arXiv:1908.03899v1 [q-fin.MF] 11 Aug 2019 applies. vlae rc n egn taeyfrvltlt deriva volatility for strategy hedging and price evaluated Benth by gro used was is model derivatives volatility volatility the to t devoted rate, exchange literature the The and u asset the the from between different correlation currency the a of in paying those as such ments, asset underlying financial different two financial using recent speculation among are swaps correlation and Covariance Introduction 1 91G80 91G10, Classification: AMS Eig Maximum Trace, Swaps, Variance, Generalized Keywords: ∗ etakIdai egpafrhlflcmet n suggest and comments helpful for Sengupta Indranil thank We bandi hspprhv motn mlctosfrtec the risk. for hedging implications for important have paper this exampl in numerical obtained with approach our demonstrate and involvpurpose c assets We volatilities. the Markov-modulated of with matrix markets covariance financial the of ge trace of measures and new value important two on swaps proposes paper This rpslfrMliastGnrlsdVrac Swaps Variance Generalised Multi-asset for Proposal A apigadOfiilSaitc Unit Statistics Official and Sampling ninSaitclIsiue Kolkata Institute, Statistical Indian ninSaitclIsiue Kolkata Institute, Statistical Indian tal. et [email protected] 20)t td oaiiyadvrac wp.BodeadJ and Broadie swaps. variance and volatility study to (2007) [email protected] iat Mukherjee Diganta uut1,2019 13, August uhjtBiswas Subhojit Abstract oswihhshle mrv h xoiinconsiderably. exposition the improve helped has which ions 1 ie nteHso qaero tcatcvltlt model volatility stochastic root square Heston the in tives stochastic Ornstein-Uhlenbeck Non-Gaussian The wing. .Freape pindpneto xhnert move- rate exchange on dependent option example, For s. ∗ i ikmyb lmntdb sn oaineswap. covariance a using by eliminated be may risk his nie utpeast nteprflofrtheoretical for portfolio the in assets multiple onsider moiysco hr uhsaswudb useful be would swaps such where sector ommodity d epieteegnrlzdvrac wp for swaps variance generalized these price We ed. rdcswihaeueu o oaiiyhdigand hedging volatility for useful are which products dryn urny aea xouet movements to exposure an have currency, nderlying stkn he tcsi h otoi.Teresults The portfolio. the in stocks three taking es eaie aine aeytemxmmeigen- maximum the namely variance, neralized envalue h sa caveat usual The i (2008a) ain and in Broadie and Jain (2008b) they compare result from various model in order to investigate the effect of jumps and discrete sampling on variance and volatility swaps. Pure jump process with independent increments return models were used by Carr et al. (2005) to price derivatives written on realized variance, and subsequent develop- ment by Carr et al. (2005). This paper also provides a good survey on volatility derivatives. Fonseca et al. (2009) analyzed the influence of variance and covariance swap in a market by solving a portfolio optimization problem in a market with risky assets and volatility derivatives. Correlation swap price has been investigated by Bossu (2005) and Bossu (2007) for component of an equity index using statistical method. By definition, all the above methods can only consider a combination of two assets at a time. But in today’s complex financial transactions, there is no reason why volatility of three or more assets will not be considered for contracting together. Thus, in this paper, we extend these methods to a situation where some generalized variance of a portfolio of assets can be contracted on. Taking cue from multivariate analysis, we look at two important measures of generalized variance, namely the maximum eigenvalue and trace of the covariance matrix of the assets involved. The objective is to price generalized variance swaps for financial markets with Markov-modulated volatilities. As an example, we consider stochastic volatility driven by a finite state continuous time Markov chain. To the best of our knowledge, this is the first attempt in extending the covariance swaps to a multidimensional situation. We outline the problem and the theoretical results is section 2. First we look at case of the trace swap and in a subsequent subsection we discuss the eigenvalue swap with a target return constraint. The numerical examples are presented with real data in section 3. Finally section 4 concludes.

2 Problem formulation

Let us consider a financial market with two types of securities, the risk free bond and the stock. Suppose that the stock prices (St )t R+ satisfy the following stochastic differential equation ∈

dSt = St (µdt + σ(xt )dwt ) where w is a standard Wiener process independent of the Markov process (xt )t . A portfolio consists of n stocks with the corresponding returns given by dS1 , dS2 ,..... dSn . The vector of individual S1 S2 Sn returns has variances and co-variances involved with it. Let the portfolio return covariance matrix be given by

Cov(r1,r1) Cov(r1,r2) Cov(r1,r3) ..... Cov(r1,rn) Cov(r2,r1) Cov(r2,r2) Cov(r2,r3) ..... Cov(r2,rn) Ω = ......     Cov(rn,r1) Cov(rn,r2) Cov(rn,r3) ..... Cov(rn,rn)     2 σ1 (xt ) ρ(12)σ1(xt )σ2(xt ) ρ(13)σ1(xt )σ3(xt ) ..... ρ(1n)σ1(xt )σn(xt ) 2 ρ σ2(xt )σ1(xt ) σ (xt ) ρ σ2(xt )σ3(xt ) ..... ρ σ2(xt )σn(xt ) Ω =  (21) 2 (23) (2n)  ......    2  ρ(n1)σn(xt )σ1(xt ) ρ(n2)σn(xt )σ2(xt ) ρ(n3)σn(xt )σ3(xt ) ..... σ (xt )   n   

Let (ΩS, F , (Ft )t R+ , P) be a filtered probability space, with a right-continuous filtration (Ft )t R+ and proba- ∈ ∈ bility P. The following two results allow us to associate (xt )t R+ , which is a Markov Process with generator Q, to ∈ a martingale and to obtain its quadratic variation Salvi and Swishchuk (2012). We refer to Elliott and Swishchuk

2 (2007) for the proofs.

