Anthony Chen’s quiet character study follows a traumatized Liberian woman (Cynthia Erivo) on a Greek island who befriends an American tour guide (Alia Shawkat).

‘Drift’ Review: Cynthia Erivo Keeps a Breakdown at Bay - The New York Times

Pod of orcas in north Japan freed themselves as gaps between the ice grew, officials say

Killer whales trapped in drift ice off Japan’s Hokkaido coast escape | Japan | The Guardian

The pod of about a dozen killer whales were first spotted stranded on Monday north of Hokkaido.

Trapped orcas escape from drift ice near Japan

An image captured off the Norwegian archipelago of Svalbard shows a young polar bear drifting to sleep on a small iceberg after carving a bed in the ice.

Polar bear sleeping on tiny iceberg drifting in Arctic sea captured in heartbreaking photo | Live Science

A group of more than a dozen orcas has disappeared after being trapped in freezing Japanese waters for about a day – it isn’t known if they escaped or have died

Orca pod disappears after being trapped in drifting ice off Japan | New Scientist

Video posted on social media shows dozens of trapped orca whales escaping after being encircled by drift ice.

Video Shows Dozens Of Trapped Whales Attempt To Escape Drift Ice | The Daily Caller

A pod of killer whales that was trapped in drift ice off Japan's northern main island of Hokkaido, prompting concern from environmental groups, has apparently safely escaped, officials said Wednesday. The killer whales, also known as orcas, were initially spotted by a local fisherman who reported ...

A pod of killer whales trapped in drift ice off northern Japan has apparently safely escaped

The coast guard is unable to rescue them as the ice is too thick, an official told local media.

Orcas gasp for air whilst trapped in drift ice off the coast of Japan

The coast guard is unable to rescue them as the ice is too thick, an official tells local media.

Orcas gasp for air whilst trapped in drift ice off the coast of Japan - BBC News

The struggling orcas appeared to be sticking their heads out of the water to breathe. - Videos from The Weather Channel | weather.com

Drone Footage Captures Killer Whales Trapped in Drift Ice off Japan's Coast; Thick Ice Hampers Rescue Efforts - Videos from The Weather Channel

Driff

driff

Difference Between Real DOM and Virtual DOM: What You Need to Know

iOS vs. Android Monetization: Why Not Understanding the Differences Could Harm Your Business

