Luca

luca

So, let’s start! Tell us a bit about yourself. For example, name, profession, and personal interests. Hey Everyone, my name is Lucas Matheus. I’m from Brazil, and I currently work in the technology area at a supercomputing and applied sciences company here in my country. I have interests in a range of topics in human knowledge; I really like culture, music and I consider programming an art. Interesting! What was your latest Hackernoon Top story about? It's called “The Surprising Lessons I've Learned as a Tech Lead,” and it's about my first adventures and learnings leading a project. Do you usually write on similar topics? If not, what do you usually write about? To be honest, it was the first time I chose to write about a topic that deals with soft skills and behaviors that are necessary for a successful project. These were things that I learned the importance of over time and that I ended up enjoying the experience of writing about. I typically focused on writing on more technical topics in business, physics, and programming. Great! What is your usual writing routine like (if you have one?) Good question! I always set aside time on the weekend to think about a topic that I'm really excited to share. I filter what is important to talk about this topic and select the most valuable learning to be exposed to. It's not a very methodical process, but it's very content-based. Depending on the subject, I like to put a context beforehand and explain the reason for writing to make sense to the audience and so as not to appear to be writing just for the sake of writing. Being a writer in tech can be a challenge. It’s not often our main role, but it's an addition to another one. What is the biggest challenge you have when it comes to writing? Beat the blank paper or, in this case, the blank computer screen. Just kidding! Although there is some truth to it and I believe it is a very complicated problem for any writer. My biggest challenge is to end up maintaining the coherence of what is written and ensuring that what was written maintains my identity. Nowadays, with artificial intelligence, which has the ability to produce text faster and maintain authenticity, the essence of writing has become increasingly complicated! In more technical texts, I always try to maintain objectivity and not leave room for other interpretations. A challenge of mine is to always keep the writing as impersonal and accessible in all cases. What is the next thing you hope to achieve in your career? I'm currently focused on being able to share as much knowledge as I've acquired in my 10-year programming career. I feel like it's time to be able to share and learn more from people. I think about evolving in technical leadership, perhaps reaching a C-level in some companies and contributing even more to the technology community around the world. I'm also thinking about improving a project for a social recommendation network that I created with some friends here in Brazil. I hope to end up evolving a lot over the years and helping many people on my journey in the technology area. Wow, that’s admirable. Now, something more casual: What is your guilty pleasure of choice? Ok, ok, I admit that from time to time, I end up reviewing films considered very bad by the general public. I have a list of these movies, and for some reason, I like the 2005 Fantastic Four and that hacker movie with Jonny Lee Miller. Do you have a non-tech-related hobby? If yes, what is it? Well, I really enjoy working out and playing soccer sometimes. Lately, I've been focusing on relaxing a bit with short walks around where I live. I also really like watching movies and playing on the PS4. Every now and then, I end up listening to an album on Spotify from some obscure 2000s rock band. What can the Hacker Noon community expect to read from you next? More stories about my experiences as a tech lead in different projects, more technical articles from the hacker world and perhaps some content more focused on the startup area. I'm already working on some related to meetings and their importance in development (I think some myths should be broken about this practice). What’s your opinion on HackerNoon as a platform for writers? One of the best platforms for sharing knowledge. It's incredible how it was thought of in every detail with a very motivating environment. I felt welcomed by the community and the fact that the article was analyzed. Thank you very much for the opportunity, and I hope to contribute more to this incredible platform Thanks for taking time to join our “Meet the writer” series. It was a pleasure. Do you have any closing words? Thanks a lot! There's a funny story from Brazil: a TV network once ran a show where they had a random guy dressed as an alien. They hyped it up, saying the alien had an important message for everyone. And when the big moment came, the alien simply said: 'Just seek it out. Knowledge.' So, I'll leave you with those words! So, let’s