Finance · 13 min read · ~28 min study · advanced
Butterfly Spread
How it works, payoff diagrams, when to use it — with Python code for both call and put butterflies.
Butterfly Spread: Strategy, Payoff & When to Use It in 2026
A clear explanation of the butterfly spread options strategy - how it works, payoff diagrams, when to use it, and Python code to model the trade. Covers both call and put butterflies.
What Is a Butterfly Spread?
A butterfly spread is a three-legged options strategy built around three strike prices that profits when the underlying asset finishes near the middle strike at expiry. It combines a bull spread and a bear spread into a single position with limited risk and limited profit potential, making it one of the most capital-efficient ways to bet on low volatility.
The structure uses four option contracts across three equally spaced strikes. You buy one contract at the lowest strike, sell two contracts at the middle strike, and buy one contract at the highest strike - all sharing the same expiry date. The result is a position that earns its maximum profit if the underlying price lands exactly at the middle strike and loses no more than the initial premium paid.
Butterfly spreads are popular among traders who expect a stock or index to stay within a tight range. Unlike a naked short straddle - which also profits from low volatility but carries theoretically unlimited risk - the butterfly caps your downside at the net premium you pay to enter the trade. This makes it attractive for defined-risk strategies in 2026, particularly around events where a trader has a strong view on where the price will settle but wants to avoid open-ended exposure.
The name comes from the payoff diagram. When you plot profit and loss against the underlying price at expiry, the shape has a peaked center and two flat wings - resembling a butterfly. The peak sits at the middle strike, and the wings extend to the lower and upper strikes where the position breaks even before flattening into maximum loss territory.
How Does a Butterfly Spread Work?
A long call butterfly spread works by buying one in-the-money call, selling two at-the-money calls, and buying one out-of-the-money call - all at equally spaced strikes and the same expiry. The net cost to enter the trade is small because the premium collected from selling two calls partially offsets the cost of buying the outer calls.
Let's walk through a concrete butterfly spread example. Suppose a stock is trading at $100 and you expect it to stay close to that price over the next 30 days. You could construct a long call butterfly as follows:
- Buy 1 call at the $95 strike (lower wing) for $6.00
- Sell 2 calls at the $100 strike (body) for $2.50 each ($5.00 total collected)
- Buy 1 call at the $105 strike (upper wing) for $0.20
The net debit is: $6.00 − $5.00 + $0.20 = $1.20
This $1.20 is your maximum possible loss per share. It occurs if the stock finishes below $95 or above $105 at expiry — in either case, all options expire worthless or cancel out to zero value.
The maximum profit occurs when the stock finishes exactly at $100 (the middle strike). At that point, the $95 call is worth $5.00 of intrinsic value, the two short $100 calls expire worthless, and the $105 call expires worthless. Your profit is $5.00 − $1.20 = $3.80 per share. That is a return of over 300% on the premium invested.
The equal spacing between strikes is critical. In this example, each strike is $5 apart. This symmetry is what gives the butterfly its characteristic shape and ensures the payoff is balanced around the center. The distance between strikes is called the wing width, and it determines both the maximum profit and the range within which the trade is profitable.
Two breakeven points exist:
- Lower breakeven = lower strike + net debit = $95 + $1.20 = $96.20
- Upper breakeven = upper strike − net debit = $105 − $1.20 = $103.80
The stock needs to finish between $96.20 and $103.80 for the trade to be profitable. This gives you a $7.60-wide profitable range on a $1.20 investment — a favorable risk-reward structure for range-bound scenarios.
For a deeper look at how option positions interact, see our guide to derivatives fundamentals.
Butterfly Spread Payoff
The butterfly spread payoff is zero outside the wing strikes, rises linearly from the lower breakeven to a peak at the middle strike, and then falls linearly back to zero at the upper breakeven. Maximum profit equals the wing width minus the net premium paid, and maximum loss is limited to the net premium.
