process with a compensated Poisson jump as

p ( t ) = p ( 0 ) + ∫ 0 t α ( s , B ) d s + ∫ 0 t σ ( s , B ) d B ( s ) + ∫ 0 t d M ( s ) (3)

where σ is adapted to a filtration but α is not necessarily adapted and it’s bounded by | α | + | σ | ≤ c , for c ∈ ℝ .

Now we let p ( t ) = p 1 ( t ) , ⋯ , p n ( t ) , satisfying

d p j ( t ) = ∑ i = 1 d σ i j ( t ) d B i ( t ) + α j ( t ) d t + d M j ( t ) , j = 1 , ⋯ , n , (4)

where B = B 1 , ⋯ , B d are independent Brownian motions on a probability space, such that σ i j and α j are random processes which are adapted to a filtration and satisfies the conditions in (QA) below;

E [ ∫ 0 T ( α i ( t ) ) 2 d t ] < ∞

(QA)

E [ ∫ 0 T ( σ i j ( t ) ) 4 d t ] < ∞ , i = 1 , ⋯ , d , j = 1 , ⋯ , n .

Definition 8. Suppose we have two assets whose prices are p j ( t ) , p k ( t ) , then its respective volatilities will be σ i j ( t ) , σ i k ( t ) (co-volatilities), hence the entries of its volatility matrix Σ j , k ( t ) , is

Σ j , k ( t ) = ∑ i = 1 d σ i j ( t ) σ i k ( t ) .

When p j ( t ) = p k ( t ) , the volatility matrix will be

Σ j , k ( t ) = ∑ i = 1 d ( σ i j ( t ) ) 2

known as the instantaneous volatilities.

Theorem 4. Suppose the function ϕ ( ν ) has Fourier transform:

F ( ϕ ) ( k ) : = 1 2π ∫ 0 2π ϕ ( ν ) e − i k ν d ν , k ∈ ℤ (5)

and its differential form:

F ( d ϕ ) ( k ) : = 1 2π ∫ 0 2π e − i k ν d ϕ ( ν ) (6)

then

F ( ϕ ) ( k ) = i k [ 1 2 π ( ϕ ( 2π ) − ϕ ( 0 ) ) − F d ϕ ( k ) ] . (7)

Proof. From Equation (5),

F ( ϕ ) ( k ) = 1 2 π [ − ϕ ( ν ) e − i k ν i k − ∫ − e − i k ν i k d ϕ ( ν ) ] 0 2π = 1 2 π i k [ − ϕ ( ν ) e − i k ν + ∫ e − i k ν d ϕ ( ν ) ] 0 2π = − 1 2 π i k [ ϕ ( ν ) e − i k ν − ∫ e − i k ν d ϕ ( ν ) ] 0 2π

From Equation (6)

F ( ϕ ) ( k ) = [ − 1 2 π i k ϕ ( ν ) e − i k ν ] 0 2π + 1 i k F d ϕ ( k ) = − 1 2 π i k [ ϕ ( 2π ) e − i k ( 2π ) − ϕ ( 0 ) ] + 1 i k F d ϕ ( k )

but e − i k ( 2π ) = cos ( 2 π k ) − i sin ( 2 π k ) , and cos ( 2 π k ) = 1 , sin ( 2 π k ) = 0 , then we have

F ( ϕ ) ( k ) = − 1 2 π i k [ ϕ ( 2π ) ( cos ( 2 π k ) + i sin ( 2 π k ) ) − ϕ ( 0 ) ] + 1 i k F d ϕ ( k ) = − 1 2 π i k [ ϕ ( 2π ) − ϕ ( 0 ) ] + 1 i k F d ϕ ( k )

that is

F ( ϕ ) ( k ) = i k [ 1 2 π ( ϕ ( 2π ) − ϕ ( 0 ) ) − F d ϕ ( k ) ]

The Identity Relation for a Complex Martingale CaseWe present the following propositions with their proofs below.

Proposition 5. The identity that relates the price process and volatility matrix with the compensated Poisson jump is

1 2π F ( Σ i j ) ( k ) + 1 2π F ( d N ) ( k ) = F ( d p i ) * B F ( d p j ) (8)

Proof. Here we establish an identity which relates the Fourier transform of the price process p ( t ) to the Fourier transform of the volatility matrix Σ i j ( t ) . The drift α does not contribute to the quadratic variation [

Suppose we have a price process which has a volatility matrix and a com- pensated Poisson jump,

d p j ( t ) = ∑ i = 1 d σ i j ( t ) σ i k ( t ) d B i ( t ) + α j ( t ) d t + d M j ( t ) , j = 1 , ⋯ , n , (9)

where σ ( t , B ( t ) ) does not depend on B ( t ) then from Equation (9), we have,

F ( d p j ) ( k ) = F ( ∑ i = 1 d σ i j σ i k d B i ) ( k ) + F ( α j d t ) ( k ) + F ( d M j ) ( k ) .

let d p σ j ( t ) = ∑ i = 1 d σ i j ( t ) σ i k ( t ) d B i ( t ) , then,

F ( d p j ) ( k ) = F ( d p σ j ) ( k ) + F ( α j d t ) ( k ) + F ( d M j ) ( k ) .