Proposition 1. (Elliott and Swishchuk,2007) Let (xt )t R+ be a Markovprocesswith generatorQ and f Domain(Q), ∈ ∈ then t f mt = f (xt ) f (x0) Q f (xs)ds − − 0 Z is a zero mean martingale with respect to the Ft : σ y(s);0 s t . { ≤ ≤ }

Proposition 2. (Elliott and Swishchuk, 2007) Let (xt )t R+ be a Markov process with generator Q, f Domain(Q) f ∈ ∈ and (mt )t R+ its associated martingale, then ∈ t f 2 < m >t := [Q f (xs) 2 f (xs)Q f (xs)]ds 0 − Z is the quadratic variation of m f .

Proposition 3. (Salvi and Swishchuk, 2012) Let (xt )t R+ be a Markov process with generatorQ, f ,g Domain(Q) f g∈ ∈ such that fg Domain(Q). Denote by (mt )t R+ ; (mt )t R+ their associated martingale.Then then ∈ ∈ ∈ t < f (x.),g(x.) >t := [Q f (xs)g(xs) f (xs)Qg(Xs) g(xs)Q f (xs)]ds 0 − − Z is the quadratic variation of f and g.

In our model the volatility is stochastic. Then it is interesting to study the property of σ and in particular how to price derivative contracts on realized variance. We consider σ as a martingale as we are assuming that, regardless of a stock’s current and past volatility, his expected volatility at any time in the future is the same as his current volatility.

Proposition 4. (Salvi and Swishchuk, 2012) Suppose that σ Domain(Q). Then, ∈ t 2 2 2 E σ (xt ) Fu = σ (xu)+ E σ (xs) Fu ds { | } 0 { | } Z for all 0 u t. The value of conditional expectation is given by ≤ ≤

2 (t u)Q 2 E σ (xt ) Fu = e − σ (xu) { | }

If we remove the conditional part then, 2 tQ 2 E σ (xt ) = e σ (x) { } where we have denoted x0 := x. For the covariance terms we can write similar value for the expectation if we remove the conditional part. Then,

tQ E σ1(xt )σ2(xt ) = e σ1(x)σ2(x). { }

These results help us to derive the probability distribution of the eigenvalue that we discuss subsequently. We first look at the derivation of the trace swap.

3 2.1 Swap using the trace of the covariance matrix

As the first proposal, we consider the investor using the trace of the covariance matrix to develop the swap. The trace is given by 2 2 2 2 tr Ω(xt )= σ1 (xt )+ σ2 (xt )+ σ3 (xt )+ ....σn (xt ).

Now the price of the swap on trace is the expected present value of the payoff in the risk neutral world for the assets we have considered

rT Ptrace(x)= E e− (tr Ω(xt ) Kstrike price) { − } rT Ptrace(x)= e− E (tr Ω(xt ) Kstrike price) { − } Example: Let us consider 3 stocks in the portfolio. Then the covariance matrix becomes

2 σ1 (xt ) ρ(12)σ1(xt )σ2(xt ) ρ(13)σ1(xt )σ3(xt ) 2 Ω = ρ(21)σ2(xt )σ1(xt ) σ2 (xt ) ρ(23)σ2(xt )σ3(xt ) ρ σ (x )σ (x ) ρ σ (x )σ (x ) σ 2(x )  (31) 3 t 1 t (32) 3 t 2 t 3 t    2 2 2 tr Ω(xt )= σ1 (xt )+ σ2 (xt )+ σ3 (xt )

rT Ptrace(x)= E e− (tr Ω(xt ) Kstrike price) { − } rT Ptrace(x)= e− E (tr Ω(xt ) Kstrike price) { − } For our Markov modulated market, this becomes

T T T rT 1 tQ 2 rT 1 tQ 2 rT 1 tQ 2 rT Ptrace(x)= e− (e σ1 (x))dt + e− (e σ2 (x))dt + e− (e σ3 (x))dt e− Kstrike price {T 0 } {T 0 } {T 0 } − Z Z Z


2.2 Swap using the largest eigenvalue

The objective here is to define and derive the price of an eigenvalue swap. But we do not address the problem without an efficiency consideration as combinations of underlying assets for unconstrained variance may not be interesting as an investment destination. So, here we assume that the investor considers the maximum eigenvalue of the covariance matrix, for a given expected mean return. For which we have to find the distribution. Let the weights associated with the given stocks be

w1(t) w2(t) w(t) = w (t)  3     ....    w (t)  n    The optimization problem can be written in the following structure

4 maximize w(t)T Ωw(t) w(t) subject to w(t)T w(t) = 1, IT w(t) = 1, E(R)T w(t) = k, R is the vector containing the expected return of the stocks. Overall the constraint can be combined as

maximize w(t)T Ωw(t) w(t) subject to w(t)T Iw(t) = 1, AT w(t) = b, where µ1 1 µ2 1 A = E(R) I = µ 1  3    h i ......    µ 1  n    k b = . "1# Here Ω is a n n and A is a n 2. We are going to simplify the first constraint and we are goingto do a QR × × decomposition of the matrix A.Gander et al. (1991)

R PT A = , "0# where P denotes an orthogonal matrix, and R is a upper triangular matrix

AT P = RT 0 h i Multiplying both the sides by PT AT PPT = RT 0 PT h i or AT = R 0 PT h i The optimization problem now becomes,

maximize w(t)T PPT ΩPPT w(t) w(t) subject to RT 0 PT w(t) = b,

w(t)hT PPT IPPi T w(t) = 1.

5 We can now use the following definitions B ΓT PT ΩP = "Γ C # where the dimensions of the matrix B, matrix Γ, matrix C will have the dimensions accordingly

q PT w(t) = . "r# Similarly the dimensions of the matrix q and matrix r will be decided accordingly

w(t)T PPT = qT rT PT h i w(t)T = qT rT PT . h i Also, C = CT , so

B ΓT q q w(t)T Ωw(t) = w(t)T PPT ΩPPT w(t) = qT rT = qT B + rT Γ qT ΓT + rT C "Γ C #"r# "r# h i h i = (qT Bq + rT Γq + qT ΓT r + rT Cr) = (qT Bq + 2rT Γq + rT Cr).

q Then, AT w(t) = RT 0 PT w(t) = RT 0 = b "r# h i h i or, RT q = b

T and finally, q = R− b (1)

The value of q helps to determine the term qT Bq, so the objective function becomes (2rT Γq + rT Cr) which now needs to be minimized. From the last constraint equation,

q w(t)T w(t) = qT rT = qT q + rT r = 1. "r# h i We define s2 = 1 qT q = rT r − and g = Γq − So the optimization problem now becomes,

maximize 2rT g + rT Cr r − subject to rT r = s2.

