From multi-dimensional black scholes to Hamilton jacobi
aa r X i v : . [ q -f i n . M F ] A p r Multi-dimensional Black-Scholes-Merton toHamilton-Jacobi
Muhammad Naqeeb
Department of Mathematics, Quaid-i-Azam University,Islamabad-44000, Pakistan.Email; [email protected]
Amjad Hussain
Department of Mathematics, Quaid-i-Azam University,Islamabad-44000, Pakistan.Email; [email protected]
Abstract
The first widely used financial model to price the evolution of optionshas footprints almost in every field. The aim of this paper is to derivethe Black-Scholes-Merton formula of multiple options, generally for ann-dimensional assets and its links to Hamilton-Jacobi equation of me-chanics with solution of black-Scholes equation in the metric of Banachspace.keywords: Multi-dimensional Black-Scholes-Merton, Hamilton-JacobiEuropean call option, Banach space.
Introduction
Dynamical systems, deterministic or stochastic are interlinked withMathematical Finance. In Dynamical Mathematical Finance we ex-amine financial models by mean of dynamical systems. Dynamical sys-tems defined over multiple independent variables categorized as multi-dimensional systems. For Black-Scholes model of Mathematical Fi-nance one or multiple underlying assets could be consider. For oneunderlying asset the model is, ∂v∂t + rs ∂ v∂s + 12 σ s ∂ v∂s − rv = 0 (1)For an n-underlying assets or multi-assets the formula, which is derivedin section one, is, ∂v∂t + 12 σ s ∂ v∂s + 12 σ s ∂ v∂s + ... + 12 σ n s n ∂ v∂s n + ∂v∂s ds + ∂v∂s ds + ... + ∂v∂s n ds n + rs ∂v∂s + rs ∂v∂s + ... + rs n ∂v∂s n + ρ n σ σ n ∂ v∂s s n + ρ n σ σ n ∂ v∂s s n + ... + ρ n − n σ σ n ∂ v∂s n − n = 0 (2)The Hamilton-Jacobi equation, an alternate formulation of ClassicalMechanics and necessary condition for describing extremal geometry in generalization of problems from the calculus of variations. For onespatial or generalized coordinate the equation is give as, ∂U∂t + H ( q , ∂U∂q , ) (3)For an n-generalized coordinates or spatial variables the above equationafter extension is give as, ∂U∂t + H ( q , q , q , ...q n , ∂U∂q , ∂U∂q , ∂U∂q , ..., ∂U∂q n ) (4)Rodrigo and Galvez in [4] presented their work on relationships betweenequation (1) and (3) for one spatial variable considered as an underlyingasset. In present work, the relationships between equation (2) and (4)are studied and explained, Which are multi-dimensional and multi-asset models. The work is presented systematically by dividing it indifferent sections.The first one is to derive and analyze Black-Scholes Merton havingan n-underlying assets, the second section is to dicuss Hamilton-Jacobiequation in multi-variable calculus. Moving to third major section thatexplains relationships between Black-Bcholes-Merton and Hamilton-Jacobi equation of mechancis with concluding remarks and suggestionsin fourth section 1. black scholes equation From everyday markets to the implications of theory of Relativityin financial mathematics [1]. This formula beyond the achievement ofNoble price in 1997 for Scholes and Merton [3] has impressed everyfield. To derive having an n-underlying assets starting from multiplestochastic process ds s = µ dt + σ dw , ds s = µ dt + σ dw ds n s n = µ n dt + σ n dw n where dw n ; n = 1 , , ... is Brownian motion. E ( dw i ) = 0; E ( dw i ) = dt By definition of random walk or wiener process [2]Correlation factor which appears to be zero in case of one underlyingassest here it will be in relation with each two distinct random walksto determine their strength from positive one, zero or negative onecorrelation factor having range -1 to +1 is quite different from risk freeinterest rate ranging from 0 to +1. These factors are vital to consider.