Again, let

F ( d p σ j ) ( k ) = ϕ m ( k ) , F ( α j d t ) ( k ) = ϕ α ( k ) and F ( d M j ) ( k ) = ϕ M ( k )

then it suffices from the theory of convolution that,

( ϕ σ + ϕ α + ϕ M ) * B ( ϕ σ + ϕ α + ϕ M ) = ( ϕ σ * B + ϕ α * B + ϕ M * B ) ( ϕ σ + ϕ α + ϕ M ) = ϕ σ * B ϕ σ + ϕ σ * B ϕ α + ϕ σ * B ϕ M + ϕ α * B ϕ σ + ϕ α * B ϕ α + ϕ α * B ϕ M + ϕ M * B ϕ σ + ϕ M * B ϕ α + ϕ M * B ϕ M

Since α = 0 then from Bohr’s convolution, the functions that convolve with α will be zero and also functions that convolve with M apart from itself will be zero, hence we have,

( ϕ σ + ϕ α + ϕ M ) ∗ B ( ϕ σ + ϕ α + ϕ M ) = ϕ σ ∗ B ϕ σ + ϕ M ∗ B ϕ M

From Definition 3, we have,

( ϕ α * B ϕ α ) ( k ) = lim n → ∞ 1 2 n + 1 ∑ s = − n n ϕ α ( s ) ϕ α ( k − s ) . (10)

From Equation (10), we define the Bohr’s convolution of the volatility and the compensated Poisson jump as:

( ϕ σ * B ϕ σ ) ( k ) = lim n → ∞ 1 2 n + 1 ∑ s = − n n ϕ σ ( s ) ϕ σ ( k − s ) .

( ϕ M * B ϕ M ) ( k ) = lim n → ∞ 1 2 n + 1 ∑ s = − n n ϕ M ( s ) ϕ M ( k − s ) .

which implies that

( ϕ σ + ϕ α + ϕ M ) * B ( ϕ σ + ϕ α + ϕ M ) = lim n → ∞ 1 2 n + 1 ∑ s = − n n ϕ σ ( s ) ϕ σ ( k − s ) + lim n → ∞ 1 2 n + 1 ∑ s = − n n ϕ M ( s ) ϕ M ( k − s )

But

F ( d p j ) * B F ( d p j ) = ( ϕ σ + ϕ α + ϕ M ) * B ( ϕ σ + ϕ α + ϕ M ) = ϕ σ * B ϕ σ + ϕ M * B ϕ M

We define

F ( d X ) * B F ( d X ) = 1 2π F ( [ d X , d X ] ) = 1 4 π 2 ∫ 0 t e − i k s d [ X , X ] ( s ) , [

which implies

F ( d p i ) * B F ( d p j ) = ( ϕ σ * B ∗ ϕ σ ) + ( ϕ M * B ∗ ϕ M ) = F ( d p σ j ) * B F ( d p σ j ) + F ( d M j ) * B F ( d M j ) = 1 2π F ( [ d p σ i , d p σ j ] ) + 1 2π F ( [ d M j , d M j ] ) = 1 4 π 2 ∫ 0 t e − i k s d [ p σ i , p σ j ] ( s ) + 1 4 π 2 ∫ 0 t e − i k s d [ M j , M j ] ( s )

where d [ M j , M j ] ( s ) = d N j ( s ) is:

F ( d p i ) * B F ( d p j ) = 1 4 π 2 ∫ 0 t e − i k s d [ p σ i , p σ j ] ( s ) + 1 4 π 2 ∫ 0 t e − i k s d N j ( s )

Let

d [ p σ i , p σ j ] ( s ) = Σ i j ( t ) ⇒ 1 4 π 2 ∫ 0 t e − i k s d [ p σ i , p σ j ] ( s ) = 1 2 π F ( Σ i j ) ( k )

and

1 4 π 2 ∫ 0 t e − i k s d N j ( s ) = 1 2π F ( d N j ) ( k )

Then we have,

1 2π F ( Σ i j ) ( k ) = F ( d p i ) * B F ( d p j ) − 1 2π F ( d N j ) ( k )

Hence the identity that relates the volatility matrix and the compensated Poisson jump is

1 2π F ( Σ i j ) ( k ) + 1 2π F ( d N j ) ( k ) = F ( d p i ) * B F ( d p j )

Proposition 6. The volatility matrix and compensated Poisson jump are related to the price process by the identity

( F ( d p i ) ∗ B F ( d p j ) ) ( q ) = 1 2π F ( Σ i j ) ( q ) + 1 2π F ( d N j ) ( q ) (11)

with the volatility independent of the stock’s Brownian motion.

Proof. In the case where σ ( t ) is independent of B t , we introduce complex martingales Γ k i ( t ) , Γ r j ( t ) for any integers r , k , where p is the price process and we have two assets, i , j = 1 , 2. The Fourier transform of the complex martingale is,

Γ k i ( t ) : = 1 2 π ∫ 0 t e − i k s d p i ( s )

Γ r j ( t ) : = 1 2 π ∫ 0 t e − i r s d p j ( s ) .

That is:

F ( d p i ) ( k ) = 1 2π ∫ 0 2π e − i k s d p i ( s ) ,

which implies Γ k i ( 2π ) = F ( d p i ) ( k ) .

Using Itô formula in Theorem 2 to solve the complex martingale we have,

d ( Γ k i Γ r j ) ( t ) = Γ k i ( t ) d Γ r j ( t ) + Γ r j ( t ) d Γ k i ( t ) + d Γ k i ( t ) d Γ r j ( t ) = Γ k i ( t ) d Γ r j ( t ) + Γ r j ( t ) d Γ k i ( t ) + ( d ( 1 2π ∫ 0 t e − i k s d p i ( s ) ) d ( 1 2π ∫ 0 t e − i r s d p j ( s ) ) )

From Equation (4), if α = 0 we have

But ∑ l = 0 d σ l i σ l j = Σ i j and ( d M j ( t ) ) 2 = d N t j , which implies,

d ( Γ k i Γ r j ) ( t ) = Γ k i ( t ) d Γ r j ( t ) + Γ r j ( t ) d Γ k i ( t ) + ( 1 2π ) 2 Σ i j e − i ( k + r ) t d t + ( 1 2π ) 2 ( e − i ( k + r ) t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) + ( 1 2π ) 2 ( e − i ( k + r ) t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) + ( 1 2π ) 2 ( e − i ( k + r ) t d N t j )