But we can see that 2rT g is a scalar quantity, therefore we can write 2rT g = 2gT r. So the optimization function

6 becomes, maximize 2gT r + rT Cr r − subject to rT r = s2. Now using the Lagrangian multiplier we write the objective as

φ(r,λ)= 2gT r + rT Cr λ(rT r s2) (2) − − −

Differentiating (2) with r and λand equating to zero we get

2g+ 2Cr 2λr = 0 − −

such that rT r = s2.

Normalizing the equations we get, Cr = g + λr (3)

and rT r = s2 (4)

T T Doing an Eigenvalue decomposition of C we get C = QDQ , where Q Q = 1 and D = diag(δ1,δ2,....,δ(n 2)). − Now substituting it in equation (3) and (4) we obtain

QDQT r = g + λQQT r.

Multiplying the entire equation by QT ,

QT QDQT r = QT g + QT λQQT r

and rT QT Qr = s2.

As QT Q = 1 therefore, DQT r = QT g + λQT r. (5)

Let us define u = QT r

and d = QT g

Thus equation (5) reduces to Du = d + λu (6) and uT u = s2 (7)

u1 r1  u2   r2  Solving equation (6) and (7), we get u = u . Once we get u we can get r as r = Q T u where r = r .  3  −  3       ....   ....      u  r   (n 2)  (n 2)  −   − 

7 q1 q1  q2   q2  r1 r1 q     1 T     We already have q = , and P w(t) =  r2  and hence w(t) = P r2 . "q2#      r   r   3   3       ....   ....      r(n 2) r(n 2)  −   −  Example: Let us take an example for 3 stocks. For the simplification of algebraic calculation we take µ1 = 0.

2 σ1 (xt ) ρ(12)σ1(xt )σ2(xt ) ρ(13)σ1(xt )σ3(xt ) 2 Then Ω = ρ(21)σ2(xt )σ1(xt ) σ2 (xt ) ρ(23)σ2(xt )σ3(xt ) ρ σ (x )σ (x ) ρ σ (x )σ (x ) σ 2(x )  (31) 3 t 1 t (32) 3 t 2 t 3 t    0 1 and A = µ2 1. µ 1  3    Doing a QR decomposition of A, we get

P = P1 P2 ,

h i 2 2 µ2 +µ3 0 2 2 2 2 √2(µ2 +µ3 µ2µ3)(µ2 +µ3 ) 2−  µ2 µ3 µ2µ3  where P1 = − √µ2+µ2 √2(µ2+µ2 µ µ )(µ2+µ2) 2 3 2 3 − 2 3 2 3  µ µ2 µ µ   3 2 − 3 2   µ2 µ2 2 µ2 µ2 µ µ µ2 µ2  √ 2 + 3 √ ( 2 + 3 2 3)( 2 + 3 )   −  µ2 µ3 2 −2 √2(µ2 +µ3 µ2µ3) µ − and P =  3 . 2 √2(µ2+µ2 µ µ ) 2 3 − 2 3  µ2   2 µ2 µ2 µ µ  −√ ( 2 + 3 2 3)   −  R A = P1 P2 "0# h i µ2 + µ2 µ2+µ3 2 3 √µ2+µ2 where R = 2 3 . q 2(µ2+µ2 µ µ )  0 2 3 − 2 3 µ2+µ2  2 3   r  We are going to calculate B, Γ and C from the below equations as previously defined

2 2 2(µ +µ µ2µ3) µ +µ 2 3 − 2 3 1 1 2 2 2 2 µ2 +µ3 −√µ +µ R− = r 2 3  detR 2 2 0 µ2 + µ3    q  2(µ2+µ2 µ µ ) 2 3 − 2 3 0 T 1 µ2+µ2 or R− = r 2 3 detR  µ +µ  2 3 µ2 + µ2 µ2 µ2 2 3  −√ 2 + 3   q 

8 2(µ2+µ2 µ µ ) 2 3 − 2 3 0 1 µ2+µ2 k and q = r 2 3 detR  µ +µ  2 3 µ2 + µ2 "1# µ2 µ2 2 3  −√ 2 + 3   q  2(µ2+µ2 µ µ ) k 2 3 − 2 3 1 µ2+µ2 or q = r 2 3 . detR  µ +µ  k 2 3 + µ2 + µ2 µ2 µ2 2 3 − √ 2 + 3   q  We finally want to compute

2 σ (xt ) ρ σ1(xt )σ2(xt ) ρ σ1(xt )σ3(xt ) T 1 (12) (13) 2 . P1 P2 ρ(21)σ2(xt )σ1(xt ) σ2 (xt ) ρ(23)σ2(xt )σ3(xt ) P1 P2 h i ρ σ (x )σ (x ) ρ σ (x )σ (x ) σ 2(x ) h i  (31) 3 t 1 t (32) 3 t 2 t 3 t    Let us consider the following definitions, 2 2 µ2 + µ3 = Y, q 2 2 2 2 2(µ + µ µ2µ3)(µ + µ )= X, 2 3 − 2 3 q 2 2 2(µ + µ µ2µ3)= Z, 2 3 − qand µ2 µ3 = V. − As r contains only 1 element, we can use the definition

s2 = 1 qT q = rT r − to calculate r 2(µ2+µ2 µ µ ) 1 k 2 3 − 2 3 detR µ2+µ2  r 2 3  w(t) = P 1 µ2+µ3 2
2 detR k 2 2 + µ2 + µ3  − √µ2 +µ3   q    r
     Using the notations 1 kZ detR Y w(t) = P 1 k µ2+µ3 +Y  detR − Y
 r 