start! Tell us a bit about yourself. For example, name, profession, and personal interests. Hey Everyone, my name is Lucas Matheus. I’m from Brazil, and I currently work in the technology area at a supercomputing and applied sciences company here in my country. I have interests in a range of topics in human knowledge; I really like culture, music and I consider programming an art. Interesting! What was your latest Hackernoon Top story about? It's called “The Surprising Lessons I've Learned as a Tech Lead,” and it's about my first adventures and learnings leading a project. Do you usually write on similar topics? If not, what do you usually write about? To be honest, it was the first time I chose to write about a topic that deals with soft skills and behaviors that are necessary for a successful project. These were things that I learned the importance of over time and that I ended up enjoying the experience of writing about. I typically focused on writing on more technical topics in business, physics, and programming. Great! What is your usual writing routine like (if you have one?) Good question! I always set aside time on the weekend to think about a topic that I'm really excited to share. I filter what is important to talk about this topic and select the most valuable learning to be exposed to. It's not a very methodical process, but it's very content-based. Depending on the subject, I like to put a context beforehand and explain the reason for writing to make sense to the audience and so as not to appear to be writing just for the sake of writing. Being a writer in tech can be a challenge. It’s not often our main role, but it's an addition to another one. What is the biggest challenge you have when it comes to writing? Beat the blank paper or, in this case, the blank computer screen. Just kidding! Although there is some truth to it and I believe it is a very complicated problem for any writer. My biggest challenge is to end up maintaining the coherence of what is written and ensuring that what was written maintains my identity. Nowadays, with artificial intelligence, which has the ability to produce text faster and maintain authenticity, the essence of writing has become increasingly complicated! In more technical texts, I always try to maintain objectivity and not leave room for other interpretations. A challenge of mine is to always keep the writing as impersonal and accessible in all cases. What is the next thing you hope to achieve in your career? I'm currently focused on being able to share as much knowledge as I've acquired in my 10-year programming career. I feel like it's time to be able to share and learn more from people. I think about evolving in technical leadership, perhaps reaching a C-level in some companies and contributing even more to the technology community around the world. I'm also thinking about improving a project for a social recommendation network that I created with some friends here in Brazil. I hope to end up evolving a lot over the years and helping many people on my journey in the technology area. Wow, that’s admirable. Now, something more casual: What is your guilty pleasure of choice? Ok, ok, I admit that from time to time, I end up reviewing films considered very bad by the general public. I have a list of these movies, and for some reason, I like the 2005 Fantastic Four and that hacker movie with Jonny Lee Miller. Do you have a non-tech-related hobby? If yes, what is it? Well, I really enjoy working out and playing soccer sometimes. Lately, I've been focusing on relaxing a bit with short walks around where I live. I also really like watching movies and playing on the PS4. Every now and then, I end up listening to an album on Spotify from some obscure 2000s rock band. What can the Hacker Noon community expect to read from you next? More stories about my experiences as a tech lead in different projects, more technical articles from the hacker world and perhaps some content more focused on the startup area. I'm already working on some related to meetings and their importance in development (I think some myths should be broken about this practice). What’s your opinion on HackerNoon as a platform for writers? One of the best platforms for sharing knowledge. It's incredible how it was thought of in every detail with a very motivating environment. I felt welcomed by the community and the fact that the article was analyzed. Thank you very much for the opportunity, and I hope to contribute more to this incredible platform Thanks for taking time to join our “ Meet the writer ” series. It was a pleasure. Do you have any closing words? Meet the writer Thanks a lot! There's a funny story from Brazil: a TV network once ran a show where they had a random guy dressed as an alien. They hyped it up, saying the alien had an important message for everyone. And when the big moment came, the alien simply said: 'Just seek it out. Knowledge.' So, I'll leave you with those words!