Welcome to the great crypto bull market of 2024–2025. The “banana zone,” as some call it. You’re here to make a TON of money. There are two ways to do this with crypto: trading and investing. Make sure you know the difference, or it might cost you! Mark, aren’t they the same? Nope. I consulted the dictionary. Trading: buying, selling, or exchanging stocks, bonds, or currency.
Investing: putting assets to use, by purchase or expenditure, in something offering potential profitable returns, such as interest, income, or appreciation in value. Trader’s mindset: cash now, not later Do you have your eye on that amazing memecoin everybody’s talking about? Is your mind racing with possibilities and the potential for huge returns? Are you expecting to cash out in November 2025 and buy some Lambos? If so, you need to trade the market. When you trade, you’re trying to make money from changes in the prices of cryptos. You don’t want crypto. You want to sell it for more of your government’s money. Buying is simply the cost of doing business. This means you will identify entries, exits, stop-losses, and take-profit targets. You may follow the news, read roadmaps and whitepapers, or research key events that make for good buying or selling opportunities. You keep as little money in the market as necessary and wait for momentum to shift. Then, after the market has gone up a little bit, you buy. After the market has gone down a little bit, you sell. While an investor will aim to get ahead of the market, you will wait for the market to act and then ride the momentum whichever way it goes. Investopedia has a nice article about trading. //www.investopedia.com/articles/trading/02/100102.asp?source=post_page-----90fb48ed9223--------------------------------&embedable=true Investor’s mindset: win no matter what happens Do you want generational wealth you can bank on for years to come? Durable financial assets that throw off income and generate capital gains? Chances are, you're an investor. You don't worry as much about what happens today or how much cash you can get out of the market. You want to make sure that you are in a position to take care of yourself for as long as you need to. When you invest, you find projects that will likely grow and succeed, then buy their tokens, stake those tokens for passive income, delegate them for rewards, or deposit them into liquidity pools for a portion of fees and perhaps incentives from using them. When you do this, it’s like earning dividends or interest and recycling the profits back into the market to compound month after month, like a snowball. You can also leverage your tokens in other ways, for example, to buy land or items in a metaverse, participate in play-to-earn games, mint NFTs, become a validator, run a node, and participate in the communities that grow around the tokens. Both make money in different ways You can do well with either approach, but you need to pick one. What drives success in one approach will drive failure in the other. For example, as an investor, you will lock up your tokens for weeks or months for passive rewards and income. Free crypto! That’s terrible if you’re a trader. It defeats the whole point! You don’t want to keep the tokens. You want to sell them as soon as possible! On the other hand, when you lock up your tokens, you have to accept all the volatility that comes your way. You have to sit through big upswings and downswings. Some altcoins will go up or down 50% in a week, sometimes in a day. When you need to sell, it may not be on your terms. As a trader, you will buy after prices go up and sell after prices drop. That may fly in the face of the adage, “buy low, sell high,” but that is not a trading strategy. When you trade, you may buy high to sell higher or sell low and then short the market. As one trader put it, you play for the middle of the field, not the red zone. This makes no sense for an investor. The whole point is to build a position before the market turns so that you can capture limitless upside. Traders can't expose their capital like that. They need to wait until there's less upside because they need clarity about the market's direction. In other words, you can't trade and invest with the same money. Pick one. Options abound Some people split their portfolios into a separate trading account and investment account. Others take an investor’s approach but try to time their entries and exits. You can do well with either approach or another path of your choosing. Whether you’re here to trade or invest, remember: the key to success is knowing your strategy and sticking to it. If it helps, subscribe to my newsletter, Crypto is Easy. I sometimes share trading setups, analysis, and commentary to help you with either approach. We look at trading charts, on-chain data, and various metrics and strategies to navigate the ups and downs so you can make the most of this market and the opportunities it brings. Mark Helfman publishes the Crypto is Easy newsletter. He is also the author of three books and a top Bitcoin writer on Medium and Hacker Noon. Learn more about him in his bio and connect with him on Tealfeed. Welcome to the great crypto bull market of 2024–2025. The “banana zone,” as some call it. You’re here to make a TON of money. There are two ways to do this with crypto: trading and investing. Make sure you know the difference, or it might cost you! Mark, aren’t they the same? Mark, aren’t they the