∫ 0 2 π d ( Γ k i Γ r j ) ( t ) = ∫ 0 2π ( Γ k i ( t ) d Γ r j ( t ) + Γ r j ( t ) d Γ k i ( t ) ) + ∫ 0 2π ( 1 2π ) 2 Σ i j e − i ( k + r ) t d t + ∫ 0 2π ( 1 2 π ) 2 ( e − i ( k + r ) t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i ( k + r ) t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i ( k + r ) t d N t j )

Since

∫ 0 2π d ( Γ k i Γ r j ) ( t ) = ( Γ k i Γ r j ) ( 2 π ) − ( Γ k i Γ r j ) ( 0 ) = Γ k i ( 2 π ) Γ r j ( 2 π ) − Γ k i ( 0 ) Γ r j ( 0 ) ,

and Γ k i ( 0 ) Γ r j ( 0 ) = 0 we have,

Γ k i ( 2 π ) Γ r j ( 2 π ) = ∫ 0 2π ( Γ k i ( t ) d Γ r j ( t ) + Γ r j ( t ) d Γ k i ( t ) ) + ∫ 0 2π ( 1 2π ) 2 Σ i j e − i ( k + r ) t d t + ∫ 0 2π ( 1 2 π ) 2 ( e − i ( k + r ) t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i ( k + r ) t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i ( k + r ) t d N t j )

Let H i j ( k , r ) = ∫ 0 2π ( Γ k i ( t ) d Γ r j ( t ) + Γ r j ( t ) d Γ k i ( t ) ) and from the definition of a Fourier transform,

we have 1 2π ∫ 0 2π Σ i j e − i ( k + r ) t d t = F ( Σ i j ) ( k + r ) , which follows that,

Γ k i ( 2 π ) Γ r j ( 2 π ) = 1 2π F ( Σ i j ) ( k + r ) + H i j ( k , r ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i ( k + r ) t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i ( k + r ) t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i ( k + r ) t d N t j )

If we have an integer n ≥ 1 , then for any integer q, where | q | ≤ n and from the Bohr’s convolution theory we have,

ϑ q i j ( n ) = 1 2 n + 1 ∑ s = − n n Γ q + s i ( 2 π ) Γ − s j ( 2 π ) (12)

Γ q + s i ( 2 π ) Γ − s j ( 2 π ) = 1 2π F ( Σ i j ) ( q + s − s ) + H i j ( q + s , − s ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i ( q + s − s ) t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i ( q + s − s ) t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i ( q + s − s ) t d N t j )

which simplifies to

Γ q + s i ( 2 π ) Γ − s j ( 2 π ) = 1 2π F ( Σ i j ) ( q ) + H i j ( q + s , − s ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d N t j )

From Equation (12) it implies that,

ϑ q i j ( n ) = 1 2 n + 1 ∑ s = − n n Γ q + s i ( 2 π ) Γ − s j ( 2 π ) = 1 2 n + 1 ∑ s = − n n [ 1 2π F ( Σ i j ) ( q ) + H i j ( q + s , − s ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) ] + 1 2 n + 1 ∑ s = − n n [ ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d N t j ) ] = 1 2 n + 1 [ 1 2π F ( Σ i j ) ( q ) ( 2 n + 1 ) + ∑ s = − n n H i j ( q + s , − s ) ] + 1 2 n + 1 [ ∑ s = − n n ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) ] + 1 2 n + 1 ∑ s = − n n [ ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d N t j ) ] = 1 2π F ( Σ i j ) ( q ) + 1 2 n + 1 ∑ s = − n n [ H i j ( q + s , − s ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) ] + 1 2 n + 1 ∑ s = − n n [ ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d N t j ) ]

Hence,

ϑ q i j ( n ) = 1 2π F ( Σ i j ) ( q ) + H n i j ( q + s , − s ) + Y n i j (13)

where

H n i j ( q + s , − s ) = 1 2 n + 1 ∑ s = − n n H i j ( q + s , − s )

but

H i j ( k , r ) = ∫ 0 2π Γ k i ( t ) d Γ r j ( t ) + Γ r j ( t ) d Γ k i ( t )

which implies

H n i j ( q + s , − s ) = 1 2 n + 1 ∑ s = − n n ∫ 0 2π Γ q + s i ( t ) d Γ − s j ( t ) + Γ − s j ( t ) d Γ q + s i ( t )

and

Y n i j ( q ) = 1 2 n + 1 ∑ s = − n n ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) + 1 2 n + 1 ∑ s = − n n [ ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d N t j ) ]

H n i j ( q + s , − s ) can also be reduced to Q n by symmetry as,

Q n = 1 2 n + 1 ∑ s = − n n ∫ 0 2π d Γ − s j ( t 2 ) ∫ 0 t 2 d Γ q + s i ( t 1 ) [

From Q n ,

∫ 0 2π d Γ − s j ( t 2 ) ∫ 0 t 2 d Γ q + s i ( t 1 ) = [ Γ − s j ( t 2 ) ] 0 2π [ Γ q + s i ( t 1 ) ] 0 t 2 = ( Γ − s j ( 2π ) − Γ − s j ( 0 ) ) ( Γ q + s i ( t 2 ) − Γ q + s i ( 0 ) )

Let Γ ( 0 ) = 0 (it refers to the Fourier transform of the initial price) then it implies that

∫ 0 2π d Γ − s j ( t 2 ) ∫ 0 t 2 d Γ q + s i ( t 1 ) = Γ − s j ( 2π ) Γ q + s i ( t 2 )

Now

∫ d ( Γ − s j ( 2π ) Γ q + s i ( t 2 ) ) = ∫ Γ − s j ( 2 π ) d Γ q + s i ( t 2 ) + ∫ Γ q + s i ( t 2 ) d Γ − s j ( 2 π )

Since ( t 1 , t 2 ) ∈ t and t ∈ ( 0,2 π ) , we have

∫ 0 2π d ( Γ − s j ( t ) Γ q + s i ( t ) ) = ∫ 0 2π Γ − s j ( t ) d Γ q + s i ( t ) + Γ q + s i ( t ) d Γ − s j ( t )

Hence H n i j can be reduced to Q n by symmetry as n → ∞ .