Meet the Writer: Hacker Noon's Contributor Lucas Matheus, Tech Entrepreneur, Tech Lead and Hacker

Discussing the future of finance and technology with Lucas Vogelsang, CEO and Co-Founder at Centrifuge Ishan Pandey: Hi Lucas, welcome to our series “Behind the Startup.” Please tell us about yourself and the story behind ? Centrifuge Lucas Vogelsang: I’ve spent my entire career working in startups, first in web2.0, where I quickly realized the inefficiencies with our current financial system. Finance gives people the power to transform ideas into reality, yet the current system is inefficient, not transparent, and serves the largest corporations at the expense of the small businesses that account for over 90% of companies globally. I was always excited about building technology to solve exciting and hard challenges. When I started diving deeper into Ethereum in 2017, I realized the potential of crypto to build a more open, transparent, and affordable financial system. Centrifuge started in 2018 with a vision to change global trade rules. We were the first protocol to bring real-world assets on-chain with MakerDAO and NewSilver and continue to remain focused on getting the entire world of structured credit on-chain. : Vested Interest Disclosure The author is an independent contributor publishing via our . Be it through direct compensation, media partnerships, or networking, the author has a vested interest in the company/ies mentioned in this story. HackerNoon has reviewed the story for quality, but the claims hereon belong to the author. #DYOR brand-as-author program Ishan Pandey: Please tell us a little bit about Centrifuge and Real-World DeFi. Lucas Vogelsang: Centrifuge connects real-world assets (RWAs) to DeFi - hence Real-World DeFi - creating a fully transparent market which allows borrowers and lenders to transact without unnecessary intermediaries. Ultimately, the protocol aims to lower the cost of borrowing for businesses worldwide, while providing DeFi users with a stable source of collateralized yield that is uncorrelated to the volatile crypto markets. Centrifuge was the first DeFi protocol to bridge RWAs on-chain and has integrated with some of the leading DeFi protocols, including MakerDAO and Aave. Underlying RWAs and liquidity providers fully collateralize centrifuge asset pools and have legal rights to the collateral in the case of a default. This makes Centrifuge unique versus many of our RWA peers that largely focus on undercollateralized lending. NFTs are used as digital representations of the real-world collateral on-chain, allowing liquidity providers to have visibility on the exact assets they finance. The protocol is asset class agnostic and has created liquidity pools for assets such as mortgages, invoices, microlending and consumer finance. Ishan Pandey: Ethereum will switch the cryptocurrency consensus method from Proof-of-Work (PoW) to Proof-of-Stake as it approaches the Merge. What will the post-merge scenario look like for Ethereum at large? Lucas Vogelsang: The switch to proof-of-stake is a great foundational change at the heart of how Ethereum works. A successful merge should be good for the price of ETH in the short term, but the real impact will take place over a much longer timeline. The new block production mechanism will directly impact Ethereum’s underlying infrastructure, especially when considering how the new MEV implications will affect the behaviour of Ethereum’s fee market. The competition for low fees should eventually translate to the benefit of users, but there will be friction as the market takes time to adjust and react to the realities of proof-of-stake. The long-term positive impact on L2s, the inclusion of sharding, and the continued diversity and innovation that the merge will encourage will be exciting. Ishan Pandey: The hoopla surrounding the new technology has been matched by growing worries that NFTs may become the focal point of traditional financial crimes like money laundering and wire fraud given the ongoing popularity of some NFTs and the possibility of their use in the Metaverse and beyond. According to you, will the industry face a regulatory crackdown in light of such issues? Lucas Vogelsang: While we will continue to see enforcement actions for fraud and sanctions violations, we don't expect a general crackdown. Key regulators in the US and EU seem to be bringing parts of the crypto space into a regulated environment. It would not make sense to launch a major crackdown while the regulatory perimeter is being discussed and defined at the legislative level. Ishan Pandey: According to you, what are the best practices that should be followed in relation to cybersecurity in the DeFi industry? Lucas Vogelsang: When designing DeFi protocols, emphasizing simplicity is key. Complex protocols are hard to develop and even harder to audit. For example, upgradeability in Solidity smart contracts is a pattern that adds a layer of indirection and complexity to a protocol, which is difficult to check and thus has led to many exploits. Beyond the design of protocols itself, teams should spend more time ensuring the security of their front end. We have seen many exploits recently, e.g. the Curve Finance exploit hack, that was caused by issues in the security of the hosting or domain management of the frontend. These kinds of supply chain issues are hard to prevent due to the reliance on centralized services such as DNS, but are more regularly becoming targets for blackhats and more time should be spent on preventing these issues. Ishan Pandey: According to you, what are the certain limitations surrounding Dapps and Defi applications running on the ethereum blockchain? Lucas Vogelsang: Any Dapp or DeFi developer knows that gas optimization is the key metric to focus on. This means reducing the amount of state stored on-chain and the number of transactions required for interacting with their dapp or protocol. Unfortunately, this can also limit the possibilities of an app. E.g. this makes many gaming use cases almost impossible on Ethereum, which has led these projects to find other ecosystems to build on. Another limitation is the ability to schedule code automatically. Everything needs to be user-triggered. Being able to run code automatically is a powerful feature that can unlock many new use cases. Ishan Pandey: Though both DeFi and CeFi provide a variety of financial services relating to cryptocurrencies, they are vastly different from one another. Kindly elaborate on some of the major differences for our readers. Lucas Vogelsang: CeFi is opaque as users have no idea how their funds are deployed and need to “trust” a third party. In the last few months, we have seen many CeFi platforms struggle following the Three Arrows Capital (3AC) collapse and reckless lending. DeFi, on the other hand, gives you visibility on the all process without introducing any “trust” assumptions or need to KYC - not your keys not your coins. Ishan Pandey: According to you, what next trends are we going to see in the blockchain and DeFi space? Lucas Vogelsang: 3 trends I am excited about: Structured Products: these are pre-packaged strategies that allow anyone without prior knowledge of finance or derivatives to access new yield opportunities. DeFi has the potential to bring these products to mainstream usage. Cross-chain dapp: It is now possible to create composable cross-chain protocols that are no longer restricted to an ecosystem, thanks to the development of new general messaging passing bridges. Privacy: Tornado crackdown highlighted how vital privacy is - I expect a new surge of privacy projects, particularly those utilizing ZK technology. And, of course, keep an eye on Real-World Assets (RWAs)! Don’t forget to like and share the story!