Here is a payoff table for the long call butterfly example above (strikes at $95, $100, $105 with a $1.20 net debit), showing the profit or loss at various stock prices at expiry:
| Stock Price at Expiry | $95 Call Value | 2× $100 Call Value | $105 Call Value | Net Payoff | Profit / Loss |
|---|---|---|---|---|---|
| $90 | $0.00 | $0.00 | $0.00 | $0.00 | −$1.20 |
| $93 | $0.00 | $0.00 | $0.00 | $0.00 | −$1.20 |
| $95 | $0.00 | $0.00 | $0.00 | $0.00 | −$1.20 |
| $96.20 | $1.20 | $0.00 | $0.00 | $1.20 | $0.00 |
| $97 | $2.00 | $0.00 | $0.00 | $2.00 | +$0.80 |
| $98 | $3.00 | $0.00 | $0.00 | $3.00 | +$1.80 |
| $99 | $4.00 | $0.00 | $0.00 | $4.00 | +$2.80 |
| $100 | $5.00 | $0.00 | $0.00 | $5.00 | +$3.80 |
| $101 | $6.00 | −$2.00 | $0.00 | $4.00 | +$2.80 |
| $102 | $7.00 | −$4.00 | $0.00 | $3.00 | +$1.80 |
| $103 | $8.00 | −$6.00 | $0.00 | $2.00 | +$0.80 |
| $103.80 | $8.80 | −$7.60 | $0.00 | $1.20 | $0.00 |
| $105 | $10.00 | −$10.00 | $0.00 | $0.00 | −$1.20 |
| $107 | $12.00 | −$14.00 | $2.00 | $0.00 | −$1.20 |
| $110 | $15.00 | −$20.00 | $5.00 | $0.00 | −$1.20 |
The payoff diagram forms a triangle: flat at −$1.20 outside the wings, rising to +$3.80 at the center, then falling back to −$1.20. This triangular shape is the defining visual feature of a butterfly spread.
Formally, the payoff at expiry for a long call butterfly with strikes ( K_1 < K_2 < K_3 ) where ( K_2 = (K_1 + K_3) / 2 ) is:
[ \text{Payoff} = \max(S_T - K_1, 0) - 2 \cdot \max(S_T - K_2, 0) + \max(S_T - K_3, 0) ]
And the profit is:
[ \text{Profit} = \text{Payoff} - \text{Net Debit} ]
The maximum payoff of ( K_2 - K_1 ) occurs at ( S_T = K_2 ). Outside the range ([K_1, K_3]), the payoff is exactly zero, and the trade's loss equals the premium paid.
Butterfly Spread in Python
You can model the butterfly spread payoff across a range of stock prices with a short Python script. This is useful for visualizing how different strike selections and premium levels affect the profit profile.
import numpy as np
import matplotlib.pyplot as plt
def butterfly_payoff(S_T, K1, K2, K3, net_debit):
"""Calculate profit/loss for a long call butterfly spread."""
long_lower = np.maximum(S_T - K1, 0)
short_middle = -2 * np.maximum(S_T - K2, 0)
long_upper = np.maximum(S_T - K3, 0)
payoff = long_lower + short_middle + long_upper
return payoff - net_debit
# --- Trade parameters ---
K1 = 95 # lower strike (buy 1 call)
K2 = 100 # middle strike (sell 2 calls)
K3 = 105 # upper strike (buy 1 call)
net_debit = 1.20
# --- Generate payoffs across a range of prices ---
stock_prices = np.linspace(85, 115, 500)
profits = butterfly_payoff(stock_prices, K1, K2, K3, net_debit)
# --- Key levels ---
max_profit = (K2 - K1) - net_debit
lower_breakeven = K1 + net_debit
upper_breakeven = K3 - net_debit
print(f"Max profit: {max_profit:.2f}")
print(f"Max loss: {net_debit:.2f}")
print(f"Lower breakeven: {lower_breakeven:.2f}")
print(f"Upper breakeven: {upper_breakeven:.2f}")
print(f"Profit range: {upper_breakeven - lower_breakeven:.2f} wide")
# --- Payoff table ---
check_prices = [90, 93, 95, 96.20, 97, 98, 99, 100,
101, 102, 103, 103.80, 105, 107, 110]
print("\nStock Price | Payoff | Profit/Loss")
print("-" * 40)
for p in check_prices:
pnl = butterfly_payoff(np.array([p]), K1, K2, K3, net_debit)[0]
payoff = pnl + net_debit
print(f" {p:>7.2f} | {payoff:>6.2f} | {pnl:>+7.2f}")
# --- Plot ---
fig, ax = plt.subplots(figsize=(10, 6))
ax.plot(stock_prices, profits, color="#2563eb", linewidth=2)
ax.axhline(y=0, color="grey", linestyle="-", alpha=0.3)
ax.fill_between(stock_prices, profits, 0,
where=(profits > 0), alpha=0.15, color="#2563eb")
ax.axvline(x=K1, color="#ef4444", linestyle="--", alpha=0.5, label=f"K1 = {K1}")
ax.axvline(x=K2, color="#22c55e", linestyle="--", alpha=0.5, label=f"K2 = {K2}")
ax.axvline(x=K3, color="#ef4444", linestyle="--", alpha=0.5, label=f"K3 = {K3}")
ax.set_xlabel("Stock Price at Expiry", fontsize=12)
ax.set_ylabel("Profit / Loss", fontsize=12)
ax.set_title("Long Call Butterfly Spread Payoff", fontsize=14)
ax.legend(fontsize=11)
ax.grid(True, alpha=0.3)
plt.tight_layout
plt.savefig("butterfly_spread_payoff.png", dpi=150)
plt.showThe script produces the characteristic triangular payoff shape. You can adjust the strike width by changing ( K_1 ), ( K_2 ), and ( K_3 ) - wider wings increase maximum profit but require the stock to finish closer to the center for the trade to work. The butterfly_payoff function is vectorised with NumPy, so it handles arrays of prices efficiently.
