We look at a special example of the exotic options - the power option - whose payoffs are non-linear functions of the underlying asset price. In addition, we find a new analytical solution for the asymmetric type of power option.
Historical background of the emergence of Lie groups
Study [61] – one of Lie's former colleagues – described him as self-taught1, but he possessed the qualities of one of the most brilliant mathematicians who ever lived. We do this based on the successes it has achieved, especially in areas where other techniques have failed.
Financial Mathematics
History of the Lie group technique in Financial Mathematics . 5
H is said to be a symmetry group of Γ if every element h∈H transforms solutions of Γ to other solutions of Γ. The infinitesimal criterion of invariance under a one-parameter Lie group of point transformations' [8, Theorem 1.2.6, page 17].
Tool chest for application of Lie symmetries in Differential Equations 10
This optimal system provides a non-unique list representing all possible combinations of the subalgebras (and their corresponding symmetries). This method is useful as long as the group classification of the Lie algebra is known.
History on the emergence of Exotic options
Derivation of the power option PDE
The geometric Brownian equation of motion is mainly used because of its flexibility in option pricing and disallowance of negative prices. Using Itˆo's lemma (and substituting equation (1.28)), the price of the power derivative is given by.
Outline
Reed [52,53], Das [18], Hart and Ross [25], and Shimko [55] provide extensive analyzes and highlight these applications of the energy options in more detail. Esser expanded this idea by considering the case of the power factor as a real number.
The Lie group technique
In Section 2.4, we apply the initial condition to the obtained symmetries, which enables us to obtain invariant solutions to the PDE. Once G is known, we can obtain the global form of the transformation by solving the system.
Analysis of the problem by the Lie group approach
Commutator Table 2.2 for these symmetries shows us that the six-dimensional Lie algebra is decomposed into A3,512 ⊕3A1 [48].
Application of the initial conditions
We will now work with Γ on both the left and right sides of the payoff V(S, T) = 0. We see that the result of the LHS is equal to that of the RHS, thereby placing no restrictions on the arbitrary parameters for its solution. This implies that the reduction of the power option will occur through all the symmetries allowed without any restrictions whatsoever.
This optimal system gives us a list representing all relevant possible combinations of the symmetries, which will yield invariant solutions. The decomposition of the Lie algebra of the power option PDE in an Abelian identity. These are constructed by adding arbitrary scalar multiples of the generator {e4} to the subalgebras in step 1.
These are considered trivial as true subalgebras of the optimal system since the Lie bracket operation required to fulfill the commutation property requires more than one element. Since the infinitesimal form of the allowed symmetries is already known, we can obtain the global form by solving some simple ODEs.
V .(2.51) Minimizing the PDE power option via this symmetry and the subsequent ODE solution gives that. Satisfying the payoff V(S, T) =Sβ−K with the initial condition t=T will require arbitrary constants of integration to take values.
Discussion
The graph shows the option value going to zero as we would expect when Sβ ≤K. Despite reducing the parameters, we were still able to determine a clear solution. The chart shows how the value of the energy option increases with the movement of the price of the underlying asset over time.
The most important feature of the Black-Scholes theory is that the payoffs are a linear function of. Power options have a payout structure depending on the price, increasing up to a certain power of the underlying asset at the expiration date. A shortcoming of the previous solutions of the power option problem is the non-provision for a scenario in which σ and r are considered to be time-dependent (and thus constantly changing throughout the life of the option).
Gazizov and Ibragimov [24] were perhaps pioneers in the use of the Lie group technique in solving problems in financial mathematics. First, we note that the power option pays the owner at expiration the value of the stock (raised to a pre-specified power) minus an exercise price - only if the value of the underlying asset is greater than the exercise price.
Brief overview of the Lie group technique
Analysis of the problem by the Lie group approach
The functions sa(t), b(t) and c(t) satisfy the system of equations. 3.22) Since the function α(t) represented in equation (3.18) is now an explicit function of the model parameters, we can solve for b(t) to give.
Application of the initial conditions
The characteristic system and solution obtained
The case r(t) = 12σ(t)2 in equation (3.27) is not investigated due to the financial insignificance of this limitation. According to the definition of the invariants in equations (3.34) and (3.35), the solution of the time-dependent case of the power potential of the PDE will now be given. However, we must be careful because setting b1 = 0 is a trivial solution. 3.40), which is a simpler but still non-trivial solution.
