Applied Functional Analysis in Finance: Theory, Applications, and a Case Study on Infosys Limited – 2025

Introduction

Functional analysis, a branch of mathematics focusing on infinite-dimensional vector spaces and linear operators, is often considered abstract and theoretical. Yet, its role in modern finance is profound. From derivative pricing models to portfolio optimization and risk management, financial mathematics frequently operates in function spaces where functional analysis offers rigor and structure. (Applied Functional Analysis in Finance)

This blog provides a deep dive into applied functional analysis in finance. We begin with the mathematical foundations (Banach and Hilbert spaces, linear operators), proceed to applications in derivative pricing, portfolio theory, and risk management, and then examine how these methods apply to a real company—Infosys Ltd. The Infosys case study includes option pricing, portfolio risk analysis, foreign exchange hedging, and valuation models, all framed through functional analysis. (Applied Functional Analysis in Finance)

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Mathematical Foundations of Functional Analysis

Financial markets are dynamic systems where asset prices evolve continuously. Modeling such processes requires the use of infinite-dimensional spaces of functions. Functional analysis provides this framework.

Banach Spaces

A Banach space is a complete normed vector space. Many payoff functions of financial instruments naturally belong to Banach spaces, especially LpL^pLp spaces. ∥f∥p=(∫∣f(x)∣pdx)1/p,1≤p<∞\| f \|_p = \left( \int |f(x)|^p dx \right)^{1/p}, \quad 1 \leq p < \infty∥f∥p​=(∫∣f(x)∣pdx)1/p,1≤p<∞

In finance, L1L^1L1 and L2L^2L2 spaces are commonly used for expected values and variance-related calculations. (Applied Functional Analysis in Finance)

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Hilbert Spaces

A Hilbert space is a Banach space with an inner product. The inner product allows us to define angles, orthogonality, and projections—crucial in portfolio optimization. (Applied Functional Analysis in Finance)

For L2L^2L2 functions: ⟨f,g⟩=∫f(x)g(x)dx,∥f∥=⟨f,f⟩\langle f, g \rangle = \int f(x) g(x) dx, \quad \| f \| = \sqrt{\langle f, f \rangle}⟨f,g⟩=∫f(x)g(x)dx,∥f∥=⟨f,f⟩​

Finance uses Hilbert spaces in:

• Mean-variance portfolio optimization.
• Stochastic processes for option pricing.
• Principal component analysis (PCA) in interest rate modeling.

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Linear Operators

A linear operator maps one function to another. In finance, many pricing and risk evaluation models can be framed as linear operators.

If fff is a payoff function and PPP is a pricing operator: P(f)=E[e−rtf(ST)]P(f) = \mathbb{E}[e^{-rt} f(S_T)]P(f)=E[e−rtf(ST​)]

This maps the random payoff into a present value. (Applied Functional Analysis in Finance)

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Applications of Functional Analysis in Finance

1. Derivative Pricing (Applied Functional Analysis in Finance)

Options and futures can be modeled as functions of underlying assets. Functional analysis ensures the existence, uniqueness, and stability of solutions to pricing models.

Black-Scholes PDE

For a European option, the value V(S,t)V(S,t)V(S,t) satisfies: ∂V∂t+12σ2S2∂2V∂S2+rS∂V∂S−rV=0\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} – rV = 0∂t∂V​+21​σ2S2∂S2∂2V​+rS∂S∂V​−rV=0

This PDE solution belongs to a Hilbert space of square-integrable functions.

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2. Portfolio Optimization

Markowitz’s mean-variance framework is naturally framed in Hilbert spaces.

• Portfolio return: Rp=wTμR_p = w^T \muRp​=wTμ
• Portfolio variance: σp2=wTΣw\sigma_p^2 = w^T \Sigma wσp2​=wTΣw

Optimization problem: min⁡wwTΣws.t.wTμ=μp,  wT1=1\min_w w^T \Sigma w \quad \text{s.t.} \quad w^T \mu = \mu_p, \; w^T \mathbf{1} = 1wmin​wTΣws.t.wTμ=μp​,wT1=1

This is essentially an inner product minimization problem.

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3. Risk Management

Risk measures like Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) can be modeled as functionals on Banach spaces.

For CVaR at confidence α\alphaα: CVaRα(X)=inf⁡t∈R[t+11−αE[(X−t)+]]\text{CVaR}_\alpha(X) = \inf_{t \in \mathbb{R}} \left[ t + \frac{1}{1-\alpha} \mathbb{E}[(X-t)^+] \right]CVaRα​(X)=t∈Rinf​[t+1−α1​E[(X−t)+]]

Such formulations provide rigorous optimization tools for downside risk control.

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Company Profile: Infosys Limited

Before moving into the case study, we introduce Infosys Limited, an Indian multinational corporation chosen for our financial modeling. Infosys serves as an excellent subject due to its global scale, financial stability, and exposure to both equity volatility and foreign exchange risks.

