Finance Quiz: Put Call Parity
20 questions · exam conditions
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Put Call ParityQuestion 1 of 20

A stock is trading at $110. A European call option with a strike price of $100 and one year to expiration is priced at $17. A corresponding European put option is priced at $2. The continuously compounded risk-free rate is 5% per annum. Assuming no transaction costs, what is the arbitrage profit available per share?

$0.12
$-0.12
$2.00
$4.88
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Finance Quiz

Finance Quiz: Put Call Parity

Practice Put Call Parity in Finance with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Put Call Parity, giving you a quick way to practice the rules, question types, and explanations that matter most for Finance.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

A stock is trading at $110. A European call option with a strike price of $100 and one year to expiration is priced at $17. A corresponding European put option is priced at $2. The continuously compounded risk-free rate is 5% per annum. Assuming no transaction costs, what is the arbitrage profit available per share?

  1. $0.12 (correct answer)
  2. $-0.12
  3. $2.00
  4. $4.88
Explanation: According to put-call parity for European options, the relationship is C+Ke−rT=P+S0C + K e^{-rT} = P + S_0. First, calculate the present value of the strike price: PV(K) = 100 \times e^{-0.05 \times 1} \approx \95.12$. Now, compare the two portfolios: Portfolio (LHS): Call + Bond = \17 + $95.12 = $112.12Portfolio(RHS):Put+Stock=Portfolio (RHS): Put + Stock =$2 + $110 = $112.00$ The left side is overpriced relative to the right side. To exploit this, an arbitrageur should sell the overpriced portfolio and buy the underpriced portfolio. Strategy:
  • Sell the call option (receive $17)
  • Sell a zero-coupon bond with face value $100 (receive $95.12)
  • Buy the put option (pay $2)
  • Buy the stock (pay $110)
Net initial cash flow (profit) = \17 + $95.12 - $2 - $110 = $0.12$. Distractor B represents the loss from executing the strategy in the wrong direction. Distractor D incorrectly uses the undiscounted strike price (17 + 100 - 2 - 110 = 5).

Question 2

Consider the put-call parity relationship for European options on a non-dividend-paying stock: C−P=S0−Ke−rTC - P = S_0 - K e^{-rT}. If the risk-free rate (r) unexpectedly increases, holding all other factors constant, what is the effect on the difference between the call price and the put price (C - P)?

  1. The difference (C - P) will increase. (correct answer)
  2. The difference (C - P) will decrease.
  3. The difference (C - P) will remain unchanged.
  4. The effect on (C - P) depends on whether the options are in- or out-of-the-money.
Explanation: The term in the put-call parity equation affected by the risk-free rate is Ke−rTK e^{-rT}, the present value of the strike price. When the risk-free rate (r) increases, the discount factor e−rTe^{-rT} decreases. Consequently, the present value of the strike price, Ke−rTK e^{-rT}, also decreases. The equation is C−P=S0−Ke−rTC - P = S_0 - K e^{-rT}. Since Ke−rTK e^{-rT} is smaller, we are subtracting a smaller number from S0S_0. This causes the entire right side of the equation, S0−Ke−rTS_0 - K e^{-rT}, to increase. To maintain parity, the left side, C−PC - P, must also increase.

Question 3

For a European put option on a stock that pays a continuous dividend yield of 2%, you are given: Stock Price = $100, Strike Price = $100, Risk-free rate = 5%, Time to expiration = 1 year, Put price = $5.82. Based on put-call parity, what should be the price of a corresponding European call option?

  1. $8.72 (correct answer)
  2. $10.70
  3. $8.70
  4. $3.94
Explanation: The put-call parity formula for a stock with a continuous dividend yield (q) is: C+Ke−rT=P+S0e−qTC + K e^{-rT} = P + S_0 e^{-qT}. We need to solve for C: C=P+S0e−qT−Ke−rTC = P + S_0 e^{-qT} - K e^{-rT} First, calculate the present value of the stock and strike: S_0 e^{-qT} = 100 \times e^{-0.02 \times 1} = 100 \times 0.98020 = \98.02 K e^{-rT} = 100 \times e^{-0.05 \times 1} = 100 \times 0.95123 = $95.12$ Now substitute these values into the formula: C = \5.82 + $98.02 - $95.12 = $8.72$ Distractor B is the result of incorrectly using the no-dividend parity formula (C=5.82+100−95.12=10.70C = 5.82 + 100 - 95.12 = 10.70). Distractor C arises from an approximation like S0(1−q)S_0(1-q). Distractor D results from a sign error in the main formula.

Question 4

A trader creates a box spread by buying a $95 strike call, selling a $95 strike put, selling a $105 strike call, and buying a $105 strike put. All options are European and expire in one year. The net cost to establish this entire position is $9.25. The continuously compounded risk-free rate is 4%. Which statement accurately describes this situation?

  1. The strategy's value depends on whether the stock price finishes between $95 and $105.
  2. An arbitrage loss of approximately $0.36 will be incurred.
  3. The position has a risk-free payoff of $10 at expiration, and its initial cost is fair.
  4. An arbitrage profit of approximately $0.36 can be locked in. (correct answer)
Explanation: When you encounter a box spread question, you're dealing with an arbitrage analysis problem. A box spread combines two vertical spreads and should always have a risk-free payoff equal to the difference between strike prices, regardless of where the stock price ends up. Let's trace through this box spread. You're long the $95 call and short the $95 put (synthetic long stock at $95), while short the $105 call and long the $105 put (synthetic short stock at $105). At expiration, this position will always pay exactly $10, no matter what the stock price is. If the stock is below $95, both calls expire worthless and you collect $10 from the put spread. If it's above $105, both puts expire worthless and you pay $10 on the call spread. Between $95 and $105, you get $10 from one spread. Since this guaranteed $10 payoff is worth $10 \times e^{-0.04 \times 1} = \9.61 today, but you only paid $9.25, you've locked in an arbitrage profit of approximately $0.36. Looking at the wrong answers: A) is incorrect because the payoff is always $10, not dependent on the stock's final price. B) reverses the arbitrage direction—you're making money, not losing it. C) correctly identifies the $10 payoff but wrongly claims the $9.25 cost is fair when it should be $9.61. Study tip: For box spreads, always calculate the present value of the guaranteed payoff and compare it to the net premium paid. Any difference represents a risk-free arbitrage opportunity.

Question 5

Consider a situation where the put-call parity relationship, S0+P=C+Ke−rTS_0 + P = C + K e^{-rT}, appears to be violated. The left side of the equation is significantly larger than the right side. Which of the following market conditions, if true, would be the most likely explanation for this observation without implying a true arbitrage opportunity?

  1. The stock's volatility is extremely low.
  2. There are high transaction costs for trading the stock and options. (correct answer)
  3. The interest rates are expected to decrease significantly.
  4. The options are European style and the stock does not pay dividends.
Explanation: Put-call parity is a fundamental relationship that must hold in efficient markets without arbitrage opportunities. When you see S0+P>C+Ke−rTS_0 + P > C + K e^{-rT}, this suggests the protective put strategy is overpriced relative to the fiduciary call strategy. In perfect market conditions, this violation would create an immediate arbitrage opportunity: sell the expensive left side (stock + put) and buy the cheaper right side (call + bond). However, real markets have frictions that can make apparent arbitrage unprofitable. High transaction costs (Answer B) are the most likely explanation. If trading fees, bid-ask spreads, and borrowing costs are substantial, they can eliminate the profit from exploiting this price discrepancy. The apparent arbitrage opportunity disappears once you account for the real costs of executing the required trades. Answer A is incorrect because low volatility doesn't systematically bias put-call parity in either direction—it affects both puts and calls similarly. Answer C misses the mark because expected interest rate changes don't directly explain current parity violations; the current risk-free rate determines the present value relationship. Answer D describes standard put-call parity assumptions, so European options and no dividends wouldn't cause any violation at all. Remember that on finance exams, when you see "apparent" arbitrage opportunities that seem too obvious, look for market frictions as the explanation. Transaction costs are often the bridge between textbook theory and real-world trading, making seemingly profitable strategies uneconomical.

Question 6

A portfolio manager believes a stock is significantly overvalued but is prohibited by the fund's charter from short-selling equities directly. The manager wishes to create a synthetic position that replicates the payoff of a short stock position using European options and risk-free borrowing/lending. Which of the following strategies would achieve this objective?

  1. Buy a put, sell a call, and borrow the present value of the strike price. (correct answer)
  2. Sell a put, buy a call, and lend the present value of the strike price.
  3. Buy a put, buy a call, and borrow the current stock price.
  4. Sell a put, sell a call, and lend the current stock price.
Explanation: The put-call parity equation is S0+P=C+Ke−rTS_0 + P = C + K e^{-rT}. We want to isolate −S0-S_0 on one side of the equation to find the replicating portfolio for a short stock position. Rearranging the formula: −S0=P−C−Ke−rT-S_0 = P - C - K e^{-rT}. This shows that a synthetic short stock position is created by:
  • A long put (+P+P)
  • A short call (−C-C)
  • Borrowing the present value of the strike price (−Ke−rT-K e^{-rT}).
Distractor B describes a synthetic long stock position. Distractors C and D describe option combinations (a straddle and a short straddle, respectively) combined with borrowing or lending that do not replicate a short stock payoff.

Question 7

Market data for European options on a stock are as follows: Stock price = $90, Strike price = $85, Time to expiration = 6 months, Continuously compounded risk-free rate = 3%. The call option trades for $8.00 and the put option trades for $2.50. Assuming no arbitrage opportunities exist, what is the implied present value of dividends expected to be paid before expiration?

  1. $0.77 (correct answer)
  2. $0.78
  3. $-0.77
  4. $0.50
Explanation: The put-call parity relationship adjusted for dividends is C+Ke−rT=P+S0−PV(Div)C + K e^{-rT} = P + S_0 - PV(Div). We can rearrange this to solve for the present value of the dividends, PV(Div)PV(Div): PV(Div)=P+S0−C−Ke−rTPV(Div) = P + S_0 - C - K e^{-rT} First, calculate the present value of the strike price: PV(K) = 85 \times e^{-0.03 \times 0.5} = 85 \times e^{-0.015} \approx \83.73$ Now, substitute the known values into the rearranged formula: PV(Div) = \2.50 + $90.00 - $8.00 - $83.73 = $0.77$ Distractor B calculates the future value of the dividend (0.77×e0.0150.77 \times e^{0.015}). Distractor C represents a common sign error in the calculation. Distractor D is a plausible but incorrect value.

Question 8

At expiration, a stock closes at $52. A European call with a $50 strike price has an intrinsic value of $2.00. An arbitrageur observes the corresponding European put, which has an intrinsic value of $0, is trading for a price of $0.10, while the call is trading at its intrinsic value of $2.00. What is the risk-free profit available from a transaction involving one share of stock?

  1. $1.90
  2. $0.00
  3. $0.10 (correct answer)
  4. $2.10
Explanation: When you encounter options pricing questions involving both calls and puts at expiration, look for arbitrage opportunities using put-call parity. At expiration, options trade at their intrinsic values, so any deviation creates risk-free profit potential. Let's analyze the situation: The stock closes at $52, making the call worth $52 - $50 = $2.00 (correctly priced) and the put worth $0 since the stock is above the strike price. However, the put is trading at $0.10 despite having zero intrinsic value. To capture this arbitrage, you would sell the overpriced put for 0.10.Sincethestockprice(0.10. Since the stock price (52) exceeds the strike price ($50), the put expires worthless, allowing you to keep the entire premium as risk-free profit. Your profit per share is exactly $0.10. Looking at the wrong answers: Answer A (1.90)incorrectlysuggestssubtractingtheputpricefromthecallprice,whichisn′trelevanttothearbitrageopportunity.AnswerB(1.90) incorrectly suggests subtracting the put price from the call price, which isn't relevant to the arbitrage opportunity. Answer B (0.00) wrongly assumes no arbitrage exists, missing that the put is mispriced above its intrinsic value. Answer D ($2.10) mistakenly adds the call and put prices together, confusing the profit calculation. The correct answer is C ($0.10). Study tip: At expiration, European options must trade at their intrinsic values. Any premium above intrinsic value represents pure arbitrage profit. Always compare market prices to intrinsic values and remember that puts with strike prices below the current stock price should be worthless at expiration.

Question 9

An analyst observes the following data for European options on a non-dividend paying stock: Stock Price = $110, Strike Price = $100, Time to Expiration = 6 months. The market price of the call exceeds the market price of the put by exactly $12.00. Based on this information, what is the implied continuously compounded risk-free interest rate?

  1. A negative rate, suggesting a market anomaly.
  2. 2.02%
  3. 8.16%
  4. 4.04% (correct answer)
Explanation: When you encounter options pricing problems involving the relationship between call and put prices, you should immediately think of put-call parity. This fundamental relationship connects the prices of European calls and puts on the same underlying asset. Put-call parity states that for European options on a non-dividend paying stock: C−P=S−Ke−rtC - P = S - Ke^{-rt}, where C is the call price, P is the put price, S is the current stock price, K is the strike price, r is the risk-free rate, and t is time to expiration. Given that the call exceeds the put by exactly $12.00, we have C - P = $12.00. Substituting the known values into the put-call parity equation: $12=110−100e−r(0.5)12 = 110 - 100e^{-r(0.5)} .Solvingforr:. Solving for r: 12=110−100e−0.5r12 = 110 - 100e^{-0.5r} ,whichgivesus, which gives us 100e−0.5r=98100e^{-0.5r} = 98 ,so, so e−0.5r=0.98e^{-0.5r} = 0.98 .Takingthenaturallogarithm:. Taking the natural logarithm: −0.5r=ln⁡(0.98)=−0.0202-0.5r = \ln(0.98) = -0.0202 ,therefore, therefore r=0.0404r = 0.0404 $ or 4.04%. Answer A suggests a negative rate, but our calculation clearly yields a positive rate. Answer B (2.02%) represents a common error where students forget to divide by the time factor of 0.5 years, giving half the correct rate. Answer C (8.16%) likely results from sign errors or incorrect manipulation of the exponential function. Remember this pattern: put-call parity problems always give you enough information to solve for the missing variable algebraically. Set up the equation methodically and solve step-by-step, being especially careful with the time factor in the exponent.

Question 10

For European options on a non-dividend-paying stock, you observe the following prices: Stock Price = $75, Strike Price = $80, Time to Expiration = 3 months. The call price is $2.00 and the put price is $6.50. Assuming these prices permit no arbitrage, what is the implied continuously compounded risk-free rate?

  1. 3.75%
  2. 3.14%
  3. 2.51% (correct answer)
  4. -0.63%
Explanation: When you encounter European option pricing problems, you're typically dealing with put-call parity, which establishes the fundamental relationship between call and put prices for options with identical strike prices and expiration dates. Put-call parity states that: C+Ke−rt=P+SC + Ke^{-rt} = P + S, where C is the call price, K is the strike price, r is the risk-free rate, t is time to expiration, P is the put price, and S is the stock price. Rearranging to solve for the risk-free rate: r=−1tln⁡(C−P+SK)r = -\frac{1}{t} \ln\left(\frac{C - P + S}{K}\right) Substituting the given values: r=−10.25ln⁡(2.00−6.50+7580)=−4ln⁡(70.5080)=−4ln⁡(0.88125)=−4(−0.1264)=0.0251r = -\frac{1}{0.25} \ln\left(\frac{2.00 - 6.50 + 75}{80}\right) = -4 \ln\left(\frac{70.50}{80}\right) = -4 \ln(0.88125) = -4(-0.1264) = 0.0251 or 2.51%. Choice A (3.75%) results from calculation errors, likely from incorrectly applying the natural logarithm or mishandling the negative sign in the formula. Choice B (3.14%) might come from using an incorrect version of put-call parity or making arithmetic mistakes in the fraction calculation. Choice D (-0.63%) occurs when you forget to multiply by the negative reciprocal of time, keeping the negative logarithm value without the proper transformation. Study tip: Master put-call parity cold—it's tested frequently and in various forms. Always double-check your signs and remember that when solving for interest rates, you're typically taking the natural logarithm of a ratio and dividing by time. Practice rearranging the formula beforehand so you can execute quickly during the exam.

Question 11

A portfolio consisting of a long European call option and a short European put option on a stock has a value equal to the present value of the difference between the stock's forward price and the options' strike price. This can be expressed as C−P=(F0,T−K)e−rTC - P = (F_{0,T} - K)e^{-rT}. If the current stock price is $100, the risk-free rate is 5%, the continuous dividend yield is 1%, and the 1-year forward price is $104.08, what is the value of a portfolio with a strike price of $100?

  1. $4.76
  2. $4.08
  3. $4.00
  4. $3.88 (correct answer)
Explanation: This question tests your understanding of put-call parity, a fundamental relationship in options pricing. When you see a portfolio combining long calls and short puts, you should immediately think of the put-call parity formula: C−P=(F0,T−K)e−rTC - P = (F_{0,T} - K)e^{-rT}, where the portfolio value equals the present value of the difference between the forward price and strike price. Using the given values, you can calculate the portfolio value directly. The forward price is $104.08, the strike price is $100, so the difference is $4.08. You then discount this at the risk-free rate: $(104.08 - 100)e^{-0.05 \times 1} = 4.08 \times e^{-0.05} = 4.08 \times 0.9512 = 3.88$. Answer A (4.76)representsacommonerrorwherestudentsmightincorrectlyusethecurrentstockprice(4.76) represents a common error where students might incorrectly use the current stock price (100) instead of the forward price, or fail to apply proper discounting. Answer B (4.08)isthetrapofforgettingtodiscountthepayoffdifference—thiswouldbecorrectonlyiftherisk−freeratewerezero.AnswerC(4.08) is the trap of forgetting to discount the payoff difference—this would be correct only if the risk-free rate were zero. Answer C (4.00) might result from rounding errors or using an approximate discount factor instead of the precise calculation. The correct answer is D ($3.88). Remember that put-call parity always requires present value calculations. The key insight is that you're comparing future values (forward price vs. strike) that must be discounted back to today. Don't forget the discounting step—it's where most students make errors on put-call parity problems.

Question 12

A delta-neutral hedge fund constructs a portfolio by buying one European call option and selling one European put option on a non-dividend-paying stock. Both options have the same strike price and expiration date. If the underlying stock price increases by a small amount, ΔS\Delta S, how will the value of the fund's portfolio change, according to the principles of put-call parity?

  1. It will increase by exactly ΔS\Delta S. (correct answer)
  2. It will increase by an amount less than ΔS\Delta S, depending on the option deltas.
  3. It will decrease by ΔS\Delta S.
  4. It will not change, as the change in the call value is offset by the change in the put value.
Explanation: The value of the portfolio is V=C−PV = C - P. According to put-call parity, C−P=S−Ke−rTC - P = S - K e^{-rT}. The sensitivity of the portfolio's value to a change in the stock price is the delta of the portfolio, which is the derivative of its value with respect to S. Δportfolio=∂V∂S=∂(C−P)∂S\Delta_{portfolio} = \frac{\partial V}{\partial S} = \frac{\partial (C - P)}{\partial S} Using the parity relationship: ∂(S−Ke−rT)∂S=∂S∂S−∂(Ke−rT)∂S=1−0=1\frac{\partial (S - K e^{-rT})}{\partial S} = \frac{\partial S}{\partial S} - \frac{\partial (K e^{-rT})}{\partial S} = 1 - 0 = 1 This means that ΔC−ΔP=1\Delta_C - \Delta_P = 1. A portfolio of a long call and a short put has a delta of 1, meaning its value moves one-for-one with the underlying stock price. Therefore, if the stock price increases by ΔS\Delta S, the portfolio's value will also increase by ΔS\Delta S.

Question 13

A risk manager holds a large stock position and wishes to use European options to convert the position's uncertain future value into a guaranteed value of $100 per share at expiration T. Which of the following option strategies, when combined with holding the stock, will achieve this goal?

  1. Buy a call with a strike price of $100 and sell a put with a strike price of $100.
  2. Buy a put with a strike price of $100 and sell a call with a strike price of $100. (correct answer)
  3. Sell both a put and a call with a strike price of $100.
  4. Buy both a put and a call with a strike price of $100.
Explanation: When you encounter questions about converting uncertain stock positions into guaranteed values, you're dealing with protective option strategies that create synthetic portfolios with predetermined payoffs. To guarantee a $100 value per share at expiration, you need to analyze what happens at different stock prices. If the stock price at expiration is above $100, you want to limit your gains to $100. If it's below $100, you want protection that brings your total value back up to $100. Option B achieves this perfectly. By buying a put with a $100 strike, you gain the right to sell your stock for $100 if the market price falls below that level. By selling a call with a $100 strike, you're obligated to sell your stock for $100 if the price rises above that level. This creates a "collar" that locks in exactly $100 regardless of where the stock price ends up. Option A does the opposite—you'd be obligated to buy more stock if the price falls (short put) and have the right to buy even more if it rises (long call), amplifying rather than eliminating risk. Option C (selling both options) creates unlimited risk exposure in both directions. Option D (buying both options) is a long straddle that increases rather than decreases volatility exposure. Remember that protective strategies typically involve buying options that move opposite to your existing position. Since you're long the stock, you need downside protection (long put) and should be willing to cap upside gains (short call) to achieve guaranteed outcomes.

Question 14

An analyst observes the following market prices for American options on a non-dividend paying stock: Stock Price = $100, Strike Price = $105, Time to Expiration = 1 year, Risk-Free Rate = 4%. The call is priced at $3.00 and the put is priced at $7.50. The analyst notes that for European options, this would represent a mispricing. Which statement is the most accurate assessment of this situation?

  1. An arbitrage profit is certain, as the parity relationship is violated regardless of option style.
  2. The stock must be paying an unknown dividend, which explains the discrepancy.
  3. No arbitrage opportunity is guaranteed, as the prices may fall within the wider no-arbitrage bounds for American options. (correct answer)
  4. The market is clearly inefficient, and the put option is significantly overpriced relative to the call.
Explanation: For European options, put-call parity states C−P=S0−Ke−rTC - P = S_0 - K e^{-rT}. Here, S0−Ke−rT=100−105e−0.04(1)=100−100.90=−0.90S_0 - K e^{-rT} = 100 - 105e^{-0.04(1)} = 100 - 100.90 = -0.90. The observed C−P=3.00−7.50=−4.50C - P = 3.00 - 7.50 = -4.50. Since −4.50≠−0.90-4.50 \neq -0.90, European parity is violated. However, for American options on a non-dividend paying stock, the relationship is an inequality: S0−K≤C−P≤S0−Ke−rTS_0 - K \leq C - P \leq S_0 - K e^{-rT}. Calculating the bounds: Lower bound: S0−K=100−105=−5.00S_0 - K = 100 - 105 = -5.00. Upper bound: S0−Ke−rT=−0.90S_0 - K e^{-rT} = -0.90. The observed difference C−P=−4.50C - P = -4.50 falls within the range [-5.00, -0.90]. Therefore, while European parity is violated, the prices are within the no-arbitrage bounds for American options due to the early exercise privilege.

Question 15

A synthetic long forward contract can be created by buying a European call and selling a European put with the same strike price K and expiration T. The initial value of this position is C0−P0C_0 - P_0. At expiration, the value of the position is ST−KS_T - K. What is the implied rate of return on this synthetic forward position?

  1. The expected return of the stock.
  2. The risk-free rate, r. (correct answer)
  3. Zero, as it is a zero-cost contract.
  4. A rate that depends on the stock's final price, S_T.
Explanation: When you encounter synthetic instruments in finance, remember that they must earn the same return as their underlying equivalent to prevent arbitrage opportunities. A synthetic long forward position (long call + short put) replicates owning a forward contract. The key insight is recognizing what return this position must earn. Since the initial investment is C0−P0C_0 - P_0 and the terminal value is ST−KS_T - K, we can set up the no-arbitrage condition. In an efficient market, this synthetic forward must earn the risk-free rate, just like any forward contract. Here's why: if the synthetic forward earned more than the risk-free rate, arbitrageurs would buy it and short risk-free bonds. If it earned less, they'd do the reverse. These actions would continue until the returns equalized at the risk-free rate. Option A is incorrect because the synthetic forward's return doesn't depend on the stock's expected return. The position's payoff structure eliminates exposure to the stock's risk premium. Option C misunderstands the concept—even though some synthetic positions might have near-zero initial cost, they still earn a rate of return based on their terminal value relative to initial investment. Option D is wrong because while the absolute dollar return depends on STS_T, the rate of return is fixed at the risk-free rate regardless of the final stock price. Study tip: Remember that synthetic instruments and their natural counterparts must earn identical returns due to no-arbitrage conditions. When you see "synthetic" combined with "rate of return," think about what prevents arbitrage opportunities.

Question 16

A company's stock undergoes a 2-for-1 split. Which statement best describes the theoretical effect on the put-call parity relationship for existing European options, assuming their terms (strike price, number of shares) are adjusted for the split?

  1. The relationship holds, but the C-P spread remains the same as its pre-split value.
  2. The relationship holds, and the C-P spread will be exactly half of its pre-split value. (correct answer)
  3. The relationship breaks down temporarily, creating arbitrage opportunities until new options are issued.
  4. The relationship holds, and the C-P spread will double its pre-split value.
Explanation: Stock splits test your understanding of how corporate actions affect option pricing relationships, particularly put-call parity. When you encounter split-related questions, remember that while individual option values change, fundamental pricing relationships must still hold after proper adjustments. In a 2-for-1 stock split, the stock price halves while the number of shares doubles, leaving total equity value unchanged. Option contracts are typically adjusted proportionally: if you held 1 call option with a $100 strike, you'd now hold 2 call options each with a $50 strike. The put-call parity relationship $C−P=S−Ke−rtC - P = S - Ke^{-rt} $ continues to hold, but with adjusted values. Since both the stock price (S) and strike price (K) are halved, the right side of the equation ( S - Ke^{-rt} ) becomes exactly half its original value. Therefore, the left side ( C - P , the call-put spread) must also halve to maintain the relationship. Answer B correctly identifies this outcome. Answer A incorrectly suggests the spread remains unchanged, which would violate put-call parity given the halved stock and strike prices. Answer C wrongly implies the relationship breaks down—properly adjusted options maintain theoretical pricing relationships immediately. Answer D suggests the spread doubles, which would create massive arbitrage opportunities since the underlying economic value ( S - Ke^{-rt} ) actually halves. Remember that corporate actions like splits preserve economic value while changing nominal prices. When analyzing their effects on derivatives, always verify that fundamental relationships like put-call parity remain intact after adjustments.

Question 17

An investor constructs a portfolio on a non-dividend paying stock by buying one share of the stock, selling one European call option, and buying one European put option. The options have the same strike price K and expiration date T. What is the value of this portfolio at the time of its creation?

  1. Zero, as the position is fully hedged.
  2. The stock price, S0S_0
  3. The present value of the strike price, Ke−rTK e^{-rT} (correct answer)
  4. The stock price minus the strike price, S0−KS_0 - K
Explanation: This question tests your understanding of portfolio construction and put-call parity, a fundamental relationship in options pricing. When you see a portfolio combining a stock with both a call and put option, think about how these positions interact. Let's analyze this portfolio step by step. You're buying one share of stock (value: S0S_0), selling one call option (you receive the call premium), and buying one put option (you pay the put premium). The key insight is recognizing that selling a call and buying a put with the same strike price and expiration creates a synthetic short position in the stock. From put-call parity, we know that: S0+P−C=Ke−rTS_0 + P - C = Ke^{-rT}, where P is the put price, C is the call price, and Ke−rTKe^{-rT} is the present value of the strike price. This equation tells us that buying a stock, buying a put, and selling a call equals the present value of the strike price. Therefore, the portfolio value at creation is Ke−rTKe^{-rT}. Answer A is incorrect because the position isn't fully hedged—it has a specific payoff structure. Answer B misses the options' impact on portfolio value; you can't just ignore the put and call components. Answer D represents the intrinsic value difference but ignores the time value of money and the options' premiums. Remember: whenever you see a combination of stock, calls, and puts with identical strikes and expirations, immediately think of put-call parity. This relationship appears frequently on finance exams and is essential for understanding synthetic positions and arbitrage opportunities.

Question 18

A portfolio manager believes a stock is significantly overvalued but is prohibited by the fund's charter from short-selling equities directly. The manager wishes to create a synthetic position that replicates the payoff of a short stock position using European options and risk-free borrowing/lending. Which of the following strategies would achieve this objective?

  1. Buy a put, sell a call, and borrow the present value of the strike price. (correct answer)
  2. Sell a put, buy a call, and lend the present value of the strike price.
  3. Buy a put, buy a call, and borrow the current stock price.
  4. Sell a put, sell a call, and lend the current stock price.
Explanation: The put-call parity equation is S0+P=C+Ke−rTS_0 + P = C + K e^{-rT}. We want to isolate −S0-S_0 on one side of the equation to find the replicating portfolio for a short stock position. Rearranging the formula: −S0=P−C−Ke−rT-S_0 = P - C - K e^{-rT}. This shows that a synthetic short stock position is created by:
  • A long put (+P+P)
  • A short call (−C-C)
  • Borrowing the present value of the strike price (−Ke−rT-K e^{-rT}).
Distractor B describes a synthetic long stock position. Distractors C and D describe option combinations (a straddle and a short straddle, respectively) combined with borrowing or lending that do not replicate a short stock payoff.

Question 19

An analyst observes the following market prices for American options on a non-dividend paying stock: Stock Price = $100, Strike Price = $105, Time to Expiration = 1 year, Risk-Free Rate = 4%. The call is priced at $3.00 and the put is priced at $7.50. The analyst notes that for European options, this would represent a mispricing. Which statement is the most accurate assessment of this situation?

  1. An arbitrage profit is certain, as the parity relationship is violated regardless of option style.
  2. The stock must be paying an unknown dividend, which explains the discrepancy.
  3. No arbitrage opportunity is guaranteed, as the prices may fall within the wider no-arbitrage bounds for American options. (correct answer)
  4. The market is clearly inefficient, and the put option is significantly overpriced relative to the call.
Explanation: For European options, put-call parity states C−P=S0−Ke−rTC - P = S_0 - K e^{-rT}. Here, S0−Ke−rT=100−105e−0.04(1)=100−100.90=−0.90S_0 - K e^{-rT} = 100 - 105e^{-0.04(1)} = 100 - 100.90 = -0.90. The observed C−P=3.00−7.50=−4.50C - P = 3.00 - 7.50 = -4.50. Since −4.50≠−0.90-4.50 \neq -0.90, European parity is violated. However, for American options on a non-dividend paying stock, the relationship is an inequality: S0−K≤C−P≤S0−Ke−rTS_0 - K \leq C - P \leq S_0 - K e^{-rT}. Calculating the bounds: Lower bound: S0−K=100−105=−5.00S_0 - K = 100 - 105 = -5.00. Upper bound: S0−Ke−rT=−0.90S_0 - K e^{-rT} = -0.90. The observed difference C−P=−4.50C - P = -4.50 falls within the range [-5.00, -0.90]. Therefore, while European parity is violated, the prices are within the no-arbitrage bounds for American options due to the early exercise privilege.

Question 20

Market data for European options on a stock are as follows: Stock price = $90, Strike price = $85, Time to expiration = 6 months, Continuously compounded risk-free rate = 3%. The call option trades for $8.00 and the put option trades for $2.50. Assuming no arbitrage opportunities exist, what is the implied present value of dividends expected to be paid before expiration?

  1. $0.77 (correct answer)
  2. $0.78
  3. $-0.77
  4. $0.50
Explanation: The put-call parity relationship adjusted for dividends is C+Ke−rT=P+S0−PV(Div)C + K e^{-rT} = P + S_0 - PV(Div). We can rearrange this to solve for the present value of the dividends, PV(Div)PV(Div): PV(Div)=P+S0−C−Ke−rTPV(Div) = P + S_0 - C - K e^{-rT} First, calculate the present value of the strike price: PV(K) = 85 \times e^{-0.03 \times 0.5} = 85 \times e^{-0.015} \approx \83.73$ Now, substitute the known values into the rearranged formula: PV(Div) = \2.50 + $90.00 - $8.00 - $83.73 = $0.77$ Distractor B calculates the future value of the dividend (0.77×e0.0150.77 \times e^{0.015}). Distractor C represents a common sign error in the calculation. Distractor D is a plausible but incorrect value.