same? Mark, aren’t they the same? Nope. I consulted the dictionary. Trading: buying, selling, or exchanging stocks, bonds, or currency. Investing: putting assets to use, by purchase or expenditure, in something offering potential profitable returns, such as interest, income, or appreciation in value. Trading: buying, selling, or exchanging stocks, bonds, or currency. Investing: putting assets to use, by purchase or expenditure, in something offering potential profitable returns, such as interest, income, or appreciation in value. Trader’s mindset: cash now, not later Do you have your eye on that amazing memecoin everybody’s talking about? Is your mind racing with possibilities and the potential for huge returns? Are you expecting to cash out in November 2025 and buy some Lambos? If so, you need to trade the market. When you trade, you’re trying to make money from changes in the prices of cryptos. You don’t want crypto. You want to sell it for more of your government’s money. Buying is simply the cost of doing business. This means you will identify entries, exits, stop-losses, and take-profit targets. You may follow the news, read roadmaps and whitepapers, or research key events that make for good buying or selling opportunities. You keep as little money in the market as necessary and wait for momentum to shift. Then, after the market has gone up a little bit, you buy. After the market has gone down a little bit, you sell. While an investor will aim to get ahead of the market, you will wait for the market to act and then ride the momentum whichever way it goes. Investopedia has a nice article about trading. //www.investopedia.com/articles/trading/02/100102.asp?source=post_page-----90fb48ed9223--------------------------------&embedable=true //www.investopedia.com/articles/trading/02/100102.asp?source=post_page-----90fb48ed9223--------------------------------&embedable=true Investor’s mindset: win no matter what happens Do you want generational wealth you can bank on for years to come? Durable financial assets that throw off income and generate capital gains? Chances are, you're an investor. You don't worry as much about what happens today or how much cash you can get out of the market. You want to make sure that you are in a position to take care of yourself for as long as you need to. When you invest, you find projects that will likely grow and succeed, then buy their tokens, stake those tokens for passive income, delegate them for rewards, or deposit them into liquidity pools for a portion of fees and perhaps incentives from using them. When you do this, it’s like earning dividends or interest and recycling the profits back into the market to compound month after month, like a snowball. You can also leverage your tokens in other ways, for example, to buy land or items in a metaverse, participate in play-to-earn games, mint NFTs, become a validator, run a node, and participate in the communities that grow around the tokens. Both make money in different ways You can do well with either approach, but you need to pick one. What drives success in one approach will drive failure in the other. For example, as an investor, you will lock up your tokens for weeks or months for passive rewards and income. Free crypto! That’s terrible if you’re a trader. It defeats the whole point! You don’t want to keep the tokens. You want to sell them as soon as possible! terrible On the other hand, when you lock up your tokens, you have to accept all the volatility that comes your way. You have to sit through big upswings and downswings. Some altcoins will go up or down 50% in a week, sometimes in a day. When you need to sell, it may not be on your terms. As a trader, you will buy after prices go up and sell after prices drop. That may fly in the face of the adage, “buy low, sell high,” but that is not a trading strategy. When you trade, you may buy high to sell higher or sell low and then short the market. As one trader put it, you play for the middle of the field, not the red zone. This makes no sense for an investor. The whole point is to build a position before the market turns so that you can capture limitless upside. Traders can't expose their capital like that. They need to wait until there's less upside because they need clarity about the market's direction. In other words, you can't trade and invest with the same money. Pick one. Options abound Some people split their portfolios into a separate trading account and investment account. Others take an investor’s approach but try to time their entries and exits. You can do well with either approach or another path of your choosing. Whether you’re here to trade or invest, remember: the key to success is knowing your strategy and sticking to it. If it helps, subscribe to my newsletter, Crypto is Easy. I sometimes share trading setups, analysis, and commentary to help you with either approach. Crypto is Easy . Crypto is Easy Crypto is Easy We look at trading charts, on-chain data, and various metrics and strategies to navigate the ups and downs so you can make the most of this market and the opportunities it brings. Mark Helfman publishes the Crypto is Easy newsletter. He is also the author of three books and a top Bitcoin writer on Medium and Hacker Noon . Learn more about him in his bio and connect with him on Tealfeed . Crypto is Easy Crypto is Easy Crypto is Easy Medium Medium Medium Hacker Noon Hacker Noon Hacker Noon bio bio Tealfeed Tealfeed

