In today's hyperconnected world, where businesses rely heavily on digital communication systems, the security of those systems is of paramount importance. One such system that remains at the core of business communication is email. As per the Radicati Group's study, more than 246 billion emails are sent per day, providing a vast playground for cybercriminals. This elucidates the criticality of email security software for businesses, large or small.

However, like any other business investment, allocating a budget for email security software can pose a significant challenge. It requires a comprehensive understanding of your business's needs, the available options, their cost and benefit analysis, and the overall impact on the organization's financial health. Here's a systematic approach to budgeting for email security software in business.

Firstly, the understanding of 'true cost' is crucial. The true cost of a software is not just its market price. It includes the initial cost of software purchase, cost of implementation, ongoing maintenance cost, and the cost of any potential downtime. Economic theory, specifically, the Total Cost of Ownership (TCO) model, can be borrowed here to calculate the true cost.

Secondly, assessing the 'value' the software would bring to the business. In this context, value equates to the level of security, efficiency of operations and, ultimately, the peace of mind provided by the software. The Cost-Value Ratio, a mathematical concept, can provide an understanding of whether the software's value justifies the cost.

Next comes the 'risk assessment'. This requires the understanding of potential threats and their likelihood. There could be varied threats like phishing, spamming, ransomware, etc., that your email system could be prone to. It is essential to consider the cost of loss if these threats were to materialize without proper security, and calculate the Value at Risk (VaR), a statistical technique used in financial risk management.

After a thorough understanding of cost, value, and risk, it's time to 'compare and contrast'. The cybersecurity market is brimming with numerous email security software options, each with its unique features, pros, and cons. Some may offer advanced threat protection features but may be complex to implement, others might be user-friendly but may lack sophisticated features. It's a barter between usability and security. A comparison of different software based on the already estimated cost-value ratio and VaR will guide towards the most suitable option.

Once we have shortlisted the software options, it's time to 'negotiate'. As simple as it may sound, this step is often overlooked. A good negotiation can significantly bring down the financial burden and make a higher-end software affordable. This is where the poker theory comes into play, where understanding the other party's position and leveraging it can tilt the outcome in your favor.

Lastly, 'planning for the future'. The landscape of cyber threats keeps evolving. Thus, the software purchased should not only suffice current needs but should be scalable to accommodate future requirements. It may involve a higher initial cost but will prevent recurrent expenditure on upgrades or replacements. Here, Game Theory can elucidate the importance of future planning.

In conclusion, budgeting for email security software is an intricate process that requires a deep understanding of not just the software and technology but also mathematical, statistical, and economic theories. It's a strategic decision that calls for a balance between financial investment and cyber risk management. After all, as Benjamin Franklin rightly said, "An ounce of prevention is worth a pound of cure."