Try modifying the parameters to see how the trade behaves with different strike spacings or net debits. For more on building financial models in Python, see our Python for quant finance guide.
Types of Butterfly Spreads
There are four main types of butterfly spread: the long call butterfly, long put butterfly, iron butterfly, and broken wing butterfly. Each uses a different combination of options but produces a similar payoff profile centered on the middle strike.
Long Call Butterfly
This is the standard version described above. You buy one lower-strike call, sell two middle-strike calls, and buy one upper-strike call. It's entered for a net debit and profits when the stock stays near the middle strike. The long call butterfly is the most common type and the one most traders think of when they hear "butterfly spread."
Long Put Butterfly
The put version works identically in terms of payoff at expiry. You buy one lower-strike put, sell two middle-strike puts, and buy one upper-strike put. At expiry, put-call parity guarantees the same profit profile as the call butterfly (assuming the same strikes and expiry). In practice, traders sometimes prefer the put butterfly when the lower-strike options have better liquidity or tighter spreads. The choice between call and put butterflies is often driven by execution costs rather than theoretical differences.
Iron Butterfly
The iron butterfly combines puts and calls. You sell one ATM put and one ATM call (creating a short straddle at the middle strike), then buy one OTM put at a lower strike and one OTM call at a higher strike for protection. Unlike the long call or put butterfly, the iron butterfly is entered for a net credit.
The structure looks like this:
- Buy 1 put at the lower strike
- Sell 1 put at the middle strike
- Sell 1 call at the middle strike
- Buy 1 call at the upper strike
Maximum profit equals the net credit received, which occurs when the stock finishes exactly at the middle strike. Maximum loss equals the wing width minus the credit received. The iron butterfly is functionally equivalent to a long butterfly - it's just constructed differently and entered as a credit rather than a debit.
Many traders prefer the iron butterfly because it collects premium upfront and can be easier to manage from a margin perspective. It's especially popular in index options trading in 2026 where the ATM straddle carries significant premium.
Broken Wing Butterfly
A broken wing butterfly (sometimes called a skip-strike butterfly) uses unequal spacing between the strikes. For example, you might set the strikes at $95, $100, and $110 instead of $95, $100, and $105. This shifts the payoff diagram: the trade may be entered for a credit on one side and has an asymmetric risk profile.
Traders use broken wing butterflies to create a directional bias while still maintaining a butterfly-like structure. The wider wing reduces risk on one side at the cost of increased exposure on the other. These are more advanced trades that require careful analysis of the risk-reward trade-off.
For background on how options pricing works across these structures, see our article on option pricing models.
When to Use a Butterfly Spread
Use a butterfly spread when you expect the underlying asset to trade within a narrow range and implied volatility is relatively high - giving you more premium to collect on the short strikes. The ideal setup is a range-bound market where you have a specific target price in mind for expiry.
Low volatility expectation. The butterfly spread is fundamentally a short volatility trade. You profit when the underlying doesn't move much. If you believe realized volatility over the option's life will be lower than what's currently priced into the market, the butterfly gives you a defined-risk way to express that view. For more on volatility dynamics, see our guide to the Greeks and volatility in options.