The Difference Between Crypto Trading and Crypto Investing

Authors:
(1) Yuwei Guo, The Chinese University of Hong Kong;
(2) Ceyuan Yang, Shanghai Artificial Intelligence Laboratory with Corresponding Author;
(3) Anyi Rao, Stanford University;
(4) Zhengyang Liang, Shanghai Artificial Intelligence Laboratory;
(5) Yaohui Wang, Shanghai Artificial Intelligence Laboratory;
(6) Yu Qiao, Shanghai Artificial Intelligence Laboratory;
(7) Maneesh Agrawala, Stanford University;
(8) Dahua Lin, Shanghai Artificial Intelligence Laboratory;
(9) Bo Dai, The Chinese University of Hong Kong and The Chinese University of Hong Kong. Table of Links Abstract and 1 Introduction 2 Work Related 3 Preliminary AnimateDiff 4.1 Alleviate Negative Effects from Training Data with Domain Adapter 4.2 Learn Motion Priors with Motion Module 4.3 Adapt to New Motion Patterns with MotionLora 4.4 AnimateDiff in Practice 5 Experiments and 5.1 Qualitative Results 5.2 Qualitative Comparison 5.3 Ablative Study 5.4 Controllable Generation 6 Conclusion 7 Ethics Statement 8 Reproducibility Statement, Acknowledgement and References 3 PRELIMINARY We introduce the preliminary of Stable Diffusion (Rombach et al., 2022), the base T2I model used in our work, and Low-Rank Adaptation (LoRA) (Hu et al., 2021), which helps understand the domain adapter (Sec. 4.1) and MotionLoRA (Sec. 4.3) in AnimateDiff. 4 ANIMATEDIFF This core of our method is learning transferable motion priors from video data, which can be applied to personalized T2Is without specific tuning. As shown in Fig. 2, at inference time, our motion module (blue) and the optional MotionLoRA (green) can be directly inserted into a personalized T2I to constitute the animation generator, which subsequently generates animations via an iterative denoising process. We achieve this by training three components of AnimateDiff, namely domain adapter, motion module, and MotionLoRA. The domain adapter in Sec. 4.1 is only used in the training to alleviate the negative effects caused by the visual distribution gap between the base T2I pre-training data and our video training data; the motion module in Sec. 4.2 is for learning the motion priors; and the MotionLoRA in Sec. 4.3, which is optional in the case of general animation, is for adapting pre-trained motion modules to new motion patterns. Sec.4.4 elaborates on the training (Fig. 3) and inference of AnimateDiff. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Yuwei Guo, The Chinese University of Hong Kong; (2) Ceyuan Yang, Shanghai Artificial Intelligence Laboratory with Corresponding Author; (3) Anyi Rao, Stanford University; (4) Zhengyang Liang, Shanghai Artificial Intelligence Laboratory; (5) Yaohui Wang, Shanghai Artificial Intelligence Laboratory; (6) Yu Qiao, Shanghai Artificial Intelligence Laboratory; (7) Maneesh Agrawala, Stanford University; (8) Dahua Lin, Shanghai Artificial Intelligence Laboratory; (9) Bo Dai, The Chinese University of Hong Kong and The Chinese University of Hong Kong. Authors: Authors: (1) Yuwei Guo, The Chinese University of Hong Kong; (2) Ceyuan Yang, Shanghai Artificial Intelligence Laboratory with Corresponding Author; (3) Anyi Rao, Stanford University; (4) Zhengyang Liang, Shanghai Artificial Intelligence Laboratory; (5) Yaohui Wang, Shanghai Artificial Intelligence Laboratory; (6) Yu Qiao, Shanghai Artificial Intelligence Laboratory; (7) Maneesh Agrawala, Stanford University; (8) Dahua Lin, Shanghai Artificial Intelligence Laboratory; (9) Bo Dai, The Chinese University of Hong Kong and The Chinese University of Hong Kong. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Work Related 2 Work Related 3 Preliminary 3 Preliminary AnimateDiff AnimateDiff 4.1 Alleviate Negative Effects from Training Data with Domain Adapter 4.1 Alleviate Negative Effects from Training Data with Domain Adapter 4.2 Learn Motion Priors with Motion Module 4.2 Learn Motion Priors with Motion Module 4.3 Adapt to New Motion Patterns with MotionLora 4.3 Adapt to New Motion Patterns with MotionLora 4.4 AnimateDiff in Practice 4.4 AnimateDiff in Practice 5 Experiments and 5.1 Qualitative Results 5 Experiments and 5.1 Qualitative Results 5.2 Qualitative Comparison 5.2 Qualitative Comparison 5.3 Ablative Study 5.3 Ablative Study 5.4 Controllable Generation 5.4 Controllable Generation 6 Conclusion 6 Conclusion 7 Ethics Statement 7 Ethics Statement 8 Reproducibility Statement, Acknowledgement and References 8 Reproducibility Statement, Acknowledgement and References 3 PRELIMINARY We introduce the preliminary of Stable Diffusion (Rombach et al., 2022), the base T2I model used in our work, and Low-Rank Adaptation (LoRA) (Hu et al., 2021), which helps understand the domain adapter (Sec. 4.1) and MotionLoRA (Sec. 4.3) in AnimateDiff. 4 ANIMATEDIFF This core of our method is learning transferable motion priors from video data, which can be applied to personalized T2Is without specific tuning. As shown in Fig. 2, at inference time, our motion module (blue) and the optional MotionLoRA (green) can be directly inserted into a personalized T2I to constitute the animation generator, which subsequently generates animations via an iterative denoising process. We achieve this by training three components of AnimateDiff, namely domain adapter, motion module, and MotionLoRA. The domain adapter in Sec. 4.1 is only used in the training to alleviate the negative effects caused by the visual distribution gap between the base T2I pre-training data and our video training data; the motion module in Sec. 4.2 is for learning the motion priors; and the MotionLoRA in Sec. 4.3, which is optional in the case of general animation, is for adapting pre-trained motion modules to new motion patterns. Sec.4.4 elaborates on the training (Fig. 3) and inference of AnimateDiff. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