Understanding the Impact of DeFi on Real Word Assets with Lucas Vogelsang

Learn how Lucas Liew scaled his online course for algorithmic trading from 0 to 30k students. Davis Baer : What’s your background, and what are you working on? Hey guys. I’m Lucas Liew, founder of , an online course for algorithmic trading (i.e. trading using code). Lucas Liew: AlgoTrading101 Essentially, we teach our students how to generate, code and execute their own trading ideas. We have almost 30k students from 160 countries at the moment. What motivated you to get started with your company? I was trading at a proprietary trading firm when I started AlgoTrading101 as a side project. The first version of the idea was to sell code templates for trading robots. Customers can then build their own trading robots by editing these templates. Bad idea. Turns out no one wants to pay for templated code (especially when there is plenty of free code floating around). We then pivoted to an online course for algorithmic trading. There were (and still are) many “get rich quick” trading courses around. Our angle was to appeal to those who were not looking for a quick buck, or who had tried those “get rich quick” schemes and failed. What went into building the initial product? We tried to validate the idea by talking to “users”, aka my friends who were in the finance industry. Their feedback was honest but bad. My friends were genuinely concerned for me pursuing this idea. In hindsight, my friends weren’t my target audience. We should have spoken to those seeking a career change, or college undergraduates looking at trading as a possible career path. We went ahead with the idea anyway. The plan was to launch fast and get feedback from the market directly. Our initial investment was $100; it was a Blue Snowball microphone. The launch was supposed to happen within 3 months, but self-doubt and a massive underestimation of the work required dragged that out to 7 months. It would have been much longer had we decided to perfect the course. We launched (with much hesitation) a scrappy version of our course. We should have launched earlier. Coding and testing a trading robot How have you attracted users and grown your company? We got lucky. We decided to launch on Udemy (an educational course marketplace) instead of our own hosted site. Moreover, we launched right before their massive Black Friday sale. We didn’t even know that they had a Black Friday sale. By leeching off Udemy’s user base, we had sales on our first day of launch. By the end of the first month we had over a thousand students. Thousands of free coupons were given away in the following months for publicity, reviews, and feedback. This helped us become one of the top courses on Udemy for algorithmic trading. What’s your business model, and how have you grown your revenue? We managed to grow our revenue by 10% month-on-month for the first 24 months. We got traction from content marketing and press exposure on other platforms (such as Quora and Investopedia) and SEO. For SEO, it took us about 2 years to rank first for “Learn algorithmic trading”. Unfortunately, revenue plateaued as we let our foot off the gas for SEO and content marketing efforts to look at new products, which didn’t contribute to growth. In those first 24 months, our pricing strategy contributed to our growth. We were raising prices regularly to find our optimal price point. For customers who were on the fence, this tipped them over. Our prices started from $49 (for lifetime access) in 2015 and now it is $570, but we have also added significant amount of paid content during this period. Prices were raised yearly. We usually see double the revenue in the month before a price increase as compared to an average month. If your product doesn’t solve a “hair-on-fire” problem, you will get plenty of resistance at the last mile. Having some sort of urgency to purchase your product will help greatly. What are your goals for the future? We joined Y Combinator’s Start Up School 4 weeks ago. With the guidance of our YC mentor , revenue has been growing at 20% week-on-week. Time to keep the momentum going! Igor The current focus is on SEO and content marketing. After which, we will be looking to create more courses and online tools in nearby verticals, like machine learning for finance. What are the biggest challenges you’ve faced and obstacles you’ve overcome? If you had to start over, what would you do differently? Changing my mentality made a big difference. AlgoTrading101 started out as a side project. I never thought of it as a “real” business until many months after its launch. Once we sat down and decided to focus on this business, we pushed it to grow more than we had ever thought possible. Some things I would do differently: Launch (much) earlier Presale my product Focus on building an email list of happy users Be consistent about content marketing Launch more courses in verticals close to my niche To those of you running a side project, treat it like a real startup! Have you found anything particularly helpful or advantageous? Former VP of Growth at Facebook, Chamath Palihapitiya, once . talked about how they thought of growth at Facebook The framework: Get users through the door Hit them with the “Aha” moment (within seconds) Deliver core product value as often as possible To stand out from other courses, our course had to deliver value as often as possible. Most other coding courses tend to teach serially, starting from tons of nitty-gritty basics, followed by mundane coding exercises. Our aim was for our students to feel good about their achievements within the first day. Thus, we arranged the course in a way so that most of our students could code their first robot in one day. This made them engaged, happy, and motivated enough to push through the harder parts that would come later. What’s your advice for entrepreneurs who are just starting out? Minimise the duration of your feedback loop. You need a fast way to know if what you’re doing works. Put something out there ASAP to gauge interest. Spending 7 months building an MVP without feedback is dangerous (I got lucky). Money is the best validation. Not sign-ups, free trials, or press coverage. When in doubt, charge now and charge more. Where can we go to learn more? To learn more about AlgoTrading101, check out . AlgoTrading101.com This interview is brought to you by , a tool to schedule and automatically repeat your posts on Facebook, Instagram, Twitter, Pinterest, LinkedIn, and Google My Business OneUp

Founder Interviews: Lucas Liew of AlgoTrading101

ADA: A Powerful Data Augmentation Technique for Improved Regression Robustness

Uni-OVSeg: A Step Towards Efficient and Bias-Resilient Vision Systems

Get Rich Building Verticalized AI Wrappers (Even If You Don’t Code)

রিচ বিল্ডিং ভার্টিক্যালাইজড এআই র‍্যাপার পান (এমনকি যদি আপনি কোড না করেন)

Werden Sie reich durch die Entwicklung vertikaler KI-Wrapper (auch wenn Sie nicht programmieren können)

Hazte rico creando contenedores de IA verticalizados (aunque no sepas programar)

Devenez riche en créant des wrappers d’IA verticalisés (même si vous ne codez pas)

वर्टिकलाइज़्ड AI रैपर बनाकर अमीर बनें (भले ही आप कोड न जानते हों)

Menjadi Kaya dengan Membangun Pembungkus AI Vertikal (Bahkan Jika Anda Tidak Membuat Kode)

垂直統合型 AI ラッパーを構築して金持ちになる (コードを書かなくても)

Rich Building вертикалдандырылған AI орауыштарын алыңыз (тіпті кодтамасаңыз да)

수직화된 AI 래퍼를 구축하여 부자가 되세요(코딩하지 않아도)

Баян барилгын босоо AI боодолтой бол (та код бичээгүй байсан ч)

Fique rico construindo wrappers de IA verticalizados (mesmo que você não codifique)

Разбогатейте, создавая вертикальные оболочки ИИ (даже если вы не пишете код)

Hel dhismo qani ah oo toosan AI ku duub (Xitaa Haddi aadan Codayn)

Get Rich Building Verticalized AI Wrappers (Kahit Hindi Ka Mag-code)

Dikey AI Sarmalayıcıları Oluşturarak Zengin Olun (Kodlamasanız Bile)

رچ بلڈنگ ورٹیکلائزڈ AI ریپر حاصل کریں (چاہے آپ کوڈ نہ بھی کریں)

Làm giàu bằng cách xây dựng các trình bao bọc AI theo chiều dọc (ngay cả khi bạn không viết mã)

通过构建垂直化 AI 包装器致富（即使你不会编码）

Table of Links Abstract and 1. Introduction 1.1 Option Pricing 1.2 Asymptotic Notation (Big O) 1.3 Finite Difference 1.4 The Black-Schole Model 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.6 Our Contribution Literature Review
Methodology 3.1 Model Assumption 3.2 Theorems and Model Discussion Result Analysis
Conclusion and References 5. CONCLUSION The research findings have demonstrated that finite difference methods offer a powerful tool for approximating option prices and hedging parameters, especially in cases where closed-form solutions are unavailable or impractical. The asymptotic analysis conducted in this study has provided valuable insights into the convergence properties and accuracy of finite difference approximations, shedding light on the optimal choice of discretization schemes and grid sizes for different option contracts. Furthermore, the combination of finite difference methods and Monte Carlo simulation with variance reduction techniques offers a comprehensive approach to hedge error analysis in option pricing. By leveraging the strengths of both methodologies, financial practitioners can obtain more robust and reliable estimates of option prices and hedge parameters, thereby enhancing their risk management capabilities and decision-making processes. It is important to acknowledge the limitations of this study and areas for future research. While the asymptotic analysis provides valuable theoretical insights, further empirical validation is warranted to assess the robustness of the findings across different market conditions and asset classes. Additionally, exploring alternative variance reduction techniques and incorporating more sophisticated models of market dynamics could yield further improvements in hedge error analysis and option pricing accuracy. this study has advanced our understanding of the factors influencing hedge errors and provided practical tools for mitigating their impact on derivative pricing and risk management. As financial markets continue to evolve, the insights gained from this research will remain valuable for academics, practitioners, and policymakers striving to enhance the efficiency and stability of global financial systems. REFERENCES [1] C. W. Smith, “Option pricing. A review,” J. financ. econ., vol. 3, no. 1–2, pp. 3–51, 1976, doi: 10.1016/0304-405X(76)90019-2. [2] S. Guernsey, “an Introduction To the Black-Scholes Pde Model,” 2013. [3] B. Peeters, C. L. Dert, and A. Lucas, “Black Scholes for Portfolios of Options in Discrete Time,” SSRN Electron. J., no. 03, 2005, doi: 10.2139/ssrn.469022. [4] J. He, “A Study on Analytical and Numerical Solutions of Three Types of Black-Scholes Models,” Int. J. Trade, Econ. Financ., vol. 13, no. 2, 2022, doi: 10.18178/ijtef.2022.13.2.720. [5] A. S. Shinde and K. C. Takale, “Study of BlackScholes model and its applications,” Procedia Eng., vol. 38, pp. 270–279, 2012, doi: 10.1016/j.proeng.2012.06.035. [6] Y. Zhang, “The value of Monte Carlo model-based variance reduction technology in the pricing of financial derivatives,” PLoS One, vol. 15, no. 2, pp. 1–13, 2020, doi: 10.1371/journal.pone.0229737. [7] P. L. Bonate, “A brief introduction to Monte Carlo simulation,” Clin. Pharmacokinet., vol. 40, no. 1, pp. 15–22, 2001, doi: 10.2165/00003088- 200140010-00002. [8] D. R. Tobergte and S. Curtis, “Importance sampling for high speed statistical Monte-Carlo simulations,” J. Chem. Inf. Model., vol. 53, no. 9, pp. 1689–1699, 2013. [9] E. C. Anderson and M. Stephens, “Importance Sampling and (MC)^3,” Sisg 2020, vol. 2, no. July, pp. 1–29, 2020. [10] I. N. Medvedev, “Vector estimators of the Monte Carlo method with a finite variance,” Russ. J. Numer. Anal. Math. Model., vol. 28, no. 3, pp. 231– 244, 2013, doi: 10.1515/rnam-2013-0014. [11] E. Saliby and R. J. Paul, “A farewell to the use of antithetic variates in monte carlo simulation,” J. Oper. Res. Soc., vol. 60, no. 7, pp. 1026–1035, 2009, doi: 10.1057/palgrave.jors.2602645. [12] P. W. Glynn and R. Szechtman, “Some New Perspectives on the Method of Control Variates,” Monte Carlo Quasi-Monte Carlo Methods 2000, pp. 27–49, 2002, doi: 10.1007/978-3-642-56046-0_3. [13] W. Gustafsson, “Evaluating the Longstaff-Schwartz method for pricing of American options,” 2015. [14] L. Andersen and M. Broadie, “Primal-dual simulation algorithm for pricing multidimensional American options,” Manage. Sci., vol. 50, no. 9, pp. 1222–1234, 2004, doi: 10.1287/mnsc.1040.0258. [15] E. Wipplinger, Jim Gatheral: The volatility surface, a practitioner’s guide, vol. 22, no. 1. 