Let D n ( t ) be a Dirichlet kernel defined as

D n ( t ) = 1 2 n + 1 ∑ s = − n n e i s t = 1 2 n + 1 sin ( n + 1 ) t sin ( t / 2 ) , (14)

By the definition of Q n we have,

Q n = 1 2 n + 1 ∑ s = − n n ∫ 0 2π d Γ − s j ( t 2 ) ∫ 0 t 2 d Γ q + s i ( t 1 ) = 1 2 n + 1 ∑ s = − n n Γ − s j ( 2π ) Γ q + s i ( t 2 ) = 1 2 n + 1 ∑ s = − n n 1 2π ∫ 0 2π e i s t 2 d p j ( t 2 ) × 1 2π ∫ 0 t 2 e − i ( q + s ) t 1 d p i ( t 1 ) = 1 4π 2 ( 1 2 n + 1 ) ∑ s = − n n ∫ 0 2π e i s t 2 d p j ( t 2 ) ∫ 0 t 2 e − i ( q + s ) t 1 d p i ( t 1 ) = 1 4π 2 ( 1 2 n + 1 ) ∑ s = − n n ∫ 0 2π e i s t 2 d p j ( t 2 ) ∫ 0 t 2 e − i q t 1 e − i s t 1 d p i ( t 1 ) = 1 4π 2 ( 1 2 n + 1 ) ∑ s = − n n ∫ 0 2π ∫ 0 t 2 e i s t 2 × e − i q t 1 × e − i s t 1 d p i ( t 1 ) d p j ( t 2 )

Q n = 1 4π 2 ( 1 2 n + 1 ) ∑ s = − n n ∫ 0 2π ∫ 0 t 2 e i s ( t 2 − t 1 ) × e − i q t 1 d p i ( t 1 ) d p j ( t 2 ) = 1 4π 2 ( 1 2 n + 1 ) ∫ 0 2π ∫ 0 t 2 e − i q t 1 × ∑ s = − n n e i s ( t 2 − t 1 ) d p i ( t 1 ) d p j ( t 2 ) = 1 4π 2 ∫ 0 2π ∫ 0 t 2 e − i q t 1 × 1 2 n + 1 ∑ s = − n n e i s ( t 2 − t 1 ) d p i ( t 1 ) d p j ( t 2 )

But from Equation (14)

D n ( t 2 − t 1 ) = 1 2 n + 1 ∑ s = − n n e i s ( t 2 − t 1 )

which implies

Q n = 1 4π 2 ∫ 0 2π ∫ 0 t 2 e − i q t 1 D n ( t 2 − t 1 ) d p i ( t 1 ) d p j ( t 2 )

Hence,

Q n = 1 4π 2 ∫ 0 2π d p j ( t 2 ) ∫ 0 t 2 e − i q t 1 D n ( t 2 − t 1 ) d p i ( t 1 )

From Equation (4), when α = 0 and we have two assets then,

Q n = 1 4 π 2 ∫ 0 2π d p j ( t 2 ) ∫ 0 t 2 ( cos ( q t 1 ) − i sin ( q t 1 ) ) D n ( t 2 − t 1 ) ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) )

| Q n | 2 = ( 1 4 π 2 ∫ 0 2π d p j ( t 2 ) ∫ 0 t 2 ( cos ( q t 1 ) − i sin ( q t 1 ) ) D n ( t 2 − t 1 ) ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) ) ) 2

| Q n | 2 = ( 1 4 π 2 ) 2 ∫ 0 2 π ( ∑ k = 1 2 σ k j ( t 2 ) d B k ( t 2 ) + d M j ( t 2 ) ) 2 × [ ∫ 0 t 2 ( cos ( q t 1 ) − i sin ( q t 1 ) ) D n ( t 2 − t 1 ) ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) ) ] 2

Let

ς 2 ( t 2 ) = ∫ 0 t 2 ( cos ( q t 1 ) D n ( t 2 − t 1 ) ) 2 ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) ) 2 , d M t ⋅ d B t = 0 = ∫ 0 t 2 ( cos ( q t 1 ) D n ( t 2 − t 1 ) ) 2 ( ( ∑ k = 1 2 σ k i ( t 1 ) ) 2 d t 1 + d N i ( t 1 ) )

(15)

and

ϖ 2 ( t 2 ) = ∫ 0 t 2 ( sin ( q t 1 ) D n ( t 2 − t 1 ) ) 2 ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) ) 2 = ∫ 0 t 2 ( sin ( q t 1 ) D n ( t 2 − t 1 ) ) 2 ( ( ∑ k = 1 2 σ k i ( t 1 ) ) 2 d t 1 + d N i ( t 1 ) ) (16)

| Q n | 2 = ( 1 16 π 4 ) ∫ 0 2π ( ( ∑ k = 1 2 σ k j ( t 2 ) ) 2 d t 2 + d N j ( t 2 ) ) [ ς 2 ( t 2 ) + ϖ 2 ( t 2 ) ]

16 π 4 | Q n | 2 = ∑ k = 1 2 ∫ 0 2π [ ς 2 ( t 2 ) + ϖ 2 ( t 2 ) ] ( ( σ k j ( t 2 ) ) 2 d t 2 + d N j ( t 2 ) )

then using Itô energy identity Equation (1) we have,

16 π 4 E [ | Q n | 2 ] = ∑ k = 1 2 E [ ∫ 0 2π [ ς 2 ( t 2 ) + ϖ 2 ( t 2 ) ] ( ( σ k j ( t 2 ) ) 2 d t 2 + d N j ( t 2 ) ) ] . (17)