Discussion
We do this based on the success it has achieved over the years, especially in areas where other techniques have failed. During the plenary meeting of the Royal Saxon Society of the Sciences, Friedrich Engel [23] ranked him as one of the leading mathematicians of all time. Twenty-five years later, Eduard Study [61] - one of Lie's former colleagues - described him as an autodidact, but with the qualities of one of the most brilliant mathematicians who ever lived.
To this end, together with Friedrich Engel, he completed the third and final volume of the massive treatise Theorie der Transformationsgruppen [36]. At the end of the nineteenth century, Lie made the profound and far-reaching discovery that all special methods of solving DEs were in fact special cases of a general integration procedure based on the invariance of DEs under a continuous set of symmetries (Here, a symmetry refers to a set of transformations that transforms the set of all solutions of the differential equation into itself.). Regarding his original idea of developing the equivalent theory of Galois theory with DEs, one researcher notes that 'the extraordinary range of applications of Lie groups to DEs in geometry, analysis, physics and engineering over the 40 recent years have revived Lie's original vision in one of the most active and rewarding areas of contemporary research' [44].
It is one of the exotic options (i.e. derivatives that have more complex characteristics than the commonly known products such as the vanilla options). We extended the results to a case of the volatility (σ) and the interest rate (r) changing with time for the terminal case [40].
The methodology of the Lie group approach
Invariance of differential equations
In this work, we try to go beyond previous experiments by looking at a situation where the power parameter (β) varies with time. Subject to Definition 4.2.1, a local group of transformations is called a symmetry of a DE system if the following definition holds. A symmetric group, G, of a system DE is a local group of transformations acting on an open subset M of the space X and U for the system, with the property that whenever u=f(x) is a solution of S, and whenever g·f is defined for g ∈G, then nu=g·f(x) is also a solution of the system.
The invariance of the subset S as defined in Definition 4.2.1 for an n-th system of DEs is explained by the next theorem [43]. Then G is a symmetry group of the system of equations in the sense of definition 4.2.2. 1A subset of an algebraic variety which is itself a variety - where an algebraic variety is defined as a generalization of dimensions of algebraic curves.
Group invariant solutions of differential equations
Symmetry analysis of the Power option PDE
4.11) The function F(S, t) is the solution of the equation and is henceforth not included in further analysis. We now proceed to obtain an explicit form of the general symmetry as none of the software packages could do this.
Application of the initial conditions
The associated Lagrange’s system and solution obtained
On the other hand, applying the initial conditionat =T when the payoff is given by V(S, T) =Sβ−K did not yield any symmetry, so we could not continue this case.
Discussion
In this work, however, we were able to reconstruct and solve these defining equations, allowing us to obtain the explicit nature of the six-parameter symmetry of the time-dependent power option PDE. Having obtained this symmetry, we now impose the terminal condition on it to obtain a symmetry invariant under the initial condition. We proceeded to construct the Lagrange system of this symmetry, which we used in the minimization of the PDE.
The advent of the Lie group technique in Financial Mathematics is indeed further strengthened by this work, by showing not only an alternative method for problem solving, but an algorithmic one that can also be applied to realistic time-dependent financial derivatives. We then discussed the field of Financial Mathematics and the history of the problem solving approach (the Lie group technique) we used. A detailed explanation of the technique followed before the definitions and theorems relevant to the field of Financial Mathematics were presented.
In chapter two we dealt with the case of the power option PDE that has constant model parameters. Chapter three dealt with the power option PDE which has time dependent volatility and interest rate.
Significance of the results
Obtaining new solutions to the time-dependent exotic option PDE via this approach has opened new avenues for solving time-dependent DEs in Financial Mathematics. The possible application of the power parameter as just a real number or a time dependent value, now offers risk hedging traders more options. In some cases, the various software packages for obtaining symmetry have failed to solve the defining equations that follow.
The reconstruction of these equations in this work provides further insight into their solution and it is hoped that the developers of these packages can integrate this technique into their packages. 21] Esser A., Prices in (in)complete markets: structural analysis and applications, (Springer-Verlag, New York 2004). 22] Esser A., Derivatives written based on the strength of the stock price: general valuation principles and application to stochastic volatility models, Swiss Soc.
In: The handbook of Exotic Options (red.. Nelken), (Irwin professional Publishing, Chicago Olver P., Applications of Lie groups to differentiaalvergelijkingen, (Springer–Verlag,.