2008. doi: 10.1007/s11408-007-0072-4. [16] M. Avellaneda and S. Stoikov, “High-frequency trading in a limit order book,” Quant. Financ., vol. 8, no. 3, pp. 217–224, 2008, doi: 10.1080/14697680701381228. [17] S. Mayoral, D. Moreno, and A. Zareei, “Using a hedging network to minimize portfolio risk,” Financ. Res. Lett., vol. 44, no. April, p. 102044, 2022, doi: 10.1016/j.frl.2021.102044. [18] S. Gayathri Devi, K. Selvam, and S. P. Rajagopalan, “An abstract to calculate big o factors of time and space complexity of machine code,” IET Conf. Publ., vol. 2011, no. 583 CP, pp. 844–847, 2011, doi: 10.1049/cp.2011.0483. [19] M. S. H. Mojumder, M. N. Haque, and M. J. Alam, “Efficient Finite Difference Methods for the Numerical Analysis of One-Dimensional Heat Equation,” J. Appl. Math. Phys., vol. 11, no. 10, pp. 3099–3123, 2023, doi: 10.4236/jamp.2023.1110204. [20] I. R. Khan and R. Ohba, “Taylor series based finite difference approximations of higher-degree derivatives,” J. Comput. Appl. Math., vol. 154, no. 1, pp. 115–124, 2003, doi: 10.1016/S0377- 0427(02)00816-6. [21] S. Danho, “Pricing Financial Derivatives with the Finite Difference Method,” p. 237, 2017. [22] G. Connor and T. Lasarte, “An Introduction to Hedge Funds Strategies. Introductory Guide,” pp. 1–16, 2004, [Online]. Available: http://eprints.lse.ac.uk/24675/1/dp477.pdf [23] M. Nurul Anwar and L. Sazzad Andallah, “A Study on Numerical Solution of Black-Scholes Model,” J. Math. Financ., vol. 08, no. 02, pp. 372–381, 2018, doi: 10.4236/jmf.2018.82024. [24] J. Akahori, F. Barsotti, and Y. Imamura, “Hedging error as generalized timing risk,” Quant. Financ., vol. 23, no. 4, pp. 693–703, 2023, doi: 10.1080/14697688.2022.2154255. [25] R. C. A. Oomen and G. J. Jiang, “Hedging Derivatives Risks - A Simulation Study,” SSRN Electron. J., no. May, 2005, doi: 10.2139/ssrn.302525. [26] K. A. Froot, “Hedging Portfolios with Real Assets,” J. Portf. Manag., vol. 21, no. 4, pp. 60–77, 1995, doi: 10.3905/jpm.1995.409527. [27] H. E. LELAND, “Option Pricing and Replication with Transactions Costs,” J. Finance, vol. 40, no. 5, pp. 1283–1301, 1985, doi: 10.1111/j.1540- 6261.1985.tb02383.x. [28] T. Hayashi and P. A. Mykland, “Evaluating hedging errors: An asymptotic approach,” Math. Financ., vol. 15, no. 2, pp. 309–343, 2005, doi: 10.1111/j.0960-1627.2005.00221.x. [29] J.-L. Prigent, R. Hentati, and A. Kaffel, “Dynamic versus static optimization of hedge fund portfolios: the relevance of performance measures,” Int. J. Bus., vol. 15, no. 156, pp. 1–17, 2010. [30] R. A. Jarrow, D. Lando, and F. Yu, “Default Risk and Diversification : Theory and Empirical Implications * Default Risk and Diversification: Theory and Empirical Implications Abstract,” no. 607, 2003. [31] S. Kim, D. Jeong, C. Lee, and J. Kim, “Finite difference method for the multi-asset black-scholes equations,” Mathematics, vol. 8, no. 3, pp. 1–17, 2020, doi: 10.3390/math8030391. [32] D. Jeong, M. Yoo, and J. Kim, “Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions,” Comput. Econ., vol. 51, no. 4, pp. 961–972, 2018, doi: 10.1007/s10614-017- 9653-0. [33] T. Guo, Black-Scholes Process and Monte Carlo Simulation-Based Options Pricing. Atlantis Press International BV, 2023. doi: 10.2991/978-94-6463- 010-7_75. [34] P. Glasserman, “Monte Carlo Methods in Financial Engineerring,” J. Chem. Inf. Model., vol. 53, no. 9, pp. 1689–1699, 2013. [35] Q. Jiang*, “Comparison of Black–Scholes Model and Monte-Carlo Simulation on Stock Price Modeling,” vol. 109, no. Icemci, pp. 135–137, 2020, doi: 10.2991/aebmr.k.191217.025. [36] E. Derman and E. K, “Stochastic Caclulus for Finance,” pp. 1–63, 2000. Authors:
(1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com);
(2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in);
