Questions & explanations
1. Compare adverse selection and inventory risk as reasons for the bid-ask spread. Which one is more important for very liquid stocks?
For very liquid stocks, like those of large companies with many analysts, adverse selection is low because information is widely available. In such stocks, the spread is mainly driven by order-processing costs and inventory risk. Inventory risk matters because the market maker may have to hold unwanted shares. However, for liquid stocks, inventory risk is also small because the market maker can offset positions easily. So the spread is narrow. For illiquid stocks, adverse selection dominates because there is more private information. Therefore, adverse selection is more important for stocks with high information asymmetry, while inventory risk plays a bigger role in less liquid stocks, but both are smaller when liquidity is high.
2. What are the three main components typically found in yield curve PCA?
The three main components from PCA of yield curves are level, slope, and curvature. The first component, level, is a roughly equal change in all yields, representing a parallel shift of the curve. The second component, slope, shows short-term rates moving opposite to long-term rates, making the curve steeper or flatter. The third component, curvature, affects intermediate maturities differently, making the curve hump-shaped or inverted in the middle. Together, these three factors typically explain over 95% of yield curve movements. Traders use these to understand risk: a 'level' shock affects all bonds, a 'slope' shock affects spread trades, and a 'curvature' shock impacts barbell vs bullet strategies.
3. Why might a real-world credit rating transition matrix not be perfectly Markovian?
Real-world rating transitions often show 'rating momentum' or 'rating drift', meaning a downgraded company is more likely to be downgraded again soon. This violates the Markov property, which says only the current rating matters. Also, business cycles cause transition probabilities to change over time; a recession increases downgrade rates, making the matrix non-stationary. Furthermore, some companies skip ratings (e.g., from AA to BBB) which the matrix may not capture well. Agencies also may revise ratings slowly, introducing autocorrelation. Therefore, simple Markov models are approximations, and practitioners often use more complex models like duration-dependent or regime-switching models.
4. How do you calibrate the Heston model to market option prices?
Calibration means finding model parameters (like initial variance, mean reversion speed, long-run variance, volatility of variance, and correlation) so that the model prices match market option prices as closely as possible. You choose a set of traded options with different strikes and maturities. Then you compute model prices using a pricing formula (like the Heston characteristic function) and compare them to market mid-prices. You adjust the parameters to minimize the difference, often using a least-squares optimization routine. The goal is to get a good fit across all options, especially for liquid ones. After calibration, the model can be used to price other options or manage risk.
5. Give an example of a market pattern that stochastic volatility models can capture.
A key pattern is the volatility smile: for options on the same underlying with the same expiry, implied volatility is higher for deep out-of-the-money puts and calls than for at-the-money options. This is seen in equity index options like the S&P 500. Black-Scholes cannot explain this because it assumes constant volatility. Stochastic volatility models like Heston generate a smile because when volatility is random, options that are far from the money become more valuable relative to at-the-money. Another pattern is the term structure of volatility smiles, where the smile flattens for longer maturities. Stochastic volatility models capture these features and help price exotic options.
6. What is an OIS and how is it used?
OIS stands for Overnight Index Swap. It is a financial contract where two parties agree to exchange interest payments. One party pays a fixed interest rate, and the other pays the average of an overnight reference rate, like the Fed funds rate or SOFR, over the same period. OIS rates are often used as a benchmark for the expected path of short-term interest rates. They are also used to hedge or speculate on changes in monetary policy. Because the overnight rate is very low risk, the OIS rate is seen as a pure measure of interest rate expectations without credit risk. Traders use OIS to measure the health of the banking system: a large gap between OIS and other rates signals stress.
7. How does OAS help compare bonds with different embedded options?
OAS lets you compare bonds with different embedded options by stripping out the option value. If a bond has a call option, its yield is higher to compensate the investor for the risk of early redemption. The OAS is the spread after removing that compensation. So you can compare the OAS of a callable bond with that of a non-callable bond to see which offers better compensation for credit risk alone. A higher OAS means more compensation for default risk and liquidity, not for the option. This helps investors decide which bond is a better value. For instance, if two similar credit bonds have OAS of 100 and 150 basis points, the one with 150 is more attractive for its credit risk.
8. What does option-adjusted spread (OAS) measure that ordinary spread does not?
Option-adjusted spread (OAS) is the constant spread added to the risk-free rate to make the present value of a bond's cash flows equal its market price, after removing the effect of embedded options. Ordinary spread, like yield spread, ignores how future interest rate changes affect the bond's cash flows if it has options. OAS adjusts for the value of the option, so it shows the pure credit spread and liquidity premium. This lets you compare bonds with different options fairly. For example, a callable bond's yield spread might look high, but its OAS could be lower because part of the yield compensates for the call risk. OAS is widely used by traders to find mispriced bonds.
9. How can you identify an arbitrage opportunity between two related securities?
You compare the theoretical price of one security based on another with its actual market price. For example, in put-call parity, if a call option, put option, stock, and bond have a known relationship, any deviation indicates an arbitrage. You calculate the synthetic price: for a call, synthetic call = put + stock - present value of exercise price. If the actual call price differs, you can buy the cheap side and sell the expensive side. Another example is index futures arbitrage: if futures price deviates from the theoretical value (spot price plus carry), you trade the basket of stocks against the futures. Identification requires constant monitoring and fast execution.
10. How does adverse selection directly cause the bid-ask spread to be larger than the market maker's order-processing costs?
The bid-ask spread has three parts: order-processing costs, inventory costs, and adverse selection costs. Adverse selection adds an extra layer because the market maker must protect against losses to informed traders. If there were no adverse selection, the spread would just cover the cost of processing orders and holding inventory. But when there is a risk of trading with someone who knows more, the market maker increases the spread. For instance, in a stock with rumored news, the spread widens even if order-processing costs are the same. Empirical studies show that adverse selection accounts for a large part of the spread, especially for small, less-covered stocks.
11. What is a common criticism of CCAPM?
A common criticism is that the model requires very high risk aversion to explain observed asset returns. For example, the equity risk premium—the extra return of stocks over bonds—is much larger than CCAPM predicts with reasonable risk aversion. This is called the equity premium puzzle. Also, consumption data is measured with error and may not capture investors' true marginal utility. The model assumes that the representative investor's consumption is the same as average consumption, but in reality, wealth is concentrated. These issues make CCAPM less successful in practice, though it remains important for understanding the link between macroeconomy and asset prices.
12. What is reinvestment risk?
Reinvestment risk is the chance that future cash flows from an investment, such as bond coupon payments or the principal at maturity, will have to be reinvested at a lower interest rate than the original yield. This can reduce the total return an investor earns over time. It is especially important for bonds that pay regular coupons, because you need to reinvest those payments to achieve the promised yield. If interest rates fall, you will earn less on reinvested money. Reinvestment risk is higher for bonds with long maturities, high coupons, or callable features. Investors often manage this risk by using strategies like laddering bonds with different maturities.