How Stable Diffusion and LoRA Work

Authors:
(1) Yuwei Guo, The Chinese University of Hong Kong;
(2) Ceyuan Yang, Shanghai Artificial Intelligence Laboratory with Corresponding Author;
(3) Anyi Rao, Stanford University;
(4) Zhengyang Liang, Shanghai Artificial Intelligence Laboratory;
(5) Yaohui Wang, Shanghai Artificial Intelligence Laboratory;
(6) Yu Qiao, Shanghai Artificial Intelligence Laboratory;
(7) Maneesh Agrawala, Stanford University;
(8) Dahua Lin, Shanghai Artificial Intelligence Laboratory;
(9) Bo Dai, The Chinese University of Hong Kong and The Chinese University of Hong Kong. Table of Links Abstract and 1 Introduction 2 Work Related 3 Preliminary AnimateDiff 4.1 Alleviate Negative Effects from Training Data with Domain Adapter 4.2 Learn Motion Priors with Motion Module 4.3 Adapt to New Motion Patterns with MotionLora 4.4 AnimateDiff in Practice 5 Experiments and 5.1 Qualitative Results 5.2 Qualitative Comparison 5.3 Ablative Study 5.4 Controllable Generation 6 Conclusion 7 Ethics Statement 8 Reproducibility Statement, Acknowledgement and References8 Reproducibility Statement, Acknowledgement and References 2 RELATED WORK Text-to-image diffusion models. Diffusion models (Ho et al., 2020; Dhariwal & Nichol, 2021; Song et al., 2020) for text-to-image (T2I) generation (Gu et al., 2022; Mokady et al., 2023; Podell et al., 2023; Ding et al., 2021; Zhou et al., 2022b; Ramesh et al., 2021; Li et al., 2022) have gained significant attention in both academic and non-academic communities recently. GLIDE (Nichol et al., 2021) introduced text conditions and demonstrated that incorporating classifier guidance leads to more pleasing results. DALL-E2 (Ramesh et al., 2022) improves text-image alignment by leveraging the CLIP (Radford et al., 2021) joint feature space. Imagen (Saharia et al., 2022) incorporates a large language model (Raffel et al., 2020) and a cascade architecture to achieve photorealistic results. Latent Diffusion Model (Rombach et al., 2022), also known as Stable Diffusion, moves the diffusion process to the latent space of an auto-encoder to enhance efficiency. eDiff-I (Balaji et al., 2022) employs an ensemble of diffusion models specialized for different generation stages. Personalizing T2I models. To facilitate the creation with pre-trained T2Is, many works focus on efficient model personalization (Shi et al., 2023; Lu et al., 2023; Dong et al., 2022; Kumari et al., 2023), i.e., introducing concepts or styles to the base T2I using reference images. The most straightforward approach to achieve this is complete fine-tuning of the model. Despite its potential to significantly enhance overall quality, this practice can lead to catastrophic forgetting (Kirkpatrick et al., 2017; French, 1999) when the reference image set is small. Instead, DreamBooth (Ruiz et al., 2023) fine-tunes the entire network with preservation loss and uses only a few images. Textual Inversion (Gal et al., 2022) optimize a token embedding for each new concept. Low-Rank Adaptation (LoRA) (Hu et al., 2021) facilitates the above fine-tuning process by introducing additional LoRA layers to the base T2I and optimizing only the weight residuals. There are also encoder-based approaches that address the personalization problem (Gal et al., 2023; Jia et al., 2023). In our work, we focus on tuning-based methods, including overall fine-tuning, DreamBooth (Ruiz et al., 2023), and LoRA (Hu et al., 2021), as they preserve the original feature space of the base T2I. Animating personalized T2Is. There are not many existing works regarding animating personalized T2Is. Text2Cinemagraph (Mahapatra et al., 2023) proposed to generate cinematography via flow prediction. In the field of video generation, it is common to extend a pre-trained T2I with temporal structures. Existing works (Esser et al., 2023; Zhou et al., 2022a; Singer et al., 2022; Ho et al., 2022b,a; Ruan et al., 2023; Luo et al., 2023; Yin et al., 2023b,a; Wang et al., 2023b; Hong et al., 2022; Luo et al., 2023) mostly update all parameters and modify the feature space of the original T2I and is not compatible with personalized ones. Align-Your-Latents (Blattmann et al., 2023) shows that the frozen image layers in a general video generator can be personalized. Recently, some video generation approaches have shown promising results in animating a personalized T2I model. Tune-a-Video (Wu et al., 2023) fine-tune a small number of parameters on a single video. Text2Video-Zero (Khachatryan et al., 2023) introduces a training-free method to animate a pre-trained T2I via latent wrapping based on a pre-defined affine matrix. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Yuwei Guo, The Chinese University of Hong Kong; (2) Ceyuan Yang, Shanghai Artificial Intelligence Laboratory with Corresponding Author; (3) Anyi Rao, Stanford University; (4) Zhengyang Liang, Shanghai Artificial Intelligence Laboratory; (5) Yaohui Wang, Shanghai Artificial Intelligence Laboratory; (6) Yu Qiao, Shanghai Artificial Intelligence Laboratory; (7) Maneesh Agrawala, Stanford University; (8) Dahua Lin, Shanghai Artificial Intelligence Laboratory; (9) Bo Dai, The Chinese University of Hong Kong and The Chinese University of Hong Kong. Authors: Authors: (1) Yuwei Guo, The Chinese University of Hong Kong; (2) Ceyuan Yang, Shanghai Artificial Intelligence Laboratory with Corresponding Author; (3) Anyi Rao, Stanford University; (4) Zhengyang Liang, Shanghai Artificial Intelligence Laboratory; (5) Yaohui Wang, Shanghai Artificial Intelligence Laboratory; (6) Yu Qiao, Shanghai Artificial Intelligence Laboratory; (7) Maneesh Agrawala, Stanford University; (8) Dahua Lin, Shanghai Artificial Intelligence Laboratory; (9) Bo Dai, The Chinese University of Hong Kong and The Chinese University of Hong Kong. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Work Related 2 Work Related 3 Preliminary 3 Preliminary AnimateDiff AnimateDiff 4.1 Alleviate Negative Effects from Training Data with Domain Adapter 4.1 Alleviate Negative Effects from Training Data with Domain Adapter 4.2 Learn Motion Priors with Motion Module 4.2 Learn Motion Priors with Motion Module 4.3 Adapt to New Motion Patterns with MotionLora 4.3 Adapt to New Motion Patterns with MotionLora 4.4 AnimateDiff in Practice 4.4 AnimateDiff in Practice 5 Experiments and 5.1 Qualitative Results 5 Experiments and 5.1 Qualitative Results 5.2 Qualitative Comparison 5.2 Qualitative Comparison 5.3 Ablative Study 5.3 Ablative Study 5.4 Controllable Generation 5.4 Controllable Generation 6 Conclusion 6 Conclusion 7 Ethics Statement 7 Ethics Statement 8 Reproducibility Statement, Acknowledgement and References8 Reproducibility Statement, Acknowledgement and References 8 Reproducibility Statement, Acknowledgement and References8 Reproducibility Statement, Acknowledgement and References 2 RELATED WORK Text-to-image diffusion models. Diffusion models (Ho et al., 2020; Dhariwal & Nichol, 2021; Song et al., 2020) for text-to-image (T2I) generation (Gu et al., 2022; Mokady et al., 2023; Podell et al., 2023; Ding et al., 2021; Zhou et al., 2022b; Ramesh et al., 2021; Li et al., 2022) have gained significant attention in both academic and non-academic communities recently. GLIDE (Nichol et al., 2021) introduced text conditions and demonstrated that incorporating classifier guidance leads to more pleasing results. DALL-E2 (Ramesh et al., 2022) improves text-image alignment by leveraging the CLIP (Radford et al., 2021) joint feature space. Imagen (Saharia et al., 2022) incorporates a large language model (Raffel et al., 2020) and a cascade architecture to achieve photorealistic results. Latent Diffusion Model (Rombach et al., 2022), also known as Stable Diffusion, moves the diffusion process to the latent space of an auto-encoder to enhance efficiency. eDiff-I (Balaji et al., 2022) employs an ensemble of diffusion models specialized for different generation stages. Text-to-image diffusion models. Personalizing T2I models. To facilitate the creation with pre-trained T2Is, many works focus on efficient model personalization (Shi et al., 2023; Lu et al., 2023; Dong et al., 2022; Kumari et al., 2023), i.e., introducing concepts or styles to the base T2I using reference images. The most straightforward approach to achieve this is complete fine-tuning of the model. Despite its potential to significantly enhance overall quality, this practice can lead to catastrophic forgetting (Kirkpatrick et al., 2017; French, 1999) when the reference image set is small. Instead, DreamBooth (Ruiz et al., 2023) fine-tunes the entire network with preservation loss and uses only a few images. Textual Inversion (Gal et al., 2022) optimize a token embedding for each new concept. Low-Rank Adaptation (LoRA) (Hu et al., 2021) facilitates the above fine-tuning process by introducing additional LoRA layers to the base T2I and optimizing only the weight residuals. There are also encoder-based approaches that address the personalization problem (Gal et al., 2023; Jia et al., 2023). In our work, we focus on tuning-based methods, including overall fine-tuning, DreamBooth (Ruiz et al., 2023), and LoRA (Hu et al., 2021), as they preserve the original feature space of the base T2I. Personalizing T2I models. Animating personalized T2Is. There are not many existing works regarding animating personalized T2Is. Text2Cinemagraph (Mahapatra et al., 2023) proposed to generate cinematography via flow prediction. In the field of video generation, it is common to extend a pre-trained T2I with temporal structures. Existing works (Esser et al., 2023; Zhou et al., 2022a; Singer et al., 2022; Ho et al., 2022b,a; Ruan et al., 2023; Luo et al., 2023; Yin et al., 2023b,a; Wang et al., 2023b; Hong et al., 2022; Luo et al., 2023) mostly update all parameters and modify the feature space of the original T2I and is not compatible with personalized ones. Align-Your-Latents (Blattmann et al., 2023) shows that the frozen image layers in a general video generator can be personalized. Recently, some video generation approaches have shown promising results in animating a personalized T2I model. Tune-a-Video (Wu et al., 2023) fine-tune a small number of parameters on a single video. Text2Video-Zero (Khachatryan et al., 2023) introduces a training-free method to animate a pre-trained T2I via latent wrapping based on a pre-defined affine matrix. Animating personalized T2Is. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Text-to-Image Diffusion Models and Personalized Animation Techniques

