The Scenario: You are a Junior Quant at a high-frequency trading firm. You observe two derivative contracts, Alpha and Beta, which track the same underlying volatility index. Usually, they move in lockstep. However, a sudden liquidity crunch has caused a temporary "dislocation."
Contract Alpha is trading at $102.
Contract Beta is trading at $98.
Your model proves that by the end of the trading day (i.e. in 2 hours from now), there is an 80% probability that both contracts will converge to the mean price of $100, and a 20% probability that they will both crash to $90 due to a systemic shock.
To capture the "spread" (the difference) while minimizing the risk of the 20% "crash" scenario, which of the following actions is the most mathematically sound?
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Think about Delta-Neutrality. In a "Pairs Trade," you want to profit from the prices coming back together (convergence). Calculate your profit/loss for both the "Convergence" scenario ($100) and the "Crash" scenario ($90) if you are "Long" the cheap asset and "Short" the expensive one.
Correct Answer: A (Buy 100 units of Beta and Sell 100 units of Alpha)
The Logic: This is a classic Long/Short Mean Reversion play.
Cost to Enter: You sell Alpha at $102 (cash inflow $102) and buy Beta at $98 (cash spend $98). You actually start with a +$4 net credit per pair.
Scenario 1: Convergence (80% prob):
Alpha drops from $102 to $100 (You gain $2 on your Short).
Beta rises from $98 to $100 (You gain $2 on your Long).
Total Profit: $4.
Scenario 2: Systemic Crash (20% prob)
Alpha drops from $102 to $90 (You gain $12 on your Short).
Beta drops from $98 to $90 (You lose $8 on your Long).
Total Profit: $4.
There is an 80% chance you will make $4 profit and 20% chance that you will make a $4 profit i.e. a 100% chance that you will make $4 profit in the above case.
The "Quant" Insight: Because you are "Market Neutral" (Shorting the overpriced and Buying the underpriced), the absolute direction of the market doesn't matter. Whether the market goes to $100 or crashes to $90, you lock in the $4 spread in the above example. This is the essence of Arbitrage!
To showcase how this works in a separate scenario, assume that the 20% probability case was one where BOTH the stocks ended up @ $105 (i.e. the convergence hypothesis breaks down) and even in that case, you would expect to end up in a profit. An 80% chance of a $4 profit and a 20% chance that you end up with a $2 profit!
Buy in Market A at ₹100, sell in Market B at ₹102.
Transaction cost = ₹1 per share (remember, this applies to both buy and sell transactions).
Net profit per share = (Sell price – Buy price – Costs) = 102 – 100 – 2 = ₹0
Conclusion: No arbitrage profit exists because transaction costs wipe out the gain. Always account for costs before jumping into what you think is an arbitrage!
The Scenario: You hold a European call option with strike price ₹50. The underlying asset has a 60% chance of ending at ₹70 and a 40% chance of ending at ₹40 at maturity.
What is the expected payoff of the option?
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Payoff = Max (Stock Price – Strike, 0). Multiply each payoff by its probability and sum them.