   T 2(µ2+µ2 µ µ ) 2(µ2+µ2 µ µ ) k 2 3 − 2 3 k 2 3 − 2 3 1 µ2 µ2 1 µ2 µ2 rT r = 1 2 + 3 2 + 3 detR  µr+µ  detR  µr+µ  − k 2 3 + µ2 + µ2 k 2 3 + µ2 + µ2 ! µ2 µ2 2 3 µ2 µ2 2 3 − √ 2 + 3  − √ 2 + 3  
q  
q 

2 2 1 2(µ + µ µ2µ3) µ + µ 2 r = 1 k2 2 3 − + k 2 3 + µ2 + µ2 detR 2 2 2 2 3 v − ( ) µ2 + µ3 − µ2 µ2 u  2 + 3 q  u

t q

2 1 Z µ + µ 2 r = 1 k2 + k 2 3 +Y s − (detR)2 Y 2 − Y 



9 Therefore we can calculate w(t)

1 kZ detR Y µ +µ w(t) = P 1 k 2 3 +Y  detR − Y
 1 2 Z2 µ2+µ3 2  1 2 k 2 +
k Y
+Y   − (detR) Y −  q


 We can denote 1 kZ detR Y µ +µ F = 1 k 2 3 +Y  detR − Y
 1 2 Z2 µ2+µ3 2  1 2 k 2 +
k Y
+Y   − (detR) Y −  q  So, w(t) becomes w(t) = PF. We can write the maximum eigenvalue as,

T λ(xt )= w(t) Ωw(t)

T T λ(xt )= F P ΩPF

We can now use the following definitions

T T B Γ P ΩP = = k1 k2 k3 "Γ C # h i With the previous definition we can simplify P as

2 2 µ2 +µ3 µ2 µ3 Y 2 V 0 − X Z 0 X Z 2 P = µ2 µ3 µ2µ3 µ3 = µ2 V µ3 µ3  Y −X Z   Y X Z  2 µ −V µ µ µ µ µ3µ2 µ 3 2 2  3 2 − 2  Y X Z  Y X − Z   −      Then, PT ΩP =

T Y 2 V 2 Y 2 V 0 X Z σ1 (xt ) ρ(12)σ1(xt )σ2(xt ) ρ(13)σ1(xt )σ3(xt ) 0 X Z µ2 V µ3 µ3 2 µ2 V µ3 µ3 ρ σ2(xt )σ1(xt ) σ (xt ) ρ σ2(xt )σ3(xt )  Y − X Z   (21) 2 (23)  Y − X Z  µ3 V µ2 µ2 2 µ3 V µ2 µ2 ρ σ3(xt )σ1(xt ) ρ σ3(xt )σ2(xt ) σ (xt )  Y X − Z   (31) (32) 3  Y X − Z       Multiplying out, we get the individual vectors k1 etc. as

µ2 µ2 2 µ3 µ3 µ2 µ3 2 Y Y σ2 (xt )+ Y ρ32σ3(xt )σ2(xt ) + Y Y ρ23σ2(xt )σ3(xt )+ Y σ3 (xt )

 2 2  µ2 Y Vµ3 2 V µ2
µ3 Y V µ3
V µ2 2 k1 = Y X ρ12σ1(xt )σ2(xt ) X σ2 (xt )+ X ρ32σ3(xt )σ2(xt ) + Y X ρ13σ1(xt )σ3(xt ) X ρ23σ2(xt )σ3(xt )+ X σ3 (xt )  − −     µ2 V µ3 2 µ2
µ3 V µ3 µ2 2
  ρ12σ1(xt )σ2(xt )+ σ (xt ) ρ32σ3(xt )σ2(xt ) + ρ13σ1(xt )σ3(xt )+ ρ23σ2(xt )σ3(xt ) σ (xt )   Y Z Z 2 − Z Y Z Z − Z 3   

10 2 Y µ2 µ3 ρ21σ2(xt )σ1(xt )+ ρ31σ3(xt )σ1(xt ) X Y Y − V µ3 µ2 2 µ3 V µ2 µ2 µ3 2  X Y σ2 (xt )+ Y ρ32σ3(xt )σ2(xt ) + X Y ρ23σ2(xt )σ3
(xt )+ Y σ3 (xt ) 

 2 2

  Y Y 2 V µ3 V µ2   x X σ1 (xt ) X ρ21σ2(xt )σ1(xt )+ X ρ31σ3(xt )σ1(xt )  k2 =  2 − 2 −  V µ3 Y x x V µ3 2 x V µ2 x x V µ2 Y x x V µ3 x x V µ2 2 x   X X ρ12σ1( t )σ2( t ) X σ2 (t )+ X ρ32σ3( t )σ2( t ) + X X ρ13σ1( t )σ3( t )
X ρ23σ2( t )σ3( t )+ X σ3 ( t )   − −  

  V V 2 V µ3 V µ2   σ (xt ) ρ21σ2(xt )σ1(xt )+ ρ31σ3(xt )σ1(xt ) +   Z Z 1 − X X   µ3 V µ3 2 µ2 µ2 V µ3 µ2 3   ρ12σ1(xt )σ2(xt )+ σ2 (xt ) ρ32σ3(xt )σ2(xt ) ρ13σ1(xt )σ3(xt )+ ρ23σ2(xt )σ3(xt ) σ (xt )   Z Z Z − Z − Z Z Z
− Z 3    V µ2
µ3
Z Y ρ21σ2(xt )σ1(xt )+ Y ρ31σ3(xt )σ1(xt ) + µ3 µ2 2 µ3 µ2 µ2 µ3 2  σ (xt )+ ρ32σ3(xt )σ2(xt ) ρ23σ2(xt )σ3(xt )+ σ (xt )  Z Y 2 Y − X Y
Y 3

 2

  V Y 2 V µ3 V µ2   Z X σ1 (xt ) X ρ21σ2(xt )σ1(xt )+ X ρ31σ3(xt )σ1(xt ) +  k3 =  2 − 2   µ3 Y ρ σ x σ x V µ3 σ 2 x V µ2 ρ σ x σ x µ2 Y ρ σ x σ x V µ3 ρ σ x σ x V µ2 σ 2 x   Z X 12 1( t ) 2( t ) X 2 ( t )+ X 32 3( t ) 2( t ) Z X 13 1( t ) 3( t )
X 23 2( t ) 3( t )+ X 3 ( t )   − − −  

