Question 1

What is the difference between a call option and a put option?

Accepted Answer

A call option gives the holder the right to buy an asset at a specified strike price before expiration. A put option gives the holder the right to sell. The call buyer profits when the asset rises above the strike; the put buyer profits when it falls below. Both have limited downside (the premium paid) and are used for speculation and hedging.

Question 2

What is the difference between a futures contract and a forward contract?

Accepted Answer

Both are agreements to buy/sell an asset at a future date for a specified price. Key differences: futures are standardized and exchange-traded, while forwards are customized and traded over-the-counter. Futures have daily mark-to-market settlement (margin calls), while forwards settle at expiration. Futures have no counterparty risk (cleared by exchange), while forwards carry credit risk. Due to daily settlement and interest rate effects, futures and forward prices can differ slightly.

Question 3

Explain what DV01 means and why it is important for a bond trader.

Accepted Answer

DV01 (Dollar Value of a Basis Point) is the change in a bond's price for a 1 basis point (0.01%) change in yield. For example, if a bond has a DV01 of $500, its price changes by $500 for every basis point move in rates. It is the primary risk metric for bond traders because it allows them to compare risk across different bonds, calculate hedge ratios, and set position limits. A trader hedging a portfolio will match the DV01 of the hedge to the DV01 of the position.

Question 4

You have two bonds: Bond A has duration 5 and Bond B has duration 10. Interest rates rise by 50 basis points. Which bond loses more value, and approximately how much does each lose?

Accepted Answer

Bond B loses more. The approximate percentage price change is -Duration × Δy. Bond A: -5 × 0.005 = -2.5%. Bond B: -10 × 0.005 = -5.0%. Bond B loses approximately twice as much in percentage terms because it has twice the duration. This is why longer-duration bonds are riskier in a rising rate environment. Convexity would make the actual losses slightly less than these linear approximations.

Question 5

What is the Sharpe ratio and what are its limitations?

Accepted Answer

The Sharpe ratio is (R_p - R_f) / σ_p, where R_p is portfolio return, R_f is the risk-free rate, and σ_p is portfolio standard deviation. It measures risk-adjusted return per unit of total risk. Limitations: it assumes returns are normally distributed (ignores skewness and kurtosis), treats upside and downside volatility equally, is sensitive to the time period chosen, can be gamed by strategies that have infrequent large losses (like selling options), and is only meaningful for comparing strategies with similar return distributions.

Question 6

A market maker quotes a bid of $99.50 and an ask of $100.50 on a stock. What is their expected profit per round-trip trade, and what risks do they face?

Accepted Answer

The spread is $1.00, so the expected profit per round-trip (buy at bid, sell at ask) is $1.00 per share. However, the market maker faces adverse selection risk: informed traders tend to trade when they know the price will move against the market maker. If the true value shifts to $101 before the maker can adjust quotes, they sell at $100.50 and lose $0.50. Other risks include inventory risk (holding a large position that moves against them), volatility risk, and competition from other market makers narrowing the spread.

Question 7

What is the Black-Scholes formula used for and what are its key assumptions?

Accepted Answer

Black-Scholes is used to price European-style options. Key assumptions: the stock price follows geometric Brownian motion with constant volatility, no dividends, continuous trading is possible, no transaction costs, the risk-free rate is constant, and markets are efficient (no arbitrage). The formula gives the theoretical price as a function of spot price, strike, time to expiry, volatility, and risk-free rate. In practice, most assumptions are violated (volatility is not constant, leading to the smile/skew), but the framework remains foundational for understanding options pricing and hedging.

Question 8

Explain the concept of convexity in bonds. Why do investors care about it?

Accepted Answer

Convexity measures the curvature of the price-yield relationship (the second derivative). Duration gives a linear approximation of price changes, but the actual relationship is curved. Positive convexity means the bond gains more when yields fall than it loses when yields rise by the same amount. Investors prefer positive convexity because it provides a better risk-return profile. Bonds with higher convexity are worth more, all else equal. Mortgage-backed securities can have negative convexity due to prepayment risk, which is why they trade at a spread premium.

Question 9

You are offered a bet: 60% chance of winning $100 and 40% chance of losing $100. The Kelly criterion says you should bet what fraction of your bankroll?

Accepted Answer

The Kelly criterion for even-money bets is f* = 2p - 1, where p is the probability of winning. Here f* = 2(0.6) - 1 = 0.20, so you should bet 20% of your bankroll. More generally, Kelly = (bp - q) / b, where b is the odds (here b = 1), p = 0.6, q = 0.4. Kelly maximizes long-run geometric growth rate of wealth. In practice, many traders use fractional Kelly (e.g., half-Kelly) to reduce variance at the cost of slightly lower expected growth.

Question 10

What is a yield curve inversion and why do economists pay attention to it?

Accepted Answer

A yield curve inversion occurs when short-term interest rates exceed long-term rates, making the yield curve slope downward. Normally, longer maturities pay higher yields to compensate for duration risk. An inversion suggests the market expects future rate cuts, typically because it anticipates an economic slowdown. Historically, yield curve inversions (particularly the 2-year vs 10-year Treasury spread going negative) have preceded most US recessions, making it one of the most watched recession indicators.