(3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Abstract and 1. Introduction 1.1 Option Pricing 1.1 Option Pricing 1.2 Asymptotic Notation (Big O) 1.2 Asymptotic Notation (Big O) 1.3 Finite Difference 1.3 Finite Difference 1.4 The Black-Schole Model 1.4 The Black-Schole Model 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.6 Our Contribution 1.6 Our Contribution Literature Review Methodology Literature Review Literature Review Literature Review Methodology Methodology 3.1 Model Assumption 3.1 Model Assumption 3.2 Theorems and Model Discussion 3.2 Theorems and Model Discussion Result Analysis Conclusion and References Result Analysis Result Analysis Result Analysis Conclusion and References Conclusion and References Conclusion and References 5. CONCLUSION The research findings have demonstrated that finite difference methods offer a powerful tool for approximating option prices and hedging parameters, especially in cases where closed-form solutions are unavailable or impractical. The asymptotic analysis conducted in this study has provided valuable insights into the convergence properties and accuracy of finite difference approximations, shedding light on the optimal choice of discretization schemes and grid sizes for different option contracts. Furthermore, the combination of finite difference methods and Monte Carlo simulation with variance reduction techniques offers a comprehensive approach to hedge error analysis in option pricing. By leveraging the strengths of both methodologies, financial practitioners can obtain more robust and reliable estimates of option prices and hedge parameters, thereby enhancing their risk management capabilities and decision-making processes. It is important to acknowledge the limitations of this study and areas for future research. While the asymptotic analysis provides valuable theoretical insights, further empirical validation is warranted to assess the robustness of the findings across different market conditions and asset classes. Additionally, exploring alternative variance reduction techniques and incorporating more sophisticated models of market dynamics could yield further improvements in hedge error analysis and option pricing accuracy. this study has advanced our understanding of the factors influencing hedge errors and provided practical tools for mitigating their impact on derivative pricing and risk management. As financial markets continue to evolve, the insights gained from this research will remain valuable for academics, practitioners, and policymakers striving to enhance the efficiency and stability of global financial systems. REFERENCES [1] C. W. Smith, “Option pricing. A review,” J. financ. econ., vol. 3, no. 1–2, pp. 3–51, 1976, doi: 10.1016/0304-405X(76)90019-2. [2] S. Guernsey, “an Introduction To the Black-Scholes Pde Model,” 2013. [3] B. Peeters, C. L. Dert, and A. Lucas, “Black Scholes for Portfolios of Options in Discrete Time,” SSRN Electron. J., no. 03, 2005, doi: 10.2139/ssrn.469022. [4] J. He, “A Study on Analytical and Numerical Solutions of Three Types of Black-Scholes Models,” Int. J. Trade, Econ. Financ., vol. 13, no. 2, 2022, doi: 10.18178/ijtef.2022.13.2.720. [5] A. S. Shinde and K. C. Takale, “Study of BlackScholes model and its applications,” Procedia Eng., vol. 38, pp. 270–279, 2012, doi: 10.1016/j.proeng.2012.06.035. [6] Y. Zhang, “The value of Monte Carlo model-based variance reduction technology in the pricing of financial derivatives,” PLoS One, vol. 15, no. 2, pp. 1–13, 2020, doi: 10.1371/journal.pone.0229737. [7] P. L. Bonate, “A brief introduction to Monte Carlo simulation,” Clin. Pharmacokinet., vol. 40, no. 1, pp. 15–22, 2001, doi: 10.2165/00003088- 200140010-00002. [8] D. R. Tobergte and S. Curtis, “Importance sampling for high speed statistical Monte-Carlo simulations,” J. Chem. Inf. Model., vol. 53, no. 9, pp. 1689–1699, 2013. [9] E. C. Anderson and M. Stephens, “Importance Sampling and (MC)^3,” Sisg 2020, vol. 2, no. July, pp. 1–29, 2020. [10] I. N. Medvedev, “Vector estimators of the Monte Carlo method with a finite variance,” Russ. J. Numer. Anal. Math. Model., vol. 28, no. 3, pp. 231– 244, 2013, doi: 10.1515/rnam-2013-0014. [11] E. Saliby and R. J. Paul, “A farewell to the use of antithetic variates in monte carlo simulation,” J. Oper. Res. Soc., vol. 60, no. 7, pp. 1026–1035, 2009, doi: 10.1057/palgrave.jors.2602645. [12] P. W. Glynn and R. Szechtman, “Some New Perspectives on the Method of Control Variates,” Monte Carlo Quasi-Monte Carlo Methods 2000, pp. 27–49, 2002, doi: 10.1007/978-3-642-56046-0_3. [13] W. Gustafsson, “Evaluating the Longstaff-Schwartz method for pricing of American options,” 2015. [14] L. Andersen and M. Broadie, “Primal-dual simulation algorithm for pricing multidimensional American options,” Manage. Sci., vol. 50, no. 9, pp. 1222–1234, 2004, doi: 10.1287/mnsc.1040.0258. [15] E. Wipplinger, Jim Gatheral: The volatility surface, a practitioner’s guide, vol. 22, no. 1. 