Π = v − △ s + △ s + ... + △ n s n d Π = dv − △ ds + △ ds + ... + △ n ds n Now, the n-dimensional ito’s lemma is given as dv = ∂v∂t + 12 σ ∂ v∂s + 12 σ ∂ v∂s + ... + 12 σ n ∂ v∂s n + ∂v∂s ds + ∂v∂s ds + ... + ∂v∂s n ds n + ρ n σ σ n ∂ v∂s s n (5) ρ n σ σ n ∂ v∂s s n + ... + ρ n − n σ σ n ∂ v∂s n − n To eliminate the risk terms from above n ito’s lemma △ = ∂v∂s , △ = ∂v∂s , ..., △ n = ∂v∂s n dv = ∂v∂t + 12 σ s ∂ v∂s + 12 σ s ∂ v∂s + ... + 12 σ n s n ∂ v∂s n + ∂v∂s ds + ∂v∂s ds + ... + ∂v∂s n ds n + ρ n σ σ n ∂ v∂s s n (6) ρ n σ σ n ∂ v∂s s n + ... + ρ n − n σ σ n ∂ v∂s n − n d Π = riskless return rate would be d Π = r Π = r { v − ∂v∂s s − ∂v∂s s − ... − ∂v∂s n s n } dt and finally we conclude Black-Scholes equation which is ∂v∂t + 12 σ s ∂ v∂s + 12 σ s ∂ v∂s + ... + 12 σ n s n ∂ v∂s n + ∂v∂s ds + ∂v∂s ds + ... + ∂v∂s n ds n + rs ∂v∂s + rs ∂v∂s + ... + rs n ∂v∂s n + ρ n σ σ n ∂ v∂s s n + ρ n σ σ n ∂ v∂s s n + ... + ρ n − n σ σ n ∂ v∂s n − n = 0 (7)This is the required derived black scholes equation having multiple of nunderlying assets, which affectively differ from black scholes equationin one underlying asset especially in appearance of correlation factorsterms having their intensity defined below( dw dw n ) = ρ n , { dw dw n } = ρ n , ..., { dw n − dw n } = ρ n − n These are correlation coefficients for an n underlying assets, after dis-cussing their terms of correlation coefficients. here number of termsin an equation could be random but still predicted which is Mathe-matically healthy in number theory like first equation having 4 termsthe next one having 4+3=7, and next one having 7+4=11 terms, then11+5=16, then 16+6=22 terms and so on. s special kind of series atpredicts terms containing in Black-Scholes equation having multipleassets conditions and solutions:
Considering space which here bestsuited is n-dimensional Euclidean space, whose discussion will followlater in third section , which will become the Banach space in whichsolution could be defined by introducing special metric is vital, to con-sider v the solution variable in n dimensional space.Considering space R n having solution of the form v ( s , s , s , ..., s n ) = f ( s , s , s , ..., s n , t )The Initial condition for an n underlying asset for an European calloption would be v ( s , s , s , ..., s n , T ) = max { max ( s , s , s , ..., s n ) − k, } While K is an expiration or exercise date on which option has to beexercised.The above model is of Multi dimensional Black scholes equation withEuropean conditions, Although our discussion proceeds considering Eu-ropean call option, The discussion could be expanded by taking condi-tions on European put options, American put and call options, Basketoptions and for exchange options. At the expiry date ’K’ the solution,By reducing Multi-dimensional Black-scholes to diffusion equation inn dimensions [1] possessing ’N’ a standard normal cumulative distribu-tion function is V ( s , s , s , ..., s n , T ) = s N ( φ , p ) + s N ( φ , p ) + s N ( φ , p ) ... + s N ( φ n , p n ) N − Ke − rT { − { N ( φ ′ )+ N ( φ ′ ) + N ( φ ′ ) + ... + N ( φ ′ n ) }} (8)2. Hamilton jacobi differential equation and itssolutions