Question 11

Explain the difference between systematic and idiosyncratic risk. How does diversification affect each?

Accepted Answer

Systematic risk (market risk) affects the entire market — things like interest rate changes, recessions, or geopolitical events. It cannot be diversified away. Idiosyncratic risk is specific to a single company or sector — like a CEO resignation or a product recall. Diversification reduces idiosyncratic risk by spreading exposure across many uncorrelated assets. As you add more stocks to a portfolio, idiosyncratic risk approaches zero, but systematic risk remains. This is why the market only compensates investors for bearing systematic risk (CAPM beta).

Question 12

Explain put-call parity and why it must hold.

Accepted Answer

Put-call parity states: C - P = S - K·e^(-rT), where C is the call price, P the put price, S the spot price, K the strike, r the risk-free rate, and T time to expiry. It must hold because otherwise you could construct a risk-free arbitrage by buying the cheap side and selling the expensive side. For example, if C - P > S - K·e^(-rT), you sell the call, buy the put and the stock, and borrow K·e^(-rT), locking in a risk-free profit.

Question 13

A stock pays no dividends and is currently at $100. The 1-year risk-free rate is 5%. What is the 1-year forward price? Now suppose it pays a $2 dividend in 6 months. What is the forward price?

Accepted Answer

Without dividends: F = S·e^(rT) = 100·e^(0.05) ≈ $105.13. With a $2 dividend in 6 months: F = (S - PV(div))·e^(rT) = (100 - 2·e^(-0.05×0.5))·e^(0.05) = (100 - 1.951)·1.0513 ≈ 98.049 × 1.0513 ≈ $103.07. The dividend reduces the forward price because the forward holder does not receive the dividend, so the cost of carry is lower.

Question 14

A stock is trading at $100. A call option with strike $100 expiring in one year costs $10. The risk-free rate is 5%. What should the put with the same strike and expiry cost?

Accepted Answer

Using put-call parity: P = C - S + K·e^(-rT) = 10 - 100 + 100·e^(-0.05) = 10 - 100 + 95.12 = $5.12. The put should cost approximately $5.12.

Question 15

What are the Greeks in options trading? Explain delta, gamma, theta, and vega.

Accepted Answer

Delta measures the option price sensitivity to a $1 move in the underlying. Gamma measures how delta changes as the underlying moves (second derivative of price). Theta measures time decay — how much the option loses per day as expiration approaches. Vega measures sensitivity to a 1% change in implied volatility. A market maker manages all four: delta-hedging to stay neutral, monitoring gamma for large moves, collecting theta, and managing vega exposure.

Question 16

You are long 100 shares of a stock at $50. How would you delta-hedge using options?

Accepted Answer

Being long 100 shares means you have a delta of +100. To delta-hedge, you need -100 delta. You could buy put options: if each put has a delta of -0.50, you need 200 puts (200 × -0.50 = -100). Alternatively, sell call options: if each call has delta +0.50, selling 200 calls gives -100 delta. The hedge must be rebalanced as delta changes with the stock price (gamma effect).

Question 17

What is implied volatility and how does it differ from historical volatility?

Accepted Answer

Historical volatility measures the actual past price fluctuations of an asset, calculated from observed returns. Implied volatility is the market's forward-looking expectation of volatility, derived from current option prices using a pricing model like Black-Scholes. Implied volatility is typically higher than historical because it includes a risk premium. When implied vol is significantly higher than realized vol, options are considered expensive.

Question 18

Explain the concept of no-arbitrage pricing. Why is it fundamental to derivatives pricing?

Accepted Answer

No-arbitrage pricing assumes that in efficient markets, there are no risk-free profits available. If two portfolios have identical future payoffs in all states of the world, they must have the same price today. If they don't, arbitrageurs will buy the cheap one and sell the expensive one, forcing prices to converge. This principle underlies all derivatives pricing: the Black-Scholes formula, put-call parity, and forward pricing all follow from the assumption that arbitrage opportunities cannot persist.

Question 19

A forward contract on a non-dividend-paying stock is priced at $105, the stock is at $100, and the risk-free rate is 3% per year. The forward expires in one year. Is there an arbitrage opportunity?

Accepted Answer

The fair forward price is S·e^(rT) = 100·e^(0.03) ≈ $103.05. The forward is priced at $105, which is too expensive. Arbitrage: borrow $100, buy the stock today, and sell the forward at $105. In one year, deliver the stock for $105, repay the loan of $103.05, and pocket $1.95 risk-free profit.

Question 20

What is the volatility smile and what does it tell us about market expectations?

Accepted Answer

The volatility smile is the pattern where implied volatility is higher for deep in-the-money and deep out-of-the-money options compared to at-the-money options. It contradicts Black-Scholes, which assumes constant volatility. The smile tells us that the market assigns higher probabilities to extreme moves (fat tails) than a normal distribution would predict. In equity markets, the smile is often a skew (higher IV for puts), reflecting demand for downside protection and the empirical observation that markets crash more than they rally.

Quantitative Finance

Quantitative Finance

Quantitative Finance

Quantitative Finance