Expressing Equation (17) using Cauchy-Schwarz inequality we have

( 16 π 4 E [ | Q n | 2 ] ) 2 ≤ 4 ∑ k = 1 2 E [ ∫ 0 2π ( σ k j ( t 2 ) ) 4 d t 2 + ( d N j ( t 2 ) ) 2 ] × { E [ ∫ 0 2π ( ∫ 0 t 2 D n ( t 2 − t 1 ) cos ( q t 1 ) ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) ) ) 4 d t 2 ] + E [ ∫ 0 2π ( ∫ 0 t 2 D n ( t 2 − t 1 ) sin ( q t 1 ) ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) ) ) 4 d t 2 ] }

But E [ ∫ 0 2π ( σ k i ( t ) ) 4 d t ] < ∞ ⇒ E [ ∫ 0 2π ( σ k j ( t 2 ) ) 4 d t 2 + ( d N i ( t 2 ) ) 2 ] < ∞ then eva- luating the rest of the terms using Burkholder-Gundy’s inequality, we have

E [ ∫ 0 2π ( ∫ 0 t 2 D n ( t 2 − t 1 ) cos ( q t 1 ) ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) ) ) 4 d t 2 ] ≤ 4 E [ ∫ 0 2π ∫ 0 t 2 D n 4 ( t 2 − t 1 ) cos 4 ( q t 1 ) ( ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) ) ) 4 d t 2 ] = 4 E [ ∫ 0 2π ∫ 0 t 2 D n 4 ( t 2 − t 1 ) cos 4 ( q t 1 ) ( ∑ k = 1 2 ( σ k i ( t 1 ) ) 4 d t 1 + ( d N i ( t 1 ) ) 2 ) d t 2 ]

− 1 ≤ c o s ( q t 1 ) ≤ 1 and cos 4 ( q t 1 ) takes the interval 0 ≤ cos 4 ( q t 1 ) ≤ 1 and since the maximum of c o s is 1, we have

E [ ∫ 0 2π ( ∫ 0 t 2 D n ( t 2 − t 1 ) cos ( q t 1 ) ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) ) ) 4 d t 2 ] ≤ 4 E [ ∫ 0 2π ∫ 0 t 2 D n 4 ( t 2 − t 1 ) ( ∑ k = 1 2 ( σ k i ( t 1 ) ) 4 d t 1 + ( d N i ( t 1 ) ) 2 ) d t 2 ]

Similarly for

E [ ∫ 0 2π ( ∫ 0 t 2 D n ( t 2 − t 1 ) sin ( q t 1 ) ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) ) ) 4 d t 2 ] ≤ 4 E [ ∫ 0 2π ∫ 0 t 2 D n 4 ( t 2 − t 1 ) sin 4 ( q t 1 ) ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) ) 4 d t 2 ] = 4 E [ ∫ 0 2π ∫ 0 t 2 D n 4 ( t 2 − t 1 ) ( ∑ k = 1 2 ( σ k i ( t 1 ) ) 4 d t 1 + ( d N i ( t 1 ) ) 2 ) d t 2 ]

we have

E [ ∫ 0 2π ( ∫ 0 t 2 D n ( t 2 − t 1 ) cos ( q t 1 ) ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) ) ) 4 d t 2 ] + E [ ∫ 0 2π ( ∫ 0 t 2 D n ( t 2 − t 1 ) sin ( q t 1 ) ( ∑ k = 1 2 σ k i ( t 1 ) d B k ( t 1 ) + d M i ( t 1 ) ) ) 4 d t 2 ] ≤ 4 E [ ∫ 0 2π ∫ 0 t 2 D n 4 ( t 2 − t 1 ) ( ∑ k = 1 2 ( σ k i ( t 1 ) ) 4 d t 1 + ( d N i ( t 1 ) ) 2 ) d t 2 ] + 4 E [ ∫ 0 2π ∫ 0 t 2 D n 4 ( t 2 − t 1 ) ( ∑ k = 1 2 ( σ k i ( t 1 ) ) 4 d t 1 + ( d N i ( t 1 ) ) 2 ) d t 2 ]

Let t 1 = u , t 2 − t 1 = ν , then by change of variables we have, t 2 = u + ν , d u d t 1 = 1 , d u d t 2 = 1 , d ν d t 2 = 1

4 E [ ∫ 0 2π ∫ 0 t 2 D n 4 ( ν ) ( ∑ k = 1 2 ( σ k i ( u ) ) 4 d u + ( d N i ( u ) ) 2 ) d ( u + ν ) ] + 4 E [ ∫ 0 2π ∫ 0 t 2 D n 4 ( ν ) ( ∑ k = 1 2 ( σ k i ( u ) ) 4 d u + ( d N i ( u ) ) 2 ) d ( u + ν ) ] = 4 E [ ∫ 0 2π ( ∑ k = 1 2 ( σ k i ( u ) ) 4 ) d u + ( d N i ( u ) ) 2 ] ∫ 0 2π D n 4 ( ν ) d ν + 4 E [ ∫ 0 2π ( ∑ k = 1 2 ( σ k i ( u ) ) 4 ) d u + ( d N i ( u ) ) 2 ] ∫ 0 2π D n 4 ( ν ) d ν = 8 E [ ∫ 0 2π ( ∑ k = 1 2 ( σ k i ( u ) ) 4 ) d u + ( d N i ( u ) ) 2 ] ∫ 0 2π D n 4 ( ν ) d ν

We have | D n ( ν ) | ≤ 1 , and

∫ 0 2π D n 4 ( ν ) d ν ≤ ∫ 0 2π D n 2 ( ν ) d ν ≤ ∫ 0 2π | D n ( ν ) | d ν ,

∫ 0 2π D n 2 ( ν ) d ν = 2 π 2 n + 1

This implies that as n → ∞ , ∫ 0 2π D n 2 ( ν ) d ν = 0 . Q n 2 = 0 ⇒ Q n = 0 . If Q n = 0 , then H n i j = 0 .