  V V 2 µ3 µ2   Z Z σ1 (xt )+ Z ρ21σ2(xt )σ1(xt ) Z ρ31σ3(xt )σ1(xt ) +   −2   µ3 V µ3 2 µ2 µ2 Y µ3 µ2 2   ρ12σ1(xt )σ2(xt )+ σ (xt ) ρ32σ3(xt )σ2(xt ) ρ13σ1(xt )σ3(xt )+ ρ23σ2(xt )σ3(xt ) σ (xt )   Z Z Z 2 − Z − Z X
Z − z 3  

 So, we can calculate the eigenvalue as,

T λ(xt )= F k1 k2 k3 F h i where T F k1 k2 k3 = d1 d2 d3 h i h i µ µ µ µ µ µ 1 kZ d = 2 2 σ 2(x )+ 3 ρ σ (x )σ (x ) + 3 2 ρ σ (x )σ (x )+ 3 σ 2(x ) + 1 Y Y 2 t Y 32 3 t 2 t Y Y 23 2 t 3 t Y 3 t detR Y  2 2   µ2 Y V µ3 2
V µ2 µ3 Y

V µ3 ρ12σ1(xt )σ2(xt ) σ (xt )+ ρ32σ3(xt )σ2(xt ) + ρ13σ1(xt )σ3(xt ) ρ23σ2(xt )σ3(xt )+ Y X − X 2 X Y X − X V µ2 2 1 µ2 + µ3
σ (xt ) k +Y + X 3 detR − Y µ2 V
 µ3 2
µ
2 µ3 V µ3 ρ12σ1(xt )σ2(xt )+ σ (xt ) ρ32σ3(xt )σ2(xt ) + ρ13σ1(xt )σ3(xt )+ ρ23σ2(xt )σ3(xt ) Y Z Z 2 − Z Y Z Z −  2
µ2 2 1 2 Z µ2 + µ3 2 σ3 (xt ) 1 k + k +Y Z s − (detR)2 Y 2 − Y
 



2 Y µ2 µ3 d2 = ρ21σ2(xt )σ1(xt )+ ρ31σ3(xt )σ1(xt ) X Y Y − V µ3 µ2 µ3 V µ2
µ2 µ3 1 kZ σ 2(x )+ ρ σ (x )σ (x ) + ρ σ (x )σ (x )+ σ 2(x ) + X Y 2 t Y 32 3 t 2 t X Y 23 2 t 3 t Y 3 t detR Y 2 2 2   Y Y 2 V µ3
V µ2 V µ3 Y

V µ3 2 σ (xt ) ρ21σ2(xt )σ1(xt )+ ρ31σ3(xt )σ1(xt ) ρ12σ1(xt )σ2(xt ) σ (xt )+ x X 1 − X X − X X − X 2  2 V µ2 V µ2 Y V µ3
V µ2 2 1 µ2 + µ3 ρ32σ3(xt )σ2(xt ) + ρ13σ1(xt )σ3(xt ) ρ23σ2(xt )σ3(xt )+ σ (xt ) k +Y + X X X − X X 3 detR − Y V V 2 V µ3
V µ2 µ3 V
µ3 2 µ2

 σ (xt ) ρ21σ2(xt )σ1(xt )+ ρ31σ3(xt )σ1(xt ) + ρ12σ1(xt )σ2(xt )+ σ (xt ) ρ32σ3(xt )σ2(xt ) Z Z 1 − X X Z Z Z 2 − Z 

11 2 µ2 V µ3 µ2 2 1 2 Z µ2 + µ3 2 ρ13σ1(xt )σ3(xt )+ ρ23σ2(xt )σ3(xt ) σ3 (xt ) 1 k + k +Y − Z Z Z − Z s − (detR)2 Y 2 − Y
 



V µ µ d = 2 ρ σ (x )σ (x )+ 3 ρ σ (x )σ (x ) + 3 Z Y 21 2 t 1 t Y 31 3 t 1 t µ3 µ2 2 µ3 µ2 µ2
µ3 2 1 kZ σ (xt )+ ρ32σ3(xt )σ2(xt ) ρ23σ2(xt )σ3(xt )+ σ (xt ) + Z Y 2 Y − X Y Y 3 detR Y 2 2   V Y 2 V µ3
Vµ2 µ3 Y

V µ3 2 σ (xt ) ρ21σ2(xt )σ1(xt )+ ρ31σ3(xt )σ1(xt ) + ρ12σ1(xt )σ2(xt ) σ (xt )+ Z X 1 − X X Z X − X 2  2 V µ2 µ2 Y V µ3
V µ2 2 1 µ2 + µ3 ρ32σ3(xt )σ2(xt ) ρ13σ1(xt )σ3(xt ) ρ23σ2(xt )σ3(xt )+ σ (xt ) k +Y + X − Z X − X X 3 detR − Y V V 2 µ3
µ2 µ3 V
µ3 2 µ2

 σ (xt )+ ρ21σ2(xt )σ1(xt ) ρ31σ3(xt )σ1(xt ) + ρ12σ1(xt )σ2(xt )+ σ (xt ) ρ32σ3(xt )σ2(xt ) Z Z 1 Z − Z Z Z Z 2 − Z −  2
2
µ2 Y µ3 µ2 2 1 2 Z µ2 + µ3 2 ρ13σ1(xt )σ3(xt )+ ρ23σ2(xt )σ3(xt ) σ3 (xt ) 1 k + k +Y Z X Z − z s − (detR)2 Y 2 − Y
 



2 1 kZ 1 µ2 + µ3 1 2 Z µ2 + µ3 2 λ(xt )= d1 + d2 k +Y + d3 1 k + k +Y detR Y detR − Y s − (detR)2 Y 2 − Y 
 

 