Range-bound markets. Stocks that have been consolidating in a tight range are classic butterfly candidates. If a stock has been trading between $95 and $105 for weeks and you expect that to continue, a butterfly centered at $100 lets you profit from that stasis.
Earnings plays. Some traders use butterflies around earnings announcements. The logic: implied volatility is typically inflated before earnings, making the short strikes valuable. If the trader believes the stock will have a muted reaction (perhaps because guidance is already well understood), the butterfly captures the post-earnings volatility crush at a low cost. The limited risk is crucial here because earnings can surprise.
Cheap directional bets. Because the butterfly costs little to enter, it can serve as a cheap way to bet on a specific price target. If you are convinced a stock will be near $110 in 30 days, you could center a butterfly at $110 for a small debit. The risk-reward ratio is excellent if you are right, and the maximum loss is capped at the premium.
Implied volatility is high. When implied volatility is elevated, the premium collected from the two short middle-strike options is larger, which reduces the net cost of the butterfly. This improves the risk-reward ratio. Conversely, in very low implied volatility environments, butterflies become less attractive because the premium collected is thin. If you're interested in strategies that sell volatility premium, the butterfly is one of the safest ways to do it.
The Greeks of a Butterfly Spread
A butterfly spread is approximately delta-neutral near the middle strike, has positive theta that accelerates as expiry approaches, and carries negative vega - meaning it benefits from falling implied volatility. These Greek characteristics make it behave like a short volatility position with capped risk.
Delta
At inception, a butterfly centered at-the-money has near-zero delta. The long lower call has positive delta, the two short middle calls have negative delta, and the long upper call has positive delta - they roughly cancel out. As the underlying moves away from the middle strike, delta becomes non-zero: the butterfly develops a directional bias. If the stock rises above the middle strike, the position becomes short delta (it benefits from the stock falling back). If the stock falls below the middle strike, it becomes long delta.
Gamma
Gamma is the rate of change of delta. A long butterfly has negative gamma when the underlying is near the middle strike and positive gamma near the wings. This is intuitive: around the middle strike, you don't want the stock to move (negative gamma works against you if it does), while near the wings, movement toward the center helps you. The gamma profile flips sign as you move across the strikes.
Theta
Theta is where the butterfly earns its keep. The position has positive theta when the underlying is near the middle strike, meaning it profits from the passage of time. The two short options at the middle strike decay faster than the two long options at the wings, generating net time decay in your favor. This theta accelerates as expiry approaches - making the final week or two the most profitable period if the stock remains range-bound.
Vega
A butterfly spread has negative vega near the middle strike. This means the position loses value if implied volatility rises and gains value if it falls. This is consistent with the trade's purpose: you're betting on low movement, so a drop in implied volatility benefits you. The vega exposure is modest compared to a naked short straddle because the long wings offset much of the short vega from the middle strikes.
| Greek | Near Middle Strike | Near Wings |
|---|---|---|
| Delta | ~0 (neutral) | Positive or negative |
| Gamma | Negative | Positive |
| Theta | Positive (earns time decay) | Negative |
| Vega | Negative (benefits from vol drop) | Positive |
Understanding how these Greeks interact is essential for managing butterfly positions. For a thorough treatment of the Greeks and their behavior, see our dedicated guide.
Butterfly Spread vs Other Strategies
The butterfly spread sits in a family of limited-risk strategies that profit from range-bound markets. Choosing between them depends on your view on the width of the expected range, the cost you're willing to pay, and how much risk you can tolerate.
| Feature | Butterfly Spread | Iron Condor | Short Straddle | Short Strangle |
|---|---|---|---|---|
| Market view | Pinpoint target price | Broad range | Low volatility | Low volatility |
| Max profit | Wing width - premium | Net credit | Premium collected | Premium collected |
| Max loss | Net premium paid | Wing width - credit | Unlimited | Unlimited |
| Risk profile | Limited | Limited | Unlimited | Unlimited |
| Breakeven range | Narrow | Wide | Moderate | Wide |
| Capital required | Low (net debit) | Moderate (margin) | High (margin) | High (margin) |
| Best volatility environment | High IV (cheap entry) | High IV | High IV | High IV |
| Typical use | Precise target | Range-bound | Strong low-vol view | Strong low-vol view |
Butterfly vs iron condor. An iron condor sells an OTM put spread and an OTM call spread simultaneously. It profits across a wider range than a butterfly but offers a lower maximum profit. Think of the iron condor as the butterfly's wider, flatter cousin - it's more forgiving on direction but pays less when you're right. If you have a precise price target, the butterfly is better. If you just think the stock will stay in a broad range, the iron condor is more appropriate.