Name: Infosys Limited
Formerly: Infosys Technologies Limited
Founded: 1981, Pune, India
Founders: Narayana Murthy, Nandan Nilekani, S. D. Shibulal, K. Dinesh, S. Gopalakrishnan, Ashok Arora, N. S. Raghavan
Headquarters: Bengaluru, India
Type: Public Company
Industry: IT Services, Consulting, Outsourcing
Employees: ~3.3 lakh (2025)
Global Presence: 50+ countries
Major Clients: Fortune 500 companies in banking, insurance, healthcare, retail, and manufacturing

Key Financials (FY 2024–25)

• Revenue: ₹1,56,800 crore (~USD 19 billion)
• Net Profit: ₹25,000 crore (~USD 3 billion)
• Market Capitalization: ~₹6.5 lakh crore (~USD 80 billion)
• Assets: ~₹1,35,000 crore

Stock Market Information

• NSE Ticker: INFY
• BSE Code: 500209
• NYSE ADR: INFY
• ISIN: INE009A01021

Leadership (2025)

• Chairman: Nandan Nilekani
• CEO & MD: Salil Parekh
• CFO: Nilanjan Roy

Business Segments

1. Digital Transformation Services
2. Core IT Services
3. Consulting
4. Outsourcing
5. Products (Finacle, EdgeVerve, Infosys Nia)

Recognition

• Listed in Forbes Global 2000
• One of the World’s Most Ethical Companies
• Pioneer of the Global Delivery Model (GDM)

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Case Study: Applying Functional Analysis to Infosys

1. Option Pricing on Infosys Stock

Assume:

• Current Price S0=₹1,650S_0 = ₹1,650S0​=₹1,650
• Strike Price K=₹1,700K = ₹1,700K=₹1,700
• Time to Maturity T=0.5T = 0.5T=0.5 years
• Risk-free Rate r=6%r = 6\%r=6%
• Volatility σ=22%\sigma = 22\%σ=22%

Using Black-Scholes: C=S0N(d1)−Ke−rTN(d2)C = S_0 N(d_1) – K e^{-rT} N(d_2)C=S0​N(d1​)−Ke−rTN(d2​)

with d1=ln⁡(S0/K)+(r+σ22)TσT,d2=d1−σTd_1 = \frac{\ln(S_0/K) + (r + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}}, \quad d_2 = d_1 – \sigma \sqrt{T}d1​=σT​ln(S0​/K)+(r+2σ2​)T​,d2​=d1​−σT​

Substituting values: d1=0.0786,d2=−0.077d_1 = 0.0786, \quad d_2 = -0.077d1​=0.0786,d2​=−0.077 N(d1)=0.5313,N(d2)=0.4693N(d_1) = 0.5313, \quad N(d_2) = 0.4693N(d1​)=0.5313,N(d2​)=0.4693 C=1650(0.5313)−1700e−0.03(0.4693)≈₹102.3C = 1650(0.5313) – 1700 e^{-0.03}(0.4693) \approx ₹102.3C=1650(0.5313)−1700e−0.03(0.4693)≈₹102.3

👉 The fair price of the Infosys call option is ₹102.3.

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2. Portfolio Risk with Infosys, TCS, and Nifty ETF

Weights:

• Infosys = 50%, TCS = 30%, Nifty ETF = 20%

Expected Returns:

• Infosys = 12%, TCS = 11%, ETF = 10%

Volatilities:

• Infosys = 22%, TCS = 20%, ETF = 15%

Correlations:

• INFY–TCS = 0.75, INFY–ETF = 0.65, TCS–ETF = 0.70

Portfolio return: μp=0.5(0.12)+0.3(0.11)+0.2(0.10)=11.3%\mu_p = 0.5(0.12) + 0.3(0.11) + 0.2(0.10) = 11.3\%μp​=0.5(0.12)+0.3(0.11)+0.2(0.10)=11.3%

Portfolio variance: σp2=0.0321⇒σp=17.9%\sigma_p^2 = 0.0321 \quad \Rightarrow \quad \sigma_p = 17.9\%σp2​=0.0321⇒σp​=17.9%

👉 Diversification reduces risk from 22% (Infosys alone) to 17.9%.

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3. Currency Risk for Infosys

Infosys earns USD 5 billion in 6 months. At INR/USD = 83, INR appreciation of 5% to 78.85 leads to: Loss=(83−78.85)×5 billion=₹2,100 crore\text{Loss} = (83 – 78.85) \times 5 \, \text{billion} = ₹2,100 \text{ crore}Loss=(83−78.85)×5billion=₹2,100 crore

Functional analysis frames this as a random variable in LpL^pLp space, allowing use of CVaR for hedging design.

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4. Equity Valuation (DCF)

Projected Cash Flows (₹ crore):

• Year 1: 26,000
• Year 2: 29,000
• Year 3: 33,000
• Growth = 7% perpetually after Year 3
• Discount rate = 10%

Valuation: P=260001.1+290001.12+330001.13+33000×1.07(0.10−0.07)(1.13)P = \frac{26000}{1.1} + \frac{29000}{1.1^2} + \frac{33000}{1.1^3} + \frac{33000 \times 1.07}{(0.10-0.07)(1.1^3)}P=1.126000​+1.1229000​+1.1333000​+(0.10−0.07)(1.13)33000×1.07​ P≈₹463,889 croreP \approx ₹463,889 \text{ crore}P≈₹463,889 crore

Compared to market cap of ₹650,000 crore, Infosys appears overvalued.

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Conclusion

Functional analysis, though abstract, plays a crucial role in financial mathematics. It provides the framework for option pricing, portfolio theory, and risk management.

By embedding Infosys’s financials into this framework, we demonstrated:

• Option pricing operators in Hilbert spaces.
• Portfolio risk minimization via inner product structures.
• Currency hedging using CVaR in Banach spaces.
• Equity valuation as a functional operator on infinite cash flows.

Infosys thus serves as a living laboratory where advanced mathematics meets practical corporate finance. As financial systems grow more complex, functional analysis will remain central in bridging theory and practice.