 Now the price of the largest eigenvalue swap is the expected present value of the payoff in the risk neutral world for this three asset we have considered

rT P(x)= E e− (λ(xt ) K) (8) { − } rT P(x)= e− E (λ(xt ) K) { − }

T T µ2 µ2 rT 1 tQ 2 µ3 rT 1 tQ P(x1) = ( e− (e σ2 (x))dt + e− (e (ρ32σ3(x)σ2(x))dt )+ Y Y {T 0 } Y {T 0 } Z Z T
T
µ3 µ2 rT 1 tQ µ3 rT 1 tQ 2 1 kZ ( e− (e (ρ23σ2(x)σ3(x))dt + e− (e σ3 (x))dt ) + Y Y {T 0 } Y {T 0 } detR Y Z Z 2 T T µ2 Y rT 1 tQ
V µ 3 rT 1 tQ 2


( e− (e (ρ12σ1(x)σ2(x))dt ) e− (e σ2 (x))dt + Y X {T 0 } − X {T 0 } Z Z T 2 T V µ2 rT 1 tQ
µ3 Y rT 1 tQ
e− (e (ρ32σ3(x)σ2(x))dt )) + ( e− (e (ρ13σ1(x)σ3(x))dt ) X {T 0 } Y X {T 0 } − Z Z T T V µ3 rT 1 tQ
V µ2 rT 1 tQ 2 1
µ2 + µ3 e− (e (ρ23σ2(x)σ3(x))dt )+ e− (e σ3 (x))dt ) k +Y X {T 0 } X {T 0 } detR − Y Z Z T T µ2V rT 1 tQ
µ3 rT 1 tQ 2



+ ( e− (e (ρ12σ1(x)σ2(x))dt )+ e− (e σ2 (x))dt Y Z {T 0 } Z {T 0 } − Z Z T T µ2 rT 1 tQ
µ3 V rT 1 tQ
e− (e (ρ32σ3(x)σ2(x))dt )) + ( e− (e (ρ13σ1(x)σ3(x))dt + Z {T 0 } Y Z {T 0 } Z Z T T µ3 rT 1 tQ
µ2 rT 1 tQ 2
e− (e (ρ23σ2(x)σ3(x))dt ) e− (e σ3 (x))dt ) Z {T 0 } − Z {T 0 } Z Z 2

1 2 Z µ2 + µ3 2 1 kZ 1 2 k 2 + k +Y s − (detR) Y − Y ! × detR Y

12 2 T T Y µ2 rT 1 tQ µ3 rT 1 tQ P(x2) = (( e− (e (ρ21σ2(x)σ1(x))dt + e− (e (ρ31σ3(x)σ1(x))dt ) X Y {T 0 } Y {T 0 } − Z Z T
T
V µ3 µ2 rT 1 tQ 2 µ3 rT 1 tQ ( e− (e σ2 (x))dt + e− (e (ρ32σ3(x)σ2(x))dt )+ X Y {T 0 } Y {T 0 } Z Z T T V µ2 µ2 rT 1 tQ
µ3 rT 1 tQ 2
1 kZ ( e− (e (ρ23σ2(x)σ3(x))dt + e− (e σ3 (x))dt )) + X Y {T 0 } Y {T 0 } detR Y Z Z 2 2 T T Y Y rT 1 tQ 2 V µ3
rT 1 tQ


( e− (e σ1 (x))dt e− (e (ρ21σ2(x)σ1(x))dt + x X {T 0 } − X {T 0 } Z Z T 2 T V µ2 rT 1 tQ
V µ3 Y rT 1 tQ
e− (e (ρ31σ3(x)σ1(x))dt ) ( e− (e (ρ12σ1(x)σ2(x))dt X {T 0 } − X X {T 0 } − Z Z T T V µ3 rT 1 tQ 2 V µ2
rT 1 tQ
e− (e σ2 (x))dt + e− (e (ρ32σ3(x)σ2(x))dt )+ X {T 0 } X {T 0 } Z Z 2 T T V µ2 Y rT 1 tQ
V µ3 rT 1 tQ
( e− (e (ρ13σ1(x)σ3(x))dt e− (e (ρ23σ2(x)σ3(x))dt + X X {T 0 } − X {T 0 } Z Z T T V µ2 rT 1 tQ 2 1
µ2 + µ3 V V rT 1 tQ
2 e− (e σ3 (x))dt ) k +Y + ( e− (e σ1 (x))dt X {T 0 } detR − Y Z Z {T 0 } − Z Z T T V µ3 rT 1 tQ

Vµ2 rT
1

tQ
e− (e (ρ21σ2(x)σ1(x))dt + e− (e (ρ31σ3(x)σ1(x))dt )+ X {T 0 } X {T 0 } Z Z T T µ3 V rT 1 tQ
µ3 rT 1 tQ 2
( e− (e (ρ12σ1(x)σ2(x))dt + e− (e σ2 (x))dt Z Z {T 0 } Z {T 0 } − Z Z T T µ2 rT 1 tQ µ
2 V rT 1 tQ
e− (e (ρ32σ3(x)σ2(x))dt ) ( e− (e (ρ13σ1(x)σ3(x))dt + Z {T 0 } Z Z {T 0 } − Z Z T T µ3 rT 1 tQ
µ2 rT 1 tQ 2
e− (e (ρ23σ2(x)σ3(x))dt e− (e σ3 (x))dt ) Z {T 0 } − Z {T 0 } Z Z 2


1 2 Z µ2 + µ3 2 1 µ2 + µ3 1 2 k 2 + k +Y k +Y (9) s − (detR) Y − Y ! × detR − Y

T T V µ2 rT 1 tQ µ3 rT 1 tQ P(x3) = (( e− (e (ρ21σ2(x)σ1(x))dt + e− (e (ρ31σ3(x)σ1(x))dt )+ Z Y {T 0 } Y {T 0 } Z Z T
T
µ3 µ2 rT 1 tQ 2 µ3 rT 1 tQ ( e− (e σ2 (x))dt + e− (e (ρ32σ3(x)σ2(x))dt ) Z Y {T 0 } Y {T 0 } − Z Z T T µ2 µ2 rT 1 tQ
µ3 rT 1 tQ 2
1 kZ ( e− (e (ρ23σ2(x)σ3(x))dt + e− (e σ3 (x))dt )) + X Y {T 0 } Y {T 0 } detR Y Z Z 2 T T V Y rT 1 tQ 2 V µ3
rT 1 tQ