Butterfly vs short straddle. A short straddle (sell ATM call and ATM put) collects the most premium and has the highest maximum profit at the middle strike. But the risk is theoretically unlimited if the stock moves sharply. The butterfly sacrifices some profit potential in exchange for capping the downside. For most retail traders and risk-conscious professionals in 2026, the butterfly's defined risk is worth the trade-off.
Butterfly vs short strangle. A short strangle sells OTM options on both sides. It has a wider profitable range than a straddle but still carries unlimited risk. The butterfly is the defined-risk alternative: you can think of it as a straddle with insurance purchased on both sides.
The pattern is clear: the butterfly trades maximum profit potential for maximum safety. Where the straddle and strangle leave you exposed to large moves, the butterfly puts a hard floor under your losses.
Practical Tips and Common Mistakes
Setting up a butterfly spread correctly involves more than picking three strikes. Execution quality, timing, and position management all affect whether the trade works in practice.
Choose liquid options. A butterfly involves four contracts across three strikes. If any of those strikes has wide bid-ask spreads, slippage eats into your edge. Stick to highly liquid underlyings - major indices, large-cap stocks, and popular ETFs. In 2026, weekly options on indices like the S&P 500 and S&P 500 offer excellent liquidity for butterfly trades.
Mind the transaction costs. Four contracts mean four commissions (or more, if you trade individual legs). On a trade that costs $1.20 to enter with a maximum profit of $3.80, even $0.40 in total commissions per leg represents a meaningful drag. Use a broker with competitive options fees and consider entering the butterfly as a single multi-leg order rather than legging in.
Watch for early assignment risk. If you're trading American-style options and your short strikes go in-the-money before expiry, you face the possibility of early assignment. This is most common with short calls on stocks approaching an ex-dividend date. Early assignment doesn't necessarily hurt you - it just changes the position into something you need to manage. European-style index options (like S&P 500 options) eliminate this risk entirely because they can only be exercised at expiry.
Choose your strike width carefully. Narrower wings (smaller spacing between strikes) produce a cheaper butterfly with a higher percentage return if you're exactly right, but the profitable range is also narrower. Wider wings cost more but give you a larger window for profit. Match the wing width to your conviction level: use narrow wings when you have a strong, precise view on the settlement price, and wider wings when you're less certain.
Time your entry. Butterflies benefit from time decay, but they also need time for the trade to develop. Entering too early (say, 60 days before expiry) means you'll hold the position through a long period where theta is modest. Many traders prefer to enter butterflies with 15 to 30 days until expiry, when theta acceleration is beginning but there's still time for the underlying to settle into the target range.
Don't hold to expiry by default. The textbook butterfly shows maximum profit at expiry, but in practice many traders close the position before expiration. If the stock is near your middle strike and you've captured 50-70% of the maximum profit with several days still remaining, it often makes sense to close and remove the risk of a last-minute move. Pin risk - the uncertainty around whether short options expire in-the-money by a tiny amount - can cause unexpected outcomes on expiry day.
Understand the profit zone, not just max profit. Beginners sometimes focus on the maximum profit at the middle strike and forget that the trade needs the stock to finish within the breakeven range to make any money at all. Before entering, calculate your breakevens and assess whether a finish within that range is realistic given the stock's recent behavior and upcoming catalysts.
Frequently Asked Questions
What is the maximum loss on a butterfly spread?
The maximum loss on a butterfly spread is the net premium paid to enter the trade (for a long call or long put butterfly) or the wing width minus the credit received (for an iron butterfly). This loss occurs if the underlying finishes below the lowest strike or above the highest strike at expiry. For example, a butterfly entered for a $1.20 net debit has a maximum loss of exactly $1.20 per share, regardless of how far the stock moves. This defined-risk characteristic is one of the butterfly's main attractions compared to unlimited-risk strategies like naked straddles.
Is a butterfly spread bullish or bearish?
A standard at-the-money butterfly spread is neither bullish nor bearish — it is a neutral strategy that profits from the stock staying near the middle strike. However, you can create directional butterflies by shifting the center strike above or below the current price. A butterfly centered at $105 on a $100 stock is a mildly bullish bet that the stock will rise to $105 by expiry. A butterfly centered at $95 is a bearish bet. The directional bias depends entirely on where you place the middle strike relative to the current price.
