Questions & explanations
1. Let M_t = W_{t^2}, where W is a Brownian motion. Is M a time-changed Brownian motion according to the Dambis-Dubins-Schwarz theorem?
No, not directly. M_t is not a local martingale with respect to its own filtration. It is a Brownian motion run at a deterministic time change, but the Dambis-Dubins-Schwarz theorem applies to continuous local martingales. M_t is not a martingale because the time change t^2 is not adapted to the filtration of M? Actually, M_t = W_{t^2} is a martingale if we use the natural filtration of W? Check: E[W_{t^2}|F_s] = W_{s^2} for s^2 < t^2? Only if we condition on the original filtration, but M_t is not a martingale in its own filtration because the time change is deterministic. So the theorem does not apply directly. However, if we define the time change τ(t)=t^2, then W_{τ(t)} is a time-changed Brownian motion, but it is not a martingale (it's a process with deterministic variance). So the answer is that M is not a continuous local martingale, so the theorem does not apply.
2. How do HFT firms balance cost and speed when choosing data transmission technology?
HFT firms decide on transmission technology by comparing cost and speed benefits. Microwave links are very fast but also very expensive to install and maintain. They are worth it only if the extra speed leads to enough profit. Many firms start with fiber optics, which is cheaper and already installed in many places. If a firm sees that being first to trade gives it a big edge, it may invest in microwave or even laser links. The firm must also consider that competitors upgrade their own technology, so the advantage may not last. Some firms build hybrid systems that use microwave for the fastest route and fiber for backup. Others focus more on better trading algorithms rather than faster hardware. Overall, the choice depends on the firm's budget, the market they trade, and how much a millisecond is worth to them.
3. What is the difference between calibration and fitting?
Calibration and fitting are often used interchangeably, but there is a subtle difference. Fitting generally means adjusting a model to make it match data as closely as possible. Calibration is a specific type of fitting where the model's parameters are set so that the model matches observed market prices exactly for a set of instruments. In finance, calibration usually implies that we match market prices of liquid options, and then use the calibrated model to price other, less liquid derivatives. Fitting can refer to any statistical adjustment, like fitting a curve to data points. Calibration often involves minimizing a pricing error function. Both aim to make the model realistic, but calibration is more focused on consistency with market quotes for risk management and arbitrage-free pricing.
4. Compare HFT activity in Germany (Xetra) and the Netherlands (Euronext) under MiFID II.
Both Xetra and Euronext are major European exchanges that host a lot of HFT activity. Xetra, run by Deutsche Börse, is known for its fast matching engine and colocation services. Under MiFID II, Xetra requires HFT firms to have frequent testing and risk controls. Euronext, based in Amsterdam, also offers colocation and has a similar regulatory approach. One difference is that Xetra has a longer history with HFT and a larger market for German stocks. Euronext covers multiple countries like France, Belgium, and the Netherlands. Both exchanges follow MiFID II rules, so the broad regulations are the same. However, Xetra's fee structure for high-volume traders may be slightly more expensive. Overall, HFT firms operate on both but may choose based on the stocks they want to trade.
5. How does the risk of market manipulation compare in emerging markets versus developed markets for HFT?
Market manipulation is a bigger risk in emerging markets because rules are weaker and supervision is less strict. In developed markets like the US or Europe, regulators use advanced tools to catch manipulation like spoofing or layering. In emerging markets, such tricks may go unnoticed for longer. For example, a large trader might place fake orders to trick HFT algorithms into moving prices, then cancel them. HFT firms in emerging markets may also face 'front running' where insiders see their orders first. To protect themselves, HFT firms use algorithms that detect unusual patterns and avoid suspicious trades. Some firms may even choose to stay out of certain emerging markets because the risk is too high. So, while profits can be good, the danger of being cheated is greater.
6. Consider the SDE dX_t = σ(X_t) dW_t with σ bounded and strictly positive, but not Lipschitz. Does Yamada-Watanabe guarantee a unique strong solution?