Hamilton jacobiIn dynamical systems the non linear Hamilton-Jacobi equation couldbe used to derive equations of motion. The Hamilton-Jacobi equationin (n+1) variables q , q , q , ...q n , t is∂U∂t + H ( q , q , q , ...q n , ∂U∂q , ∂U∂q , ∂U∂q , ..., ∂U∂q n ) (9)The Black-Scholes model and Hamilton-Jacobi, Both models are likedto each other in terms of Initial conditions, Spaces, Energy relations,Domains, Spatial variables. Employing mathematical techniques thedynamical systems to some extent could be linked to models in financeand economics, With the help of related concepts and different fields.[4] The Hamilton-Jacobi in Variational calculus is a cauchy problem[5]which is represented by ∂U∂t + H ( q , q , q , ...q n , ∂U∂q , ∂U∂q , ∂U∂q , ..., ∂U∂q n ) = 0where R n × (0 , ∞ ) , with Initial condition U = ( g , g , ...g n ) , and ( g , g , ...g n ) ∈ R ⋉ × ( t = 0)Here we have U : R ⋉ × (0 , ∞ ) → R and Hamiltonian defined by H : R n → R, where ( g , g , ...g n ) : R n → R solution of Hamilton-Jacobi equation. The solution of theHamilton-Jacobi equation in n-dimensions, Or In variational calculuswill be:
Let L : R n → R, name it Lagrangian function satisfying conditions T he mapping q L ( q ) is convexlim L ( q ) | q | = ∞ as q → ∞ The convexity implies L is continuous lagrangian function. We canobtain Hamiltonian from Lagrangian by the Legendre transformationof L. Hamiltonian and Lagrangian are both dual complex functions,interpreting, H=L, The form of Hamiltonian and Lagrangian is almostsame. To discuss solution we try to minimize the action introduced as I { W ( . ) } = Z t L ( ˙ w ( s ) ds over W : [0 , t ] → R n where W ( . ) = { W ( . ) , W ( . ) W ( . ) , ..., W ( . ) } Modifying the above equation to include the function g evaluated atW(0) I { W ( . ) } = Z t L { ˙ W ( s ) ds } + g { w (0) } The solution of the Hamilton-Jacobi equation in terms of Variationalprinciple entailing this modified action is U ( x, t ) = inf ( Z t L { ˙ W ( s ) } ds + g ( y ) | W (0) = y, W ( t ) = x Whereas infimum is taken over all W ( . ) ∈ C with W ( t ) = x. H is smooth and Convex, While ( g , g , ...g n ) must be Lipschitz contin-uous. The above minimization problem could be simplified as U ( x, t ) = min y ∈ R n ( tL ( x − yt ) + g ( y )The expression on the right hand side is called Hopf-Lax formula. Thisformula is a unique weak solution of the initial value problem for the Hamilton-Jacobi equation. While in variational analysis the solutionof the cauchy problem ’U’ is defined in the metric of Banach space, L isinvariant[7] while H must be of the class of second differential functions,Hamilton-Jacobi equation is an evolution equation3.
On Relations of Multi-dimensionalBlack-Scholes-Merton and Hamilton-Jacobi V = F ( s , s , s , ..., s n , T )If we express Black-Scholes-Merton in the form ∂v∂t + 12 σ s ∂ v∂s + 12 σ s ∂ v∂s + ... + 12 σ n s n ∂ v∂s n + ∂v∂s ds + ∂v∂s ds + ... + ∂v∂s n ds n + rs ∂v∂s + rs ∂v∂s + ... + rs n ∂v∂s n + ρ n σ σ n ∂ v∂s s n − rv = 0 (10)for n underlying assets we conclude that; H ( DV ) = 12 σ s ∂ v∂s + 12 σ s ∂ v∂s + ... + 12 σ n s n ∂ v∂s n + ∂v∂s ds + ∂v∂s ds + ... + ∂v∂s n ds n + rs ∂v∂s + rs ∂v∂s + ... + rs n ∂v∂s n + ρ n σ σ n ∂ v∂s s n − rv (11)where H(DV) being Hamiltonian interlinked with Black-Scholes-Merton.Here, our conclusion would be following V t + H ( DV ) = 0 where V ∈ R n , t ∈ (0 , ∞ ) ,V = ( g , g , g ...g n ) , in ( t = 0)3.3. Initial condition (t=0), for Hamilton Jacobi U = ( g , g , g , ..., g n )Whereas Initial condition (t=0), for Black-Scholes-Merton for an euro-pean call option is V ( s , s , s , ...s n , T ) = { max ( max ( s , s , s , ...s n ) − K } s , s , s , ...s n )denotes an n-underlying assets, whereas spatial variables are consideredhere as underlying assets or actives.3.5. By interlinking Hamiltonian of both equations. The dynamicalaspects of Black-scholes-Merton allows to evaluate options, represent-ing behaviour of options as physical objects.3.6. As showed above the multi-dimensional Black-Scholes-Merton isa dynamical system, possessing an associated energy, which preservesby