How does the iron butterfly differ from a regular butterfly?
The iron butterfly produces the same payoff at expiry as a long call or long put butterfly with the same strikes. The structural difference is that the iron butterfly combines puts and calls (selling an ATM straddle and buying protective wings), while a regular butterfly uses all calls or all puts. The practical difference is in how the trade is entered: an iron butterfly is entered for a net credit, while a long call or put butterfly is entered for a net debit. Many traders prefer the iron butterfly because collecting premium upfront feels more intuitive, and the margin treatment can be more favorable with some brokers. Functionally, they're equivalent trades.
When should I close a butterfly spread before expiry?
Consider closing a butterfly spread early when you've captured a significant portion of the maximum profit - typically 50% to 70% - and there's still time remaining until expiry. The reason is asymmetric risk: once you've captured most of the available profit, the remaining potential gain is small relative to the risk that the stock moves away from your target. Additionally, close early if the underlying has moved significantly away from the middle strike and the remaining value is minimal - there's little point holding a position that's already near maximum loss in the hope of a reversal. Pin risk around expiry is another reason to close early, especially with American-style options.
Can I use a butterfly spread around earnings announcements?
Yes, and it's a common application. The strategy works well when you believe implied volatility is overpriced relative to the actual move the stock will make post-earnings. Before earnings, implied volatility is typically elevated as the market prices in uncertainty. If the stock reacts less than expected, the post-earnings volatility crush benefits the butterfly's negative vega exposure. The key risk is that earnings surprises can produce large gaps that push the stock well outside the butterfly's profitable range. Because your loss is capped at the premium paid, earnings butterflies offer a defined-risk way to express a view that the market is overestimating the size of the earnings move.
How does the butterfly spread relate to the Black-Scholes model?
Under the Black-Scholes framework, a butterfly spread's fair value can be calculated by pricing each of the four option legs individually and summing them. The net theoretical value depends on the underlying price, time to expiry, risk-free rate, and implied volatility. Interestingly, a very narrow butterfly spread (with strikes close together) approximates a bet on the probability density at the middle strike - it's proportional to the second derivative of the option price with respect to strike. Quantitative researchers use butterfly quotes to extract information about the implied probability distribution. In practice, the Black-Scholes model is a starting point, but real butterfly pricing also reflects the volatility smile, skew, and term structure effects.
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What You Will Learn
- Explain what is a butterfly spread.
- Build how does a butterfly spread work.
- Calibrate butterfly spread payoff.
- Compute butterfly spread in Python.
- Design types of butterfly spreads.
- Implement when to use a butterfly spread.
Prerequisites
- Derivatives intuition — see Derivatives intuition.
- Options Greeks — see Options Greeks.
- Comfort reading code and basic statistical notation.
- Curiosity about how the topic shows up in a US trading firm.
Mental Model
Markets are auctions for risk. Every product, model, and strategy in this section is a way of pricing or transferring some piece of risk between counterparties — and US markets give you the deepest, most regulated, most algorithmic version of that auction in the world. For Butterfly Spread, frame the topic as the piece that how it works, payoff diagrams, when to use it — with Python code for both call and put butterflies — and ask what would break if you removed it from the workflow.
Why This Matters in US Markets
US markets are the deepest, most algorithmic, most regulated capital markets in the world. The SEC, CFTC, FINRA, and Federal Reserve govern equities, options, futures, treasuries, and OTC derivatives. The big buy-side (Bridgewater, AQR, Citadel, Two Sigma, Renaissance) and the major sell-side (GS, MS, JPM, Citi, BofA) hire heavily against the material in this section.
In US markets, Butterfly Spread tends to surface during onboarding, code review, and the first incident a junior quant gets pulled into. Questions on this material recur in interviews at Citadel, Two Sigma, Jane Street, HRT, Jump, DRW, IMC, Optiver, and the major bulge-bracket banks.
Common Mistakes
- Quoting risk-free rates without saying which curve (T-bill, OIS, fed funds futures).
- Treating implied volatility as a forecast instead of a market-clearing quantity.
- Using realized correlation as a hedge ratio without accounting for regime change.
- Treating Butterfly Spread as a one-off topic rather than the foundation it becomes once you ship code.