( ( e− (e σ1 (x))dt e− (e (ρ21σ2(x)σ1(x))dt + Z X {T 0 } − X {T 0 } Z Z T 2 T V µ2 rT 1 tQ
µ3 Y rT 1 tQ
e− (e (ρ31σ3(x)σ1(x))dt )+ ( e− (e (ρ12σ1(x)σ2(x))dt X {T 0 } Z X {T 0 } − Z Z T T V µ3 rT 1 tQ 2 V µ2
rT 1 tQ
e− (e σ2 (x))dt + e− (e (ρ32σ3(x)σ2(x))dt ) X {T 0 } X {T 0 } − Z Z 2 T T µ2 Y rT 1 tQ
V µ3 rT 1 tQ
( e− (e (ρ13σ1(x)σ3(x))dt e− (e (ρ23σ2(x)σ3(x))dt + Z X {T 0 } − X {T 0 } Z Z T T V µ2 rT 1 tQ 2 1
µ2 +µ3 V V rT 1 tQ
2 e− (e σ3 (x))dt )) k +Y + ( ( e− (e σ1 (x))dt + X {T 0 } detR − Y Z Z {T 0 } Z Z T T µ3 rT 1 tQ
µ2 rT 1


tQ
e− (e (ρ21σ2(x)σ1(x))dt e− (e (ρ31σ3(x)σ1(x))dt )+ Z {T 0 } − Z {T 0 } Z Z

13 T T µ3 V rT 1 tQ µ3 rT 1 tQ 2 ( e− (e (ρ12σ1(x)σ2(x))dt + e− (e σ2 (x))dt Z Z {T 0 } Z {T 0 } − Z Z T 2 T µ2 rT 1 tQ
µ2 Y rT 1 tQ
e− (e (ρ32σ3(x)σ2(x))dt ) ( e− (e (ρ13σ1(x)σ3(x))dt + Z {T 0 } − Z X {T 0 } Z Z T T µ3 rT 1 tQ
µ2 rT 1 tQ 2
e− (e (ρ23σ2(x)σ3(x))dt e− (e σ3 (x))dt )) Z {T 0 } − z {T 0 } Z Z 2
2
1 2 Z µ2 + µ3 2 1 2 Z µ2 + µ3 2 1 2 k 2 + k +Y 1 2 k 2 + k +Y s − (detR) Y − Y ! × s − (detR) Y − Y





So our P(x) can be written as,

rT P(x)= P(x1)+ P(x2)+ P(x3) e− Kstrikeprice − ×
3 Numerical Example

Stock of “CMS Energy Corporation” Quantopian (2018) is chosen as S1, stock of “American Electric Power

Company Inc” is chosen as S2 and stock of “Entergy Corporation” is chosen as S3. The data used are daily closing price of S1, S2 and S3 in the time range 3rd May, 2018 till 2nd May, 2019. To create our finite state Markov chain, we consider two states for each individual stock, defined as follows.

Considering stock S1, Up, when return > µ1 128 observations x1r = ( Down, when return µ1 123 observations ≤ Similarly, for the stock S2 we can divide the data points as,

Up, when return > µ2 138 observations x2r = ( Down, when return µ2 113 observations ≤ Similarly, for the stock S3 we can divide the data points as,

Up, when return > µ3 130 observations x3r = ( Down, when return µ3 121 observations ≤ Now combining all the data points and considering that the combined state random variable takes three values, we get the following division:

Up, when x1 = x2 = x3 = Up 93 observations xdata =  Middle, otherwise 78 observations  Down, when x1 = x2 = x3 = Down 80 observations  Using the functions from R studio we can calculate the transition probability matrix (Π) as,

State Down Middle Up Down 0.3250000 0.2875000 0.3875000 Middle 0.3636364 0.3376623 0.2987013 Up 0.2795699 0.3010753 0.4193548 and the corresponding standard error of the probability matrix is given by

14 State Down Middle Up Down 0.06373774 0.05994789 0.06959705 Middle 0.06872081 0.06622103 0.06228353 Up 0.05482817 0.05689788 0.06715052

The stationary probability matrix p can be calculated from the formula,

pΠ = p where

p = pD pM pU . h i Solving the equations on the variables we get,

pD 0.32000

pM 0.30783

pU 0.37217 The mean returns of the stocks and the daily interest rate (assuming 10 % annual interest rate) is calculated as,

mu1 0.000664

mu2 0.000873

mu3 0.0.000725 r 0.0004

We can evaluate the variance terms as,

T rT 1 2 2 2 P2var(i)= e− (piDσ2D + piMσ2M + piUσ2U )dt {T 0 } Z where i = D,M,U is the initial state of the Markov chain. If we are uncertain about the initial state and we have only an initial probability distribution, say (pD, pM, pU ) then the expression will be

P2var = pDP2var(D)+ pMP2var(M)+ pUP2var(U)

Similarly other variance terms are

T rT 1 2 2 2 P1var(i)= e− (piDσ1D + piMσ1M + piUσ1U )dt {T 0 } Z

P1var = pDP1var(D)+ pMP1var(M)+ pUP1var(U)

T rT 1 2 2 2 P3var(i)= e− (piDσ3D + piMσ3M + piUσ3U )dt {T 0 } Z

P3var = pDP3var(D)+ pMP3var(M)+ pUP3var(U)

The values are shown in Table 1. We can evaluate the covariance terms as,

15 6 Table 1: All the figures are in 10−

P1var 42.978 P2var 43.275 P3var 40.240

6 Table 2: All the figures are in 10−

PCov(23) = PCov(32) 41.234 PCov(31) = PCov(13) 39.477 PCov(12) = PCov(21) 40.911

T rT 1 PCov(23)(i)= e− (piDCov23D + piMCov23M + piUCov23U )dt {T 0 } Z where i = D,M,U is the initial state of the Markov chain. If we are uncertain about the initial state and we have only a probability distribution, let say (pD, pM, pU ) then the price is going be

PCov(23) = pDPCov(23)(D)+ pMPCov(23)(M)+ pUPCov(23)(U)

Similarly other covariance terms are evaluated as

T rT 1 PCov(31)(i)= e− (piDCov31D + piMCov31M + piUCov31U )dt {T 0 } Z

PCov(31) = pDPCov(31)(D)+ pMPCov(31)(M)+ pUPCov(31)(U) and T rT 1 PCov(12)(i)= e− (piDCov12D + piMCov12M + piUCov12U )dt {T 0 } Z

PCov(12) = pDPCov(12)(D)+ pMPCov(12)(M)+ pUPCov(12)(U)

These values are shown in Table 2.

3.1 Numerical example for swap given by trace

As mentioned in the introduction, our first candidate measure of generalized variance is the trace of the covariance matrix which is nothing but the sum of the individual variances. Intuitively, this is the variance of the return of a portfolio comprising one unit of each of the stocks, assuming them to be uncorrelated. We have already calculated the expected value of the variances in Table 1. So we can calculate the price of the swap as,

rT Ptrace(x)= P1var + P2var + P3var e− Kstrike price −
rT Ptrace(x)= 42.978 + 43.275 + 40.240 e− Kstrike price −

16 rT Ptrace(x)= 126.493 e− Kstrike price − The swap is written in terms of 1 million units of the trace. Considering the strike price as 90 and duration is for 3 months i.e. T = 63 we can finally calculate the trace swap as,

(0.0004) 63 Ptrace = 126.493 e− × 90 = 126.493 87.760 = 38.733 − × −
Ptrace = 38.733

3.2 Numerical example for swap given by largest eigenvalue

The second candidate measure of generalised variance considered here is the maximum eigenvalue which is the magnitude of the biggest component of the orthogonalised system for the return covariance matrix. As the return distributions are correlated (the covariance matrix is not diagonal), this biggest component will be significantly larger than the individual variances. This is considered as we are interested in managing the variance, so swapping for the biggest component is a safe strategy to adopt. Considering T = 63 we do the following calculations,

0.000664 1 A = 0.000873 1 0.000725 1     Doing a QR decomposition of A, we get

0.505023 0.6551103

P1 = 0.6639542 0.7108661  −  0.5514677 0.2559293     0.001315 1.720445 R = " 0 0.2001735#

0.561945 P2 =  0.232022  0.793967 −    Therefore P is, 0.505023 0.6551103 0.561945 P = 0.6639542 0.7108661 0.232022  −  0.5514677 0.2559293 0.793967  −    T 760.4563 0 R− = " 6536.689733 4.995656# − Given that,

0.0007 b = " 1 #

17 0.5322577 q = "0.4210340#

r = 0.7344604 h i Therefore we can calculate w(t) 0.9569597 w(t) =  0.2239916  0.1822877 −    Note that, since the expected returns of the portfolio are quite spread out, in the optimal solution we are getting short position for one stock which has a higher expected return and long positions for the other two stocks.

P P P T 1var Cov(12) Cov(13) rT 1 tQ e− (e (Ω)dt = PCov(21) P2var PCov(23) {T 0 }   Z P P P
 Cov(31) Cov(32) 3var    Thus the price of the eigenvalue swap is given by,

T 0.9569597 P1var PCov(12) PCov(13) 0.9569597 rT Peigenvalue = 0.2239916 P P P 0.2239916 e− Kstrikeprice    Cov(21) 2var Cov(23)  − × 0.1822877 PCov(31) PCov(32) P3var 0.1822877 −   − 
     Putting the values we get,

T 0.9569597 42.978 40.911 39.477 0.9569597 rT Peigenvalue = 0.2239916 40.911 43.275 41.234 0.2239916 e− Kstrikeprice      − × 0.1822877 39.477 41.234 40.240 0.1822877 −   − 
     0.9569597 rT = 43.095 41.327 39.678 0.2239916 e− Kstrikeprice   − × h i 0.1822877 − 
  rT Peigenvalue = 43.264 e− Kstrikeprice − × The swap is written in terms of 1 million units of the eigenval ue. Considering
the strike price as 30 we can finally calculate the eigenvalue swap as,

(0.0004) 63 Peigenvalue = 43.264 e− × 30 = 43.264 29.253 = 14.011 − × −
Peigenvalue = 14.011

4 Conclusion

In this paperwe have presented a new approachfor pricing swaps defined on two important measures of generalized variance, namely the maximum eigenvalue and trace of the covariance matrix of the returns on assets involved. The objective is to price generalized variance swaps for financial markets with Markov-modulated volatilities. We have considered multiple assets in the portfolio for theoretical purpose and demonstrated the theoretical approach

18 with the help of numerical examples taking three stocks in the portfolio. The results derived in this paper are the comparison between the swaps defined by the trace and the eigenvalue. In the numerical examples of swaps priced, the price of the trace swap is more than that of the value for the eigenvalue swap. This can be justified as in the maximum eigenvalue swap it is not only the variance of the stocks which is responsible for the price determination, it is also the covariance terms and the expected value of the stocks’ return which are present in the price of the maximum eigenvalue swap which makes the price for the maximum eigenvalue swap less. So, we can say that for the same stocks we should prefer the maximum eigenvalue swap as comparedto trace swap as the price of the swap is less. Moreover, the results obtained in this paper have important implications for their use in the commodity sector as volatility in the commodity markets, agricultural in particular, are often related through natural causes. This would be an important area where such swaps would be useful for hedging risk. In our future work we also aim to incorporate an often observed phenomenon in the returns, namely jumps Broadie and Jain (2008b). This would render the usual Ito formulation unsatisfactory. One can apply the well known Levy process to the returns in such a case Habtemicael and SenGupta (2016). It would be an important extension to define and price swaps on measures of generalized variance in this scenario.

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