Not automatically. Yamada-Watanabe requires pathwise uniqueness. For this SDE, pathwise uniqueness may fail if σ is not Lipschitz or satisfies another condition like the Yamada-Watanabe condition. For example, if σ(x)=√x, then pathwise uniqueness holds? Actually, the SDE dX = √X dW has a unique solution? There is known counterexample: dX = |X|^α dW with α < 1/2 gives pathwise uniqueness? The Yamada-Watanabe condition ensures uniqueness when the diffusion coefficient has a modulus of continuity like |x-y| times a function that integrates to infinity near zero. If σ is bounded and strictly positive but only continuous, uniqueness may not hold. So the answer is that we need to check pathwise uniqueness; Yamada-Watanabe only provides strong existence if uniqueness holds.
7. Give an example of a parameter you might calibrate besides volatility.
In models more complex than Black-Scholes, other parameters may be calibrated. For example, in the Heston stochastic volatility model, we calibrate parameters like the long-term mean volatility, the speed of mean reversion, and the correlation between asset returns and volatility changes. Another example is the jump-diffusion model, where we calibrate the jump intensity (how often jumps occur) and jump size distribution. Similarly, in local volatility models, we calibrate a whole function of volatility against asset price and time. By calibrating these extra parameters, the model can better match market prices, especially for options with different strikes and maturities. This improves pricing and hedging performance compared to using fixed theoretical values.
8. Give an example of a regulatory difference between Western and emerging option markets.
In the United States, options are traded on centralized exchanges like the CBOE (Chicago Board Options Exchange) with standardized contracts. Many emerging markets, like India, also have standardized exchange-traded options (e.g., NSE options). However, some emerging markets allow only over-the-counter (OTC) options, which are private contracts between two parties. OTC options have less transparency and can be customized. Another difference is in settlement: Western markets often settle options cash or physically, but emerging markets may have different settlement cycles. Also, margin rules can be stricter. For example, China requires upfront margin for option sellers, while the US allows portfolio margining. These differences affect pricing and liquidity.
9. How does the volatility smile affect calibration?
The volatility smile is the pattern that implied volatility varies with strike price and time to expiry. This violates the Black-Scholes assumption of constant volatility. When calibrating, we cannot use a single volatility for all options. Instead, we often fit a whole volatility surface, where each option has its own implied volatility. The calibration process must capture the smile to match market prices. For example, we might use a model like SABR (stochastic alpha, beta, rho) that can produce a smile. The challenge is to find parameters that make the model's implied volatility curve fit the observed one. If we ignore the smile, calibration will be poor and prices inaccurate. Therefore, calibration methods must be flexible enough to handle the smile.
10. What happens if the model is poorly calibrated?
If the model is poorly calibrated, its prices will differ from market prices. This can lead to wrong trading decisions and risk mispricing. For example, a trader might buy an option thinking it is cheap, but it is actually fairly priced relative to the model's mistake. Poor calibration also affects hedging: the model may suggest wrong sensitivities (Greeks), so hedges do not work well. This increases risk. In extreme cases, it can cause losses if the market moves against the model's assumptions. Additionally, a poorly calibrated model may leave arbitrage opportunities unexploited. Therefore, it is critical to test calibration quality and ensure the model fits the data well. Regularly recalibrating to new market data helps keep the model accurate.
11. Why are satellites less commonly used for HFT than microwave or fiber optic links?
Satellites are not common for HFT because their signals take much longer to travel. A satellite in low earth orbit still has a delay of about 10-20 milliseconds for a round trip. This is much slower than microwave or fiber (which can be under 10 milliseconds). Also, satellite signals are affected by weather, can be jammed, and need special dishes. The cost of launching and maintaining satellites is very high. Another issue is that HFT firms want the shortest possible path, but satellites have to go up and down, adding distance. Some firms use satellites for data that is not time-sensitive, like market analysis. For trade orders, speed is king, so microwave and fiber are better choices. In short, satellites are too slow for most HFT strategies.
12. Why might liquidity be lower in emerging market options?
Liquidity is lower because there are fewer market participants. Emerging markets often have less developed financial systems, so fewer institutions trade options. Local investors may be less familiar with options, reducing demand. Additionally, the underlying assets (like stocks) themselves may have lower liquidity, making it hard to hedge. Foreign investors may be hesitant due to currency risk or regulatory uncertainty. This lower participation means fewer orders, so it takes longer to find a counterparty. As a result, bid-ask spreads are wider, making trading expensive. Low liquidity also makes it risky for market makers to provide quotes, further reducing liquidity. These factors combine to create a less liquid options market.