- Skipping the US-market context — copying European or Asian conventions and getting bitten by US tick sizes, settlement, or regulator expectations.
- Optimizing for elegance instead of auditability; trading regulators care about reproducibility, not cleverness.
- Confusing model output with reality — the tape is the source of truth, the model is a hypothesis.
Practice Questions
- Compute the delta of an at-the-money call on SPY with one month to expiry under Black-Scholes (σ=18%, r=5%).
- Why does the implied volatility surface for SPX exhibit a skew rather than a flat smile?
- Define the Sharpe ratio and explain why it is annualized.
- Why does delta-hedging a sold straddle on SPY produce P&L proportional to realized minus implied variance?
- What does a 100 bps move in the 10-year Treasury yield typically do to a 30-year fixed-rate mortgage rate?
Answers and Explanations
- Δ = N(d1) where d1 = (ln(S/K) + (r + σ²/2)T) / (σ√T). With S=K, T=1/12, σ=0.18, r=0.05: d1 ≈ (0 + (0.05 + 0.0162)·0.0833) / (0.18·0.2887) ≈ 0.106; N(0.106) ≈ 0.542. Delta ≈ 0.54.
- Because investors pay a premium for downside protection (left tail) and equity returns are negatively correlated with volatility; out-of-the-money puts therefore trade rich relative to OTM calls.
- Sharpe = (excess return) / (volatility). Annualization (multiply by √252 for daily returns) puts strategies of different frequencies on comparable footing — a key requirement for comparing US asset managers.
- Because the hedger captures gamma·dS² over time; integrating gives Σ gamma·(dS)², and theta paid over the life is set by implied variance. Net P&L tracks σ_realized² − σ_implied² scaled by gamma exposure.
- Roughly 75-100 bps move the same direction; mortgages are priced off the 10y plus a spread that includes prepayment risk and originator margin, which both move with rates.
Glossary
- Delta — first derivative of option price with respect to underlying.
- Gamma — second derivative; rate of change of delta.
- Vega — sensitivity of option price to implied volatility.
- Theta — time decay; daily P&L from holding the option as expiry approaches.
- Implied volatility — the σ that, when plugged into Black-Scholes, recovers the market price.
- Skew — variation of implied volatility across strikes.
- Spread — the difference between two prices; a yield curve, an option spread, or a cross-instrument arb.
- Sharpe ratio — annualized excess return divided by annualized volatility; the standard performance metric in US asset management.
Further Study Path
- Understanding Financial Markets — Equity, fixed income, FX, derivatives — how markets actually work and where quants fit in.
- Time Value of Money — Present value, future value, discounting, NPV — the concept that underpins all of finance.
- Bonds and Fixed Income — Pricing, yield to maturity, duration, convexity — the fixed-income concepts behind interest-rate modeling.
- Python for Quant Finance: Fundamentals — Variables, functions, data structures, classes, and error handling — the core Python every quant role expects.
- Advanced Python for Financial Applications — Decorators, generators, and context managers — the patterns that separate beginner Python from production quant code.
Key Learning Outcomes
- Explain what is a butterfly spread.
- Apply how does a butterfly spread work.
- Recognize butterfly spread payoff.
- Describe butterfly spread in Python.
- Walk through types of butterfly spreads.
- Identify when to use a butterfly spread.
- Articulate the greeks of a butterfly spread.
- Trace options as it applies to butterfly spread.
- Map strategies as it applies to butterfly spread.
- Pinpoint how butterfly spread surfaces at Citadel, Two Sigma, Jane Street, or HRT.
- Explain the US regulatory framing — SEC, CFTC, FINRA — relevant to butterfly spread.
- Apply a single-paragraph elevator pitch for butterfly spread suitable for an interviewer.
- Recognize one common production failure mode of the techniques in butterfly spread.
- Describe when butterfly spread is the wrong tool and what to use instead.
- Walk through how butterfly spread interacts with the order management and risk gates in a US trading stack.
- Identify a back-of-the-envelope sanity check that proves your implementation of butterfly spread is roughly right.
- Articulate which US firms publicly hire against the skills covered in butterfly spread.
- Trace a follow-up topic from this knowledge base that deepens butterfly spread.
- Map how butterfly spread would appear on a phone screen or onsite interview at a US quant shop.
- Pinpoint the day-one mistake a junior would make on butterfly spread and the senior's fix.