both double convex functions, Hamilton and Lagrangian, Startingfrom Hamiltonian H = P i ˙ x − L (cid:0) t, ˙ x j , ˙ x j (cid:1) , where P i is the generalized momenta and taking P i ˙ x i = φ a potential function, and depicting L = φ − H Black-Scholes equation has lagrangian depicted by equation, L ( s, ˙ s, t ) = φ − { σ s ∂ v∂s + 12 σ s ∂ v∂s + ... + 12 σ n s n ∂ v∂s n + ∂v∂s ds + ∂v∂s ds + ... + ∂v∂s n ds n + rs ∂v∂s + rs ∂v∂s + ... + rs n ∂v∂s n + ρ n σ σ n ∂ v∂s s n with φ = P ˙ s By considering that ˙ s is the variation of the price of the action.3.7. For a curious researcher it may be wealthy, to find methods whichare applicable to problems in dynamics especially Hamilton-Jacobimethod, to solve problems in financial Mathematics to understand theessence of interlinking of both equations.3.8. Multi-dimensional Black-scholes-Merton having solutions in theMetric of Banach space defined by, || V | | = | V − V | whereas V and V are solutions at time t and t respectively , f or t ,V ( s , s , ...s n ) = s N ( φ , p ) + s N ( φ , p ) + s N ( φ , p ) ... + s N ( φ n , p n ) N − Ke − rT { − { N ( φ ′ ) + N ( φ ′ ) + N ( φ ′ ) + ... + N ( φ ′ ) }} (12) For time t V { s , s , ...s n } = s N ( φ , p ) + s N ( φ , p ) + s N ( φ , p ) ... + s N ( φ n , p n ) N − Ke − rT { − { N ( φ ′ ) + N ( φ ′ ) + N ( φ ′ ) + ... + N ( φ ′ n ) }} (13)putting in above metric p | v − v | = p s | N ( φ ) − N ( φ ) | + s | N ( φ ) − N ( φ ) | + p s | N ( φ ) − N ( φ ) | ... + s n | N ( φ n ) − N ( φ n ) | + Ke − rt p { N ( φ ′ ) − N ( φ ′ ) + N ( φ ′ ) − N ( φ ′ ) + ... + N ( φ ′ n ) − N ( φ ′ n ) } (14)whereas p | v − v | = √△ v and N ( φ ) = R φ −∞ e − w / dw , So we have N ( φ ) − N ( φ )= √ π R φ −∞ e − w / dw - √ π R φ −∞ e − w / dw . The similar pro-cedure with integrals will continue until the nth difference in normaldistribution functions, N ( φ n ) − N ( φ n ) = ( 1 √ π ) Z φ n −∞ e − w / dw − ( 1 √ π ) Z φ n −∞ e − w / dw p | v − v | = s ( s √ π ) Z φ e − w / dw + ( s √ π ) Z φ e − w / dw + ... + s ( s n √ π ) Z φ n n e − w / dw + 1 + ( Ee − rT √ π ) { Z φ ′ −∞ e − w / dw + Z φ ′ −∞ e − w / dw s + ... + Z φ n ′ −∞ e − w / dw + Z φ n ′ −∞ e − w / dw } (15)After squaring both sides and using triangle inequality It becomes | v − v | ≤ ( s √ π ) Z φ e − w / dw + ( s √ π ) Z φ e − w / dw + ... +( s n √ π ) Z φ n n e − w / dw + 1 + ( Ee − rT √ π ) { Z φ ′ −∞ e − w / dw + Z φ ′ −∞ e − w / dw + ... + Z φ n ′ −∞ e − w / dw + Z φ n ′ −∞ e − w / dw } V ( s , s , ...s n , T ) = max { max ( s , s , ...s n ) − K, } choose x , x , x , ..., x n , y , y , y , ..., y n ∈ R n | V ( x , x , x , ..., x n , t ) − V ( y , y , y , ..., y n , t ) |≤ E | ( x , x , x , ..., x n ) − ( y , y , y , ..., y n ) | F or ( s , s , ...s n ) ≥ V ( s , s , ...s n , T ) = ( s − s − s , ... − s n − K ) , f or ( s , s , ...s n ) ≥ Kand , f or ( s , s , ...s n ) ≤ K = ⇒ | V ( x , x , x , ..., x n , T ) − V ( y , y , y , ..., y n , T ) |≤ E | ( x , x , x , ..., x n ) − ( y , y , y , ..., y n ) | f or ( x , x , ..., x n ) − ( y , y , ...y n ) ≥ Kand , f or ( x , x , ..., x n ) − ( y , y , ...y n ) ≤ K | V ( x , x , x , ..., x n , t ) − V ( y , y , y , ..., y n , t ) | = | ( x , x , x , ..., x n ) − ( y , y , y , ..., y n ) | E=1, While V appears to be short map.4. conclusion
We prove that Multi-dimensional Blac-scholes-Merton model is oneof the dynamic systems. which is in this case Hamilton-Jacobi in n-variables equivalent. we have found various relationships between bothmodels by analyzing the solutions of Black-Scholes-Merton in suitablemetric. Further studies could be conducted as an applications to solvefinancial problems by the methods of dynamics.