Differentially Private Data Science - What, Why And How?

 Validator or Node Operator? Understanding Key Advantages and Differences with Ethereum’s DVT

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: 10.1080/381228. [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: //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.);
(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: 10.1080/381228. [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: //eprints.lse.ac.uk/24675/1/dp477.pdf //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.); (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.); (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

Harnessing Finite Difference Methods for Enhanced Option Pricing

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 1.3 FINITE DIFFERENCE Finite difference methods are numerical techniques used to approximate derivatives and solve differential equations by discretizing the domain into a grid of points. This approach is based on Taylor series expansions, where derivatives are expressed as combinations of function values at nearby points. Consider a function 𝑓(𝑥) and its derivative 𝑓′(𝑥) then finite difference approximation of the derivative at a point 𝑥𝑖 is given by: In this plot, the tangent line represents the true derivative 𝑓 ′ (𝑥0) , Meanwhile, the secant line represents the finite difference approximation. As the step size ℎ decreases, the secant line becomes closer to the tangent line, demonstrating the convergence of the finite difference approximation to the true derivative as ℎ approaches zero. The application of finite difference methods in option pricing allows for more accurate simulations of market dynamics and estimation of option prices. These methods enable practitioners to account for various factors affecting option values, such as changes in asset prices, volatility, and interest rates. Finite difference methods offer flexibility and scalability, making them suitable for pricing various types of options and constructing hedging strategies. Despite their computational complexity, advancements in numerical algorithms and computing power have made finite difference methods increasingly accessible and efficient for option pricing applications [4]. Authors:
(1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.);
(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 1.3 FINITE DIFFERENCE Finite difference methods are numerical techniques used to approximate derivatives and solve differential equations by discretizing the domain into a grid of points. This approach is based on Taylor series expansions, where derivatives are expressed as combinations of function values at nearby points. Consider a function 𝑓(𝑥) and its derivative 𝑓′(𝑥) then finite difference approximation of the derivative at a point 𝑥𝑖 is given by: In this plot, the tangent line represents the true derivative 𝑓 ′ (𝑥0) , Meanwhile, the secant line represents the finite difference approximation. As the step size ℎ decreases, the secant line becomes closer to the tangent line, demonstrating the convergence of the finite difference approximation to the true derivative as ℎ approaches zero. The application of finite difference methods in option pricing allows for more accurate simulations of market dynamics and estimation of option prices. These methods enable practitioners to account for various factors affecting option values, such as changes in asset prices, volatility, and interest rates. Finite difference methods offer flexibility and scalability, making them suitable for pricing various types of options and constructing hedging strategies. Despite their computational complexity, advancements in numerical algorithms and computing power have made finite difference methods increasingly accessible and efficient for option pricing applications [4]. Authors: (1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.); (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.); (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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