Evaluating Y n i j ( q ) ;

Y n i j ( q ) = 1 2 n + 1 ∑ s = − n n ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) + 1 2 n + 1 ∑ s = − n n [ ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) + ∫ 0 2π ( 1 2 π ) 2 ( e − i q t d N t j ) ] = ( 1 4 π 2 ) ( 1 2 n + 1 ) [ ∫ 0 2π ( e − i q t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) ( 2 n + 1 ) ] + ( 1 4 π 2 ) ( 1 2 n + 1 ) [ ∫ 0 2π ( e − i q t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) ( 2 n + 1 ) + ∫ 0 2π e − i q t d N t j ( 2 n + 1 ) ] = ( 1 4 π 2 ) ∫ 0 2π [ ( e − i q t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) + ( e − i q t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) + e − i q t d N t j ]

but

1 4 π 2 ∫ 0 2π e − i q t d N ( t ) = 1 2π F ( d N ) ( q )

Hence,

Y n i j ( q ) = 1 4 π 2 ∫ 0 2π [ ( e − i q t d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) + ( e − i q t d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) ] + 1 2π F ( d N j ) ( q )

We express e − i q t as c o s ( q t ) + i s i n ( q t )

Y n i j ( q ) = 1 4 π 2 ∫ 0 2π [ ( ( c o s ( q t ) + i s i n ( q t ) ) d M j ( t ) ∑ l = 0 d σ l i ( t ) d B l ( t ) ) ] + 1 4 π 2 ∫ 0 2π [ ( ( c o s ( q t ) + i s i n ( q t ) ) d M j ( t ) ∑ l = 0 d σ l j ( t ) d B l ( t ) ) ] + 1 2π F ( d N j ) ( q )

When i = j , we have

Y n i j ( q ) = 1 4 π 2 ∫ 0 2π [ ( c o s ( q t ) + i s i n ( q t ) ) d M j ( t ) ( ∑ l = 0 d σ l j ( t ) d B l ( t ) + ∑ l = 0 d σ l j ( t ) d B l ( t ) ) ] + 1 2π F ( d N j ) ( q )

Let

η ( t ) = ( c o s ( q t ) + i s i n ( q t ) ) d M j ( t ) ( ∑ l = 0 d σ l j ( t ) d B l ( t ) + ∑ l = 0 d σ l j ( t ) d B l ( t ) ) .

We have,

η ( t ) = ( c o s ( q t ) + i s i n ( q t ) ) ( 2 ∑ l = 0 d σ l j ( t ) d B l ( t ) ) d M j ( t )

Taking the square of both sides

η 2 ( t ) = ( c o s ( q t ) + i s i n ( q t ) ) 2 ( 4 ∑ l = 0 d σ l j ( t ) d B l ( t ) d M j ( t ) ) 2

From De-moivre’s formula, ( cos ( x ) + i sin ( x ) ) n = cos ( n x ) + i sin ( n x ) it im- plies

η 2 ( t ) = ( cos ( 2 q t ) + i sin ( 2 q t ) ) ( 4 ∑ l = 0 d ( σ l j ( t ) ) 2 d t ( d N t j ) )

but d t ⋅ d N t j = 0 ⇒ η 2 ( t ) = 0 ⇒ η ( t ) = 0 . It follows that,

Y n i j = 1 2π F ( d N j ) ( q )

Now,

ϑ q i j ( N ) = 1 2π F ( Σ i j ) ( q ) + 1 2π F ( d N j ) ( q )

From Equation (12),

ϑ q i j ( N ) = 1 2 n + 1 ∑ s = − n n Γ q + s i ( 2π ) Γ − s j ( 2π ) = 1 2 n + 1 ∑ s = − n n F ( d p i ) ( − s ) F ( d p j ) ( q + s ) = ( F ( d p i ) ∗ B F ( d p j ) ) ( q )

This implies

( F ( d p i ) ∗ B F ( d p j ) ) ( q ) = 1 2π F ( Σ i j ) ( q ) + 1 2π F ( d N j ) ( q )

where F ( d N j ) ( q ) is defined as

F ( d N j ) ( q ) = 1 2π ∫ 0 t e − i k s d N j ( s ) .

Theorem 7. Suppose p is the price process satisfying Equation(3), then the instantaneous volatility function with a compensated Poisson jump is

V o l ( p ) = lim n → ∞ ∑ | q | < n ( 1 − | q | n ) ( 2π 2 n + 1 ∑ s = − n n F ( d p j ) ( s ) F ( d p j ) ( q − s ) + q i F ( N ) ( q ) − 1 2π ( N ( 2π ) − N ( 0 ) ) ) exp ( i q t )

where V o l ( p ) is the volatility of the price process p ( t ) at time t and q , n are integers.

