Delta Hedging - Becoming Delta-Neutral

Level 2: Intermediate | Module 2.5 | Time: 2.5 hours

🎯 Learning Objectives

By the end of this module, you will:

Understand what delta hedging is and why it matters
Learn when to hedge (market makers, volatility traders)
Master rehedging frequency trade-offs
Calculate transaction costs vs risk reduction
Implement dynamic hedging strategies

Prerequisites: The Greeks

What is Delta Hedging?

Delta hedging is the practice of offsetting the directional (delta) risk of an options position by taking an opposing position in the underlying asset.

The Core Concept

Problem:
You sold 100 calls with delta = 0.40 each
Portfolio delta = 100 × 0.40 = +40
If Bitcoin rises $1,000, you lose $40,000 ❌

Solution: Delta hedge
Buy 40 Bitcoin (each has delta = 1.00)
Hedge delta = 40 × 1.00 = -40
Net portfolio delta = +40 - 40 = 0 ✅

Result: Delta-neutral
Bitcoin rises $1,000 → No net gain/loss from price move
Now you profit from theta, gamma, vega - not direction!

Why Delta Hedge?

You isolate specific Greeks you want to trade:

Without hedge:
✅ Profit from theta (time decay)
❌ Exposed to delta (directional risk)
❌ Exposed to gamma (acceleration risk)

With delta hedge:
✅ Profit from theta (time decay)
✅ Neutral to price direction (delta = 0)
✅ Can profit from gamma (via rehedging)
✅ Clean volatility exposure (vega)

Who Uses Delta Hedging?

1. Market Makers

Market makers MUST hedge to stay in business.

Market Maker Day:

9:00 AM - Client buys 50 calls
You sold 50 calls → Delta = -20 (short delta)
Action: Immediately buy 20 BTC to hedge ✅

10:30 AM - Client sells 30 puts
You bought 30 puts → Delta = +9 (now net -11)
Action: Sell 9 BTC to rehedge ✅

2:00 PM - Client buys 20 more calls
You sold 20 calls → Delta = -8 (now net -19)
Action: Buy 8 BTC to rehedge ✅

End of day: Delta ≈ 0
Market maker profits from bid-ask spread + theta
NOT from directional bets!

2. Volatility Traders

Trade volatility, not direction.

Volatility Arbitrage Strategy:

Observation:
Implied Vol: 80%
Expected Realized Vol: 60%

Trade:
Sell options (short vega, short gamma)
Delta hedge → Remove directional risk
Profit from: IV > RV (collect theta faster than hedging costs)

This REQUIRES delta hedging to isolate volatility bet.

3. Structured Product Issuers

Banks issuing PPNs, autocallables, etc.

Bank issues:
$100M Principal Protected Note on Bitcoin

Components:
- $95M in bonds (no delta)
- $5M in call options (positive delta)

Bank's delta exposure: +positive

Action: Short Bitcoin to delta hedge
Result: Bank profits from fees, not market direction

4. Risk-Averse Option Sellers

Collect premium without directional risk.

Investor sells covered calls:

Position:
- Long 10 BTC (delta = +10)
- Short 10 calls, delta = -0.35 each (delta = -3.5)
- Net delta = +6.5 (still bullish!)

To be truly neutral:
Sell 6.5 BTC (or short via derivatives)
Net delta = 0
Collect theta without directional exposure

Delta Hedging Mechanics

Step 1: Calculate Current Delta

Portfolio:
- Short 100 calls at $55k, delta = 0.40 each
- Portfolio delta = 100 × 0.40 = +40

Interpretation:
Acts like owning 40 Bitcoin
If BTC rises $1,000 → Gain ~$40,000

Step 2: Determine Hedge Ratio

To neutralize +40 delta:

Option A: Buy 40 BTC spot
Hedge delta = 40 × 1.00 = -40
Net = 0 ✅

Option B: Buy 40 BTC futures
Same effect, potentially lower costs

Option C: Buy 200 calls at $60k (delta = 0.20)
Hedge delta = 200 × 0.20 = -40
But this adds gamma/vega exposure!

Best Practice: Hedge with underlying asset (simplest, cleanest).

Step 3: Execute Hedge

Execute:
Buy 40 BTC at market

Result:
- Short calls: delta = +40
- Long BTC hedge: delta = -40
- Net delta = 0 ✅

Portfolio is now delta-neutral!

Step 4: Monitor and Rehedge

Delta changes as underlying price moves (gamma effect!).

Initial:
BTC = $50,000
Portfolio delta = 0 (perfect hedge)

BTC rises to $51,000:
Calls now delta = 0.45 (gamma increased delta)
Portfolio delta = (100 × 0.45) + (-40) = +5

Action: Buy 5 more BTC to rehedge
New hedge: -45 BTC
Net delta = 0 again ✅

This is continuous rehedging!

Rehedging Frequency

The Trade-Off

More frequent rehedging:

Pros:
✅ Delta stays closer to 0 (less directional risk)
✅ Better gamma capture (buy low, sell high)

Cons:
❌ Higher transaction costs (fees, slