In the highly competitive world of quantitative trading and financial engineering, interviews at prestigious firms like Susquehanna International Group (SIG) test not just your technical skills, but also your understanding of real-world market mechanisms. One common theme in these interviews is the pricing of contracts when parties have asymmetric information. In this article, we will provide a comprehensive, step-by-step solution to a classic SIG Quantitative Trading Intern interview question: How do you price a contract when a buyer can choose to purchase an asset after observing its true value, but the seller must set the contract price in advance?

SIG Quantitative Trading Intern Interview Question: Pricing a Contract with Asymmetric Information

Understanding the Interview Question

Let's begin by carefully parsing the question:

An asset is worth $15,000 with 90% probability and $2,000 with 10% probability. A buyer can choose to purchase it for $6,000 after observing its value. What is the fair price of this contract?

This is an excellent example of an asymmetric information problem, a concept that lies at the heart of many real-world trading scenarios. Let’s break down the key elements:

• Asset Value: The asset has two possible values: $15,000 (with probability 90%) or $2,000 (with probability 10%).
• Contract Structure: The buyer can choose to purchase the asset for $6,000 after seeing its value. This is crucial: the buyer gets to make the purchase decision with full knowledge of the asset’s value, while the seller must set the contract price without knowing which state will occur.
• Question: What is the fair price of the contract that allows the buyer this purchase option?

Key Concepts: Asymmetric Information, Fair Pricing, and Optionality

What is Asymmetric Information?

In finance and economics, asymmetric information occurs when one party to a transaction has more or better information than the other. In this case, the buyer has the advantage: after the contract is struck, the buyer observes the true value of the asset before deciding whether to buy. The seller must set the contract price without this information.

What is a "Fair Price"?

The fair price of a contract, in this context, is the amount the buyer should pay upfront to acquire the right (but not the obligation) to buy the asset for $6,000 after observing its value. This is analogous to the price of a financial option in the markets.

The Role of Optionality

The buyer is being sold an option: the right to buy the asset for $6,000 after learning its true value. This embedded optionality is valuable, especially because the buyer can avoid purchasing if the asset turns out to be worth less than $6,000.

Step 1: Modeling the Problem

Let’s formalize the problem with some notation:

• Let \( V \) be the random variable representing the asset’s value. \( V = \$15,000 \) with probability 0.9 and \( V = \$2,000 \) with probability 0.1.
• The buyer can purchase for \( K = \$6,000 \) after seeing the value.
• The contract gives the buyer the option to buy; they will only do so when \( V \geq K \).
• We are to find the fair price of the contract: the expected value of this option.

This is structurally similar to a call option with strike price \( K \), but with only two discrete possible outcomes.

Step 2: The Buyer's Optimal Strategy

Because the buyer gets to see the asset’s value before deciding, their optimal strategy is simple:

• If \( V \geq K \), they buy (since the asset is worth more than it costs).
• If \( V < K \), they do not buy (since the asset is worth less than it costs).

In our case:

• If \( V = \$15,000 \), they will buy for \$6,000 and receive a net gain of \( \$15,000 - \$6,000 = \$9,000 \).
• If \( V = \$2,000 \), they will not buy, since \( \$2,000 - \$6,000 = -\$4,000 \) (a loss).

Step 3: Calculating the Expected Payoff

Now, let’s compute the expected payoff for the buyer:

\[ \text{Expected Payoff} = P(V = \$15,000) \times (\$15,000 - \$6,000) + P(V = \$2,000) \times 0 \]

Substituting the numbers:

\[ \text{Expected Payoff} = 0.9 \times (\$15,000 - \$6,000) + 0.1 \times 0 = 0.9 \times \$9,000 = \$8,100 \]

Therefore, the fair price of the contract is \$8,100.

Step 4: Interpretation – Why is the Price So High?

At first glance, \$8,100 may seem high, given that the asset is only worth \$15,000 in the best case. However, remember:

• The buyer only exercises the contract when it is clearly profitable (the asset is worth \$15,000).
• If the asset is worth \$2,000, the buyer simply walks away.
• This lack of downside risk (thanks to the option structure) makes the contract very valuable.

This is a classic example of adverse selection: the seller is at a disadvantage, since the buyer will only buy when it is favorable to them.

Step 5: Generalizing the Solution

Let’s generalize this to understand how the fair price changes with different probabilities and values.

Let:

• \( V_1 \) = high value, with probability \( p \)
• \( V_2 \) = low value, with probability \( 1-p \)
• Strike price \( K \)

The buyer will buy only if \( V_1 \geq K \). Therefore, the expected value of the contract is:

\[ \text{Fair Price} = p \times (V_1 - K) + (1-p) \times 0 \]

In our specific case:

• \( V_1 = \$15,000 \), \( p = 0.9 \), \( K = \$6,000 \)
• \( V_2 = \$2,000 \), \( 1-p = 0.1 \)

So, \[ \text{Fair Price} = 0.9 \times (\$15,000 - \$6,000) = 0.9 \times \$9,000 = \$8,100 \]

Step 6: Comparing to Traditional Expected Value

If there were no option (i.e., the buyer must buy before knowing the value), the expected value of the asset would be:

\[ \text{Expected Value} = 0.9 \times \$15,000 + 0.1 \times \$2,000 = \$13,500 + \$200 = \$13,700 \]

The buyer would only agree to pay up to \$13,700 for the asset in that case. However, in our contract, the buyer is paying for the optionality—the ability to avoid bad outcomes.

Step 7: Real-World Insights – Why Asymmetric Information Matters

This problem is a stylized version of many real-world financial and trading scenarios:

• Option pricing: The contract here is exactly like a call option with strike price \$6,000.
• Adverse selection: In markets where one party knows more than the other, prices are driven by the actions of the better-informed party.
• Principal-agent problems: The seller (principal) must design contracts that account for the buyer’s (agent’s) informational advantage.

In practice, firms must carefully price such contracts to avoid being "picked off" by better-informed counterparties.

Step 8: Extending the Problem – What if the Buyer Must Decide Blindly?

Suppose the buyer did not get to observe the value and had to decide whether to buy at \$6,000 before knowing the asset’s state. The expected profit from buying would be:

\[ \text{Expected Profit} = 0.9 \times (\$15,000 - \$6,000) + 0.1 \times (\$2,000 - \$6,000) \]

\[ = 0.9 \times \$9,000 + 0.1 \times (-\$4,000) = \$8,100 - \$400 = \$7,700 \]

So, if the buyer had to commit in advance, the value of the contract would be \$7,700—not \$8,100. The extra \$400 comes from the buyer’s ability to avoid the downside scenario. This is called the option’s time value.

Step 9: Coding the Solution in Python

Let’s implement the expected value calculation in Python for clarity.

# Parameters
V_high = 15000
V_low = 2000
p_high = 0.9
p_low = 0.1
K = 6000

# Buyer's optimal strategy
payoff_high = max(V_high - K, 0)
payoff_low = max(V_low - K, 0)

# Expected payoff (fair price)
fair_price = p_high * payoff_high + p_low * payoff_low
print("Fair price of the contract: $%.2f" % fair_price)

This will output:

Fair price of the contract: $8,100.00

Step 10: Visualizing the Payoff Scenarios

Let’s tabulate the possible outcomes for clarity.

Step 11: Theoretical Perspective – Connection to Option Pricing

This problem is a discrete version of the classic European call option. In the Black-Scholes world, the fair price of a call option is given by the expected value of its payoff, discounted to present value.

For a general call option:

\[ \text{Call Option Payoff} = \max(V - K, 0) \]

The expected value, given discrete outcomes, is:

\[ \text{Expected Payoff} = \sum_{i} P_i \cdot \max(V_i - K, 0) \]

Which is precisely what we've calculated above.

Step 12: Common Pitfalls in Interview Settings

• Ignoring Optionality: Some candidates mistakenly compute the expected value of the asset (\$13,700), failing to account for the buyer’s right to refuse bad deals.
• Not Modeling the Buyer’s Strategy: Always consider what the buyer would rationally do after receiving information.
• Forgetting Probability Weights: Correctly assign probabilities to each scenario.

Step 13: Further Extensions – Multiple Asset Values

Suppose the asset could take on more than two values. The approach is the same:

1. For each possible \( V_i \), compute \( \max(V_i - K, 0) \).
2. Multiply by the probability of each \( V_i \).
3. Sum to get the fair price.

For example, if there were three possible values:

• \$15,000 (80%)
• \$8,000 (15%)
• \$2,000 (5%)

Then: \[ \text{Fair Price} = 0.8 \times (\$15,000 - \$6,000) + 0.15 \times (\$8,000 - \$6,000) + 0.05 \times 0 \] \[ = 0.8 \times \$9,000 + 0.15 \times \$2,000 + 0 = \$7,200 + \$300 = \$7,500 \]

Step 14: Practice Problem

Try this variant:

• Asset is worth \$20,000 (70%) or \$5,000 (30%).
• Buyer can purchase for \$10,000 after seeing value.
• What is the fair price?

Solution: \[ \text{Expected Payoff} = 0.7 \times (\$20,000 - \$10,000) + 0.3 \times 0 =

\[ \text{Expected Payoff} = 0.7 \times (\$20,000 - \$10,000) + 0.3 \times 0 = 0.7 \times \$10,000 + 0 = \$7,000 \]

So, the fair price of this contract is $7,000.

Step 15: Summary Table of Examples

Let’s compile a table with multiple examples for quick reference.

Step 16: Economic Intuition – The Seller’s Perspective

From the seller’s perspective, selling such a contract is risky because the buyer will only exercise when it’s in their favor. The seller is essentially taking on the downside risk without compensating for the upside. This is why, in real financial markets, option sellers charge a premium—known as the option premium—for writing such contracts.

If the seller were to offer the asset blindly at a fixed price, the expected loss would be greater. By requiring an upfront payment (the contract price), the seller compensates for the buyer’s informational advantage.

Step 17: Asymmetric Information in Financial Markets

Asymmetric information is a central concern in finance:

• Insider trading: Some traders know more about the true value of an asset than others, leading to adverse selection and market inefficiencies.
• Market making: Market makers face the risk that traders will only transact when the price is favorable to them, so spreads are widened to account for this risk.
• Insurance and lending: Policyholders and borrowers often know more about their own risk than the insurer or lender, leading to higher premiums or interest rates.

Understanding how to price contracts under asymmetric information is therefore a critical skill for quants and traders.

Step 18: Theoretical Frameworks – Adverse Selection and Moral Hazard

In economic theory, adverse selection describes situations where sellers have information that buyers do not, or vice versa, about some aspect of product quality. In our scenario, the buyer’s ability to act after observing the asset’s value creates an adverse selection problem for the seller.

Moral hazard arises when one party takes more risks because they do not bear the full consequences of those risks. While not directly at play in this problem, it’s closely related in real-world contract design.

Step 19: Mathematical Generalization – N-Outcome Case

If there are \( n \) possible asset values \( V_1, V_2, ..., V_n \) with probabilities \( p_1, ..., p_n \), and a strike price \( K \), the fair price of the contract is:

\[ \text{Fair Price} = \sum_{i=1}^n p_i \cdot \max(V_i - K, 0) \]

This is the discrete version of the expectation of a call option’s payoff.

Step 20: Coding a Generalized Solution

def fair_option_price(values, probabilities, strike):
    return sum(p * max(v - strike, 0) for v, p in zip(values, probabilities))

# Example: V = [15000, 2000], p = [0.9, 0.1], K = 6000
values = [15000, 2000]
probabilities = [0.9, 0.1]
strike = 6000

print("Fair contract price: $%.2f" % fair_option_price(values, probabilities, strike))

This function can handle any number of discrete outcome scenarios and is a powerful tool for similar interview questions.

Step 21: Real Interview Tips for Quant Interns

• Clarify the scenario: Restate the problem in your own words to ensure you understand the contract’s rules and information flow.
• Model step-by-step: Write out the possible states and the buyer’s optimal decisions.
• Compute expected payoffs: Use probability-weighted outcomes and remember to multiply by the payoff only when the buyer exercises the option.
• Generalize: Show you can apply the method to variants of the problem—this demonstrates deeper understanding.
• Relate to real-world finance: Mention connections to option pricing, adverse selection, or market making if time allows.

Step 22: Frequently Asked Questions

Q: Why does the buyer get the "max" payoff instead of always buying?

A: The buyer only purchases when it’s profitable (\( V - K > 0 \)); otherwise, they do nothing. This is the essence of optionality.

Q: How does this relate to American options?

A: In this problem, the buyer can exercise after observing the value, which is like a European option with perfect information at expiration. An American option allows early exercise; the logic is similar, but with more exercise opportunities.

Q: What if the probabilities or payoffs are continuous?

A: Integrate over the probability density function: \[ \text{Fair Price} = \int_{K}^{\infty} (v - K) f(v) dv \] where \( f(v) \) is the probability density function of \( V \).

Step 23: Conceptual Recap and Takeaways

• Asymmetric information creates value for the informed party (the buyer).
• The fair price of a contract is the expected value of its payoff, given optimal buyer behavior.
• Always model the incentives and possible actions of each party.
• Connect the scenario to real financial contracts like options for deeper understanding.

Step 24: Conclusion

Pricing contracts under asymmetric information is a foundational skill for quantitative trading and financial engineering. The SIG Quantitative Trading Intern interview question we explored demonstrates the interplay between probability, strategy, and economic incentives. By carefully modeling the buyer’s option to act after observing the asset’s value, we calculated the fair price as the expected value of the positive payoffs, weighted by their probabilities.

In our original problem, the fair contract price was $8,100. This method generalizes to any discrete or continuous scenario and is directly applicable both in interviews and in real-world trading situations. Mastering these concepts will give you a strong edge in your quant interviews and future finance career.

Further Reading and Resources

• Wikipedia: Asymmetric Information
• Wikipedia: Option (Finance)
• Investopedia: Adverse Selection
• SIG Interview Preparation

Practice, model, and always consider the information set of each party—that’s the quant edge!