Proof. From Equation (7)

F ( ϕ ) ( k ) = i k [ 1 2π ( ϕ ( 2π ) − ϕ ( 0 ) ) − F d ϕ ( k ) ]

F d ϕ ( k ) = − k i F ( ϕ ) ( k ) + 1 2π ( ϕ ( 2π ) − ϕ ( 0 ) )

then it implies that

F ( d N ) ( q ) = − q i F ( N ) ( q ) + 1 2π ( N ( 2π ) − N ( 0 ) ) . (18)

Then from Proposition 2 and 3, the identity that relates the Fourier transform of the price process with a compensated Poisson jump and the volatility is simplified as,

( F ( d p i ) ∗ B F ( d p j ) ) ( q ) = 1 2π F ( Σ i j ) ( q ) + 1 2π F ( d N j ) ( q )

2π ( F ( d p i ) ∗ B F ( d p j ) ) ( q ) = F ( Σ i j ) ( q ) + F ( d N j ) ( q )

2π ( F ( d p i ) ∗ B F ( d p j ) ) ( q ) = F ( Σ i j ) ( q ) − q i F ( N ) ( q ) + 1 2π ( N ( 2π ) − N ( 0 ) )

F ( Σ i j ) ( q ) = 2π ( F ( d p i ) ∗ B F ( d p j ) ) ( q ) + q i F ( N ) ( q ) − 1 2π ( N ( 2π ) − N ( 0 ) )

But,

( F ( d p i ) ∗ B F ( d p j ) ) ( q ) = lim n → ∞ 1 2 n + 1 ∑ s = − n n F ( d p j ) ( s ) F ( d p j ) ( q − s )

then it implies that

F ( Σ i j ) ( q ) = lim n → ∞ 2π 2 n + 1 ∑ s = − n n F ( d p j ) ( s ) F ( d p j ) ( q − s ) + q i F ( N ) ( q ) − 1 2π ( N ( 2π ) − N ( 0 ) )

The Fourier-Fejer summation function for Σ i j ( t ) which is continuous is given as;

Σ i j ( t ) = lim n → ∞ ∑ | q | < n ( 1 − | q | n ) ( 2π 2 n + 1 ∑ s = − n n F ( d p j ) ( s ) F ( d p j ) ( q − s ) + q i F ( N ) ( q ) − 1 2π ( N ( 2π ) − N ( 0 ) ) ) exp ( i q t )

From Definition 8, to obtain the instantaneous volatility function V o l ( p ) ≡ σ 2 ( t ) , we extract the diagonals of the matrix and sum them, that is when i = j , Σ j j ( t ) = ∑ i = 1 d ( σ i j ( t ) ) 2 = σ 2 ( t ) , which gives the Fourier estimator of the instantaneous volatility with a compensated Poisson jump as,

V o l ( p ) = lim n → ∞ ∑ | q | < n ( 1 − | q | n ) ( 2π 2 n + 1 ∑ s = − n n F ( d p j ) ( s ) F ( d p j ) ( q − s ) + q i F ( N ) ( q ) − 1 2π ( N ( 2π ) − N ( 0 ) ) ) exp ( i q t )

where F ( d p j ) ( s ) = − s i F ( p j ) ( s ) + 1 2π ( p j ( 2π ) − p j ( 0 ) ) .

Comparing the results obtained in Theorem 2.1 of Malliavin and Mancino’s [

1 2π F ( Σ i j ) ( q ) = ( F ( d p i ) ∗ B F ( d p j ) ) ( q ) .

In comparison with the one obtained in this paper, the identity that relates the price process and the volatility matrix with the addition of compensated Poisson jump is

1 2π F ( Σ i j ) ( q ) = ( F ( d p i ) ∗ B F ( d p j ) ) ( q ) − 1 2π F ( d N j ) ( q ) .

Also, the instantaneous multivariate volatility obtained by Malliavin and Man- cino [

Σ i j ( t ) = lim n → ∞ ∑ | q | < n ( 1 − | q | n ) ( 1 2 n + 1 ∑ s = − n n F ( d p j ) ( s ) F ( d p j ) ( q − s ) ) exp ( i q t )

and when compensated Poisson jump was added to it we obtained

Σ i j ( t ) = lim n → ∞ ∑ | q | < n ( 1 − | q | n ) ( 2π 2 n + 1 ∑ s = − n n F ( d p j ) ( s ) F ( d p j ) ( q − s ) + q i F ( N ) ( q ) − 1 2π ( N ( 2π ) − N ( 0 ) ) ) exp ( i q t ) .

This means that the addition of compensated Poisson jump had an effect on the volatility of the price process.

Suppose the price process follows the process,

d p j ( t ) = ∑ i = 1 d σ i j ( t ) σ i k ( t ) d B i ( t ) + α j ( t ) d t + β j ( t ) d M j ( t ) , j , k = 1 , ⋯ , n ,

then its Fourier transform is

F ( d p j ) ( q ) = F ( ∑ i = 1 d σ i j σ i k d B i ) ( q ) + F ( α j d t ) ( q ) + F ( β j d M j ) ( q ) , j = 1 , ⋯ , n

but d p σ j ( t ) = ∑ i = 1 d σ i j σ i k d B i ( t ) , then,

F ( d p j ) ( q ) = F ( d p σ j ) ( q ) + F ( α j d t ) ( q ) + F ( β j d M j ) ( q ) .

Let F ( β j d M j ) ( q ) = ϕ M 1 ( q ) , then from the theory of convolution,

( ϕ σ + ϕ α + ϕ M 1 ) * B ( ϕ σ + ϕ α + ϕ M 1 ) = ( ϕ σ * B + ϕ α * B + ϕ M 1 * B ) ( ϕ σ + ϕ α + ϕ M 1 ) = ϕ σ * B ϕ σ + ϕ M * B ϕ M 1

From Equation (10), the Bohr’s convolution of the jump diffusion process is:

( ϕ M 1 * B ϕ M 1 ) ( q ) = lim n → ∞ 1 2 n + 1 ∑ s = − n n ϕ M 1 ( s ) ϕ M 1 ( q − s ) .

