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.

Question 21

You have purchased a call option on an underlying asset, denoted as $ S $, with its current price set at $ S_0 = \$5 $. The call option has an initial price of $ C_0 = \$0.3 $ and a strike price of $ K = \$25 $. The option is set to expire in an hour and has a charm value of $-0.04$. Assuming the underlying asset's price decreases by $\$0.5$ in the next minute, what would be an approximate price for the option?

Accepted Answer

To approximate the option's price, we first recognize that for out-of-the-money (OTM) options nearing expiry, both Delta ($\Delta$) and Gamma ($\Gamma$) are close to 0. The charm, in this case $-0.04$, indicates the rate of decay in $\Delta$. This rate suggests that $\Delta$ is already very near 0 and would decrease by 0.04 over the next hour, reaching 0 if the price remains static. Therefore, we can approximate $\Delta$ as 0.04, given the extremely short expiry period.

Since $\Gamma$ is approximately 0, a further move away from the strike price would mean the option price is expected to react to any change in the underlying by the $\Delta$ times that amount, which in this case is $\$0.5$. Hence, the new price of the option can be estimated as $ \$0.3 - 0.04 	imes  \$0.5 = \$0.28 $.

Question 22

Claire and Jeff have contrasting beliefs about the future of gold prices, driven by potential geopolitical events.
Claire believes that the uncertainty in global markets will drive the price of gold upwards, but she is also wary of a potential sudden price drop. To capitalize on her belief but also to protect herself, she decides to buy a gold futures contract and simultaneously purchase a put option on gold with a strike price of $1,800 per ounce. She pays a premium of $50 for this option.
Jeff believes the opposite. He thinks the geopolitical events will be resolved without major disruptions and thus, the price of gold will stay fairly stable but with a potential for a minor uptick. He decides to sell/write a gold futures contract and simultaneously sell a call option with a strike price of $1,850 per ounce, receiving a premium of $40.
Fast forward six months, the geopolitical events escalate, causing gold prices to spike to $1,900 per ounce, but then they resolve rapidly, leading gold to settle at $1,820 per ounce.
Who made a profit?
How did the futures and options positions impact their overall performance?

Accepted Answer

For Claire:

Claire went long on a gold futures contract. As the price increased from an assumed $1,800 to $1,820 per ounce, she secured a gain of $20 per ounce on the futures. Additionally, she purchased a put option with a strike of $1,800 for a $50 premium. Since the final gold price of $1,820 is above the put's strike, the option expired worthless, resulting in a loss of the premium. Therefore, her net result is a loss of $30 per ounce ($50 premium cost minus $20 futures gain).

For Jeff:

Jeff shorted a gold futures contract. With the price rising from $1,800 to $1,820 per ounce, he incurred a $20 loss per ounce on the futures. He also sold a call option with a $1,850 strike, collecting a $40 premium. Despite a brief spike in gold prices, the call option ultimately expired worthless at the final price of $1,820 because it was not in the money at settlement. Thus, Jeff’s net outcome was a gain of $20 per ounce (the $40 premium offset by the $20 loss on the futures).

Summary:

Claire ended up with a net loss because her protective put option did not provide any benefit at the final price level, even though she was correct about the upward movement. In contrast, Jeff managed to profit as the premium he earned from selling the call option more than offset the loss from his futures position.

Question 23

Consider a European call on a stock that will begin paying dividends before expiration. The option is in the money with 90 days to expiry. Initially, the risk-free rate is 4% and implied volatility is 30%. Now the company’s dividend initiation becomes known, the risk-free rate rises to 6%, and implied volatility falls to 20%. Qualitatively, how should the call’s value change, and why?

Accepted Answer

When you buy a call option, you're acquiring the right to purchase the underlying stock at a fixed strike price at some point in the future. If interest rates rise, the present value of that future payment decreases. In other words, the strike price you'll need to pay later is worth less in today's dollars. Since you're effectively getting a discount on the future cash outflow in present value terms, the call option becomes more valuable to you today. This is similar to how any future liability becomes less burdensome when interest rates rise: you can invest less money today to cover that same future obligation, making the option to defer payment more attractive.

If implied volatility decreases, the market expects smaller price swings in the future, which reduces the probability of large upward moves that would make your call option profitable. Since there's less uncertainty and less potential for the stock to make dramatic gains, the call option becomes less valuable. Options thrive on uncertainty, and when the expected range of price movements shrinks, the option loses value regardless of whether you hold a call or a put.

Introducing dividends decreases the expected future stock price, since dividends reduce a company's retained earnings and are usually priced in when they're announced. For a call option, a lower future stock price expectation reduces its value.

Combining these effects: The increase in the risk-free interest rate would likely push the call option's value up, but the combined effects of reduced volatility and the introduction of dividends would exert downward pressure on the option's value. The net effect would depend on the magnitude of each influence, but given the presented scenario, one might expect a decrease in the call option's value due to the pronounced impacts of reduced volatility and dividend introduction.

Question 24

You are analyzing a European-style put option on a non-dividend-paying stock. This put option is currently at-the-money with 60 days to expiration. The implied volatility stands at 25%, and the risk-free interest rate is 3%.
Now, imagine two simultaneous changes:
The risk-free interest rate drops to 1%.
The implied volatility of the option jumps to 35%.
Given these changes, how would you anticipate the value of the put option to be affected, and why?

Accepted Answer

There are two main effects to consider here:

Interest Rate Impact: When interest rates decrease, the present value of the strike price (which you'd receive if you exercised the put option) goes up. This effect tends to increase the value of the put option. If interest rates are lower, receiving an amount sooner (like the strike price from a put option) is relatively more valuable since you wouldn't be able to earn as much on it if you were to invest it given the lower rates.

Volatility Impact: An increase in implied volatility typically increases the value of options, both calls and puts. Higher volatility implies a greater range of potential stock price movements, increasing the probability that the option could end up in-the-money by expiration. This makes the option more valuable due to the increased uncertainty and potential for profit.

Combining these effects, both the decrease in the risk-free interest rate and the increase in implied volatility would likely result in an increase in the value of the European-style at-the-money put option on a non-dividend-paying stock.

Question 25

What are the most important elements of stochastic calculus used for derivatives pricing?

Accepted Answer

Brownian Motion: A continuous-time stochastic process that represents the random movement of a particle in a fluid, foundational for modeling stock prices.

Ito's Lemma: A key result in stochastic calculus that helps find the differential of a function of a stochastic process, typically applied to option pricing.

Stochastic Differential Equations (SDEs): Equations that describe how processes evolve over time, taking into account both deterministic and random influences.

Martingales: A type of stochastic process that models fair games, essential for understanding the no-arbitrage condition in finance.

Risk-neutral Measure: A change of probability measure that simplifies option pricing by converting the discounted stock price process into a martingale.

Girsanov's Theorem: This facilitates the change of measure, which is essential for moving from the real-world probability measure to the risk-neutral measure.

Feynman-Kac Formula: Connects partial differential equations with expected values of functionals of stochastic processes.

Quantitative Finance

Quantitative Finance

Quantitative Finance

Quantitative Finance