2008. doi: 10.1007/s11408-007-0072-4. [16] M. Avellaneda and S. Stoikov, “High-frequency trading in a limit order book,” Quant. Financ., vol. 8, no. 3, pp. 217–224, 2008, doi: 10.1080/14697680701381228. [17] S. Mayoral, D. Moreno, and A. Zareei, “Using a hedging network to minimize portfolio risk,” Financ. Res. Lett., vol. 44, no. April, p. 102044, 2022, doi: 10.1016/j.frl.2021.102044. [18] S. Gayathri Devi, K. Selvam, and S. P. Rajagopalan, “An abstract to calculate big o factors of time and space complexity of machine code,” IET Conf. Publ., vol. 2011, no. 583 CP, pp. 844–847, 2011, doi: 10.1049/cp.2011.0483. [19] M. S. H. Mojumder, M. N. Haque, and M. J. Alam, “Efficient Finite Difference Methods for the Numerical Analysis of One-Dimensional Heat Equation,” J. Appl. Math. Phys., vol. 11, no. 10, pp. 3099–3123, 2023, doi: 10.4236/jamp.2023.1110204. [20] I. R. Khan and R. Ohba, “Taylor series based finite difference approximations of higher-degree derivatives,” J. Comput. Appl. Math., vol. 154, no. 1, pp. 115–124, 2003, doi: 10.1016/S0377- 0427(02)00816-6. [21] S. Danho, “Pricing Financial Derivatives with the Finite Difference Method,” p. 237, 2017. [22] G. Connor and T. Lasarte, “An Introduction to Hedge Funds Strategies. Introductory Guide,” pp. 1–16, 2004, [Online]. Available: http://eprints.lse.ac.uk/24675/1/dp477.pdf http://eprints.lse.ac.uk/24675/1/dp477.pdf [23] M. Nurul Anwar and L. Sazzad Andallah, “A Study on Numerical Solution of Black-Scholes Model,” J. Math. Financ., vol. 08, no. 02, pp. 372–381, 2018, doi: 10.4236/jmf.2018.82024. [24] J. Akahori, F. Barsotti, and Y. Imamura, “Hedging error as generalized timing risk,” Quant. Financ., vol. 23, no. 4, pp. 693–703, 2023, doi: 10.1080/14697688.2022.2154255. [25] R. C. A. Oomen and G. J. Jiang, “Hedging Derivatives Risks - A Simulation Study,” SSRN Electron. J., no. May, 2005, doi: 10.2139/ssrn.302525. [26] K. A. Froot, “Hedging Portfolios with Real Assets,” J. Portf. Manag., vol. 21, no. 4, pp. 60–77, 1995, doi: 10.3905/jpm.1995.409527. [27] H. E. LELAND, “Option Pricing and Replication with Transactions Costs,” J. Finance, vol. 40, no. 5, pp. 1283–1301, 1985, doi: 10.1111/j.1540- 6261.1985.tb02383.x. [28] T. Hayashi and P. A. Mykland, “Evaluating hedging errors: An asymptotic approach,” Math. Financ., vol. 15, no. 2, pp. 309–343, 2005, doi: 10.1111/j.0960-1627.2005.00221.x. [29] J.-L. Prigent, R. Hentati, and A. Kaffel, “Dynamic versus static optimization of hedge fund portfolios: the relevance of performance measures,” Int. J. Bus., vol. 15, no. 156, pp. 1–17, 2010. [30] R. A. Jarrow, D. Lando, and F. Yu, “Default Risk and Diversification : Theory and Empirical Implications * Default Risk and Diversification: Theory and Empirical Implications Abstract,” no. 607, 2003. [31] S. Kim, D. Jeong, C. Lee, and J. Kim, “Finite difference method for the multi-asset black-scholes equations,” Mathematics, vol. 8, no. 3, pp. 1–17, 2020, doi: 10.3390/math8030391. [32] D. Jeong, M. Yoo, and J. Kim, “Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions,” Comput. Econ., vol. 51, no. 4, pp. 961–972, 2018, doi: 10.1007/s10614-017- 9653-0. [33] T. Guo, Black-Scholes Process and Monte Carlo Simulation-Based Options Pricing. Atlantis Press International BV, 2023. doi: 10.2991/978-94-6463- 010-7_75. [34] P. Glasserman, “Monte Carlo Methods in Financial Engineerring,” J. Chem. Inf. Model., vol. 53, no. 9, pp. 1689–1699, 2013. [35] Q. Jiang*, “Comparison of Black–Scholes Model and Monte-Carlo Simulation on Stock Price Modeling,” vol. 109, no. Icemci, pp. 135–137, 2020, doi: 10.2991/aebmr.k.191217.025. [36] E. Derman and E. K, “Stochastic Caclulus for Finance,” pp. 1–63, 2000. Authors: (1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com); (2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in); (3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). Authors: Authors: (1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com); (2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in); (3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

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