Then,

F ( d p i ) * B F ( d p j ) = ( ϕ σ * B ∗ ϕ σ ) + ( ϕ M 1 * B ∗ ϕ M 1 ) = F ( d p σ i ) * B F ( d p σ j ) + F ( β j d M j ) * B F ( β j d M j ) = 1 2π F ( [ d p σ i , d p σ j ] ) + 1 2π F ( [ β j d M j , β j d M j ] ) = 1 4 π 2 ∫ 0 t e − i q s d [ p σ i , p σ j ] ( s ) + 1 4 π 2 ∫ 0 t e − i q s [ β j d M j , β j d M j ] ( s )

[ β j d M j , β j d M j ] ( s ) = β 2 j ( s ) d N j ( s ) , then,

F ( d p i ) * B F ( d p j ) = 1 4 π 2 ∫ 0 t e − i q s d [ p σ i , p σ j ] ( s ) + 1 4 π 2 ∫ 0 t e − i q s β 2 j d N j ( s )

Then we have,

1 2π F ( Σ i j ) ( q ) + 1 2 π F ( β 2 j d N j ) ( q ) = F ( d p i ) * B F ( d p j )

which gives the identity relating the price process to the volatility matrix and the jump diffusion process. It implies that,

( F ( d p i ) * B F ( d p j ) ) ( q ) = 1 2π F ( Σ i j ) ( q ) + 1 2 π F ( β 2 j d N j ) ( q ) .

Then,

F ( Σ i j ) ( q ) + F ( β 2 j d N j ) ( q ) = lim n → ∞ 2π 2 n + 1 ∑ s = − n n F ( d p j ) ( s ) F ( d p j ) ( q − s ) .

Its Fourier-Fejer summation function is given as;

Σ i j ( t ) = lim n → ∞ ∑ | q | < n ( 1 − | q | n ) ( 2π 2 n + 1 ∑ s = − n n F ( d p j ) ( s ) F ( d p j ) ( q − s ) − F ( β 2 j d N j ) ( q ) ) e x p ( i q t )

Its instantaneous volatility function V o l ( p ) is given as

V o l ( p ) = lim n → ∞ ∑ | q | < n ( 1 − | q | n ) ( 2π 2 n + 1 ∑ s = − n n F ( d p j ) ( s ) F ( d p j ) ( q − s ) − F ( β 2 j d N j ) ( q ) ) e x p ( i q t )

For one asset we have,

V o l ( p ) = lim n → ∞ ∑ | q | < n ( 1 − | q | n ) ( 2π 2 n + 1 ∑ s = − n n F ( d p ) ( s ) F ( d p ) ( q − s ) − F ( β 2 d N ) ( q ) ) e x p ( i q t ) (19)

where

F ( d p ) ( s ) = − s i F ( p ) ( s ) + 1 2π ( p ( 2π ) − p ( 0 ) ) .

Numerical ExampleWhen β = 0.1 ,

V o l ( p ) = lim n → ∞ ∑ | q | < n ( 1 − | q | n ) ( 2π 2 n + 1 ∑ s = − n n F ( d p ) ( s ) F ( d p ) ( q − s ) − F ( 0.1 2 d N ) ( q ) ) e x p ( i q t )

but

F ( d N ) ( q ) = − q i F ( N ) ( q ) + 1 2π ( N ( t ) − N ( 0 ) )

and N has a jump height of 1 and 0 otherwise and N ( 0 ) = 0 , then we have,

F ( d N ) ( q ) = − q i F ( N ) ( q ) + 1 2π ( N ( t ) ) = − q 2 π i ∫ 0 t N ( s ) e − i q s d s + 1 2π ( N ( t ) )

From Definition 1, without making any assumptions in the jump, if N ( t ) = 0 then F ( d N ) ( q ) = 0 but if N ( t ) = 1 , then

F ( d N ) ( q ) = − q 2 π i ( − 1 i q e − i q s ) 0 t + 1 2π = 1 π − 1 2 π e − i q t

V o l ( p ) = lim n → ∞ ∑ | q | < n ( 1 − | q | n ) ( 2π 2 n + 1 ∑ s = − n n F ( d p ) ( s ) F ( d p ) ( q − s ) − 0.01 π + 0.01 2 π e − i q t ) e x p ( i q t )

where

F ( d p ) ( s ) = − s i F ( p ) ( s ) + 1 2π ( p ( 2π ) − p ( 0 ) ) .

This example shows that the price process is directly proportional to the volatility meaning that with all other parameters constant, an increase in volatility, ( V o l ( p ) ) will increase the price process and vice versa.

Comparing the result obtained in Theorem 7 to Equation (19), the parameter β ( t ) added to the jump component had an effect on the volatility. An increase or decrease in the parameter β ( t ) will affect the volatility in direct pro- portionality.

We have shown the theoretical basis for estimating stochastic volatility with the presence of a compensated Poisson jump. This has been achieved by use of Fourier transforms incorporating Bohr’s convolution. We further established an identity relation for the Fourier transform of the price process with a compen- sated Poisson jump and the volatility, and also estimated the instantaneous volatility. We have also shown the identity relation for a specific case when β ( t ) was added to the jump and concluded that it was directly proportional to the volatility with all other parameters constant. This estimate can be used for both univariate and multivariate volatility settings. As a future work, it will be interesting to know how other types of jump diffusion processes together with microstructure noise effect the dynamics of volatility in the financial markets.

This work was carried out with financial support from the government of Canada’s International Development Research Centre (IDRC), and within the framework of AIMS Research for Africa Project. Also our gratitude to Dr. Edward Prempeh, Department of Mathematics(KNUST), University of Cape Town, African Institute for Mathematical Science, Ghana and the reviewers.

Andam, P.S., Ackora-Prah, J. and Mataramvura, S. (2017) Theories on the Relationship between Price Process and Stochastic Volatility Matrix with Compensated Poisson Jump Using Fourier Transforms. Journal of Mathema- tical Finance, 7, 633-656. https://doi.org/10.4236/jmf.2017.73033