Why Do Quants like Closed-Form Solutions?
Because quants are quick to compute and simple to grasp.
A Short Example
The Black-Scholes formulas, despite their shortcomings, are frequently used to unintended products due to their simplicity and closed-form structure.
A Deep Dive into Quants
There are various pressures on a quant when it comes to choosing a model. What he’d really like is a model that is
- Robust: Small adjustments in the random process for the underlying do not matter too much.
- Fast: Prices and Greeks must be quickly computed for many reasons to facilitate trade. To avoid losing out to competitors, it’s important to execute trades quickly and manage positions as part of a larger portfolio.
- Accurate:Scientifically speaking, the prices should be reasonable and consistent with historical data. This is not the same as robust.
- Easy To Calibrate: Banks prefer models that mirror the trading pricing of simple contracts.
There is some overlap between these. Fast does not always imply easy to calibrate. Accuracy and robustness may be equivalent, but not usually.
Accuracy is crucial for scientific purposes. The least important is speed. For scientists, the subject of calibration relates to the existence of arbitrage. If you are a hedge fund seeking prop trading chances with vanillas, avoid calibrating. Although robustness is desirable, the financial world may be too turbulent for models to be truly robust.
The practitioner’s ability to price fast is crucial for completing deals and managing risks. When selling exotic contracts, it’s common for sellers to calibrate their prices to match those of vanilla contracts. He is satisfied with a model that is not too inaccurate or sensitive, and has a sufficient profit margin. Practitioners prioritise speed and market adaptability.
The scientist and the practitioner have competing interests. And the practitioner typically wins.
Speed Over Simplicity
And what is faster than a closed-form solution? This is why practitioners prefer closed formats.
Intuitive solutions are more understandable than numerical ones. The Black-Scholes calculations provide a straightforward interpretation of expectations and utilise the cumulative distribution function for Gaussian distributions.
People frequently utilise simple formulas for incorrect products due to their quest for simplicity. Suppose you wish to price certain Asian options using an arithmetic average. In the Black-Scholes model, solving a three-dimensional partial differential equation or running a Monte Carlo simulation is required. Assuming geometric averaging rather than arithmetic averaging, straightforward closed-form solutions are often available. Use them, even if they’re erroneous. These assumptions are likely to be more accurate than others, such as future volatility.
The Closed-Form Mirage: Why Option Pricing Needs More than Formulas
The definition of closed form varies by each interpretation. Is it valid to price an option based on an infinite sum of hypergeometric functions? Some Asian alternatives can be priced this way. Consider a closed form with complex plane integration that requires numerical computation. This is the Heston stochastic volatility model.
Is it worthwhile to spend time searching for closed forms if they are highly valued? Probably not. New goods and pricing models are constantly being developed, yet they are often complex and difficult to address clearly. To tackle these problems, you may need to use numerical methods or approximate them with something comparable. Approximations like Black ’76 offer the highest opportunity of finding closed-form solutions for new products.
Related Readings
- Put-Call Parity: All You Need To Know
- Modelling Approaches in Quantitative Finance: All You Need To Know
- Central Limit Theorem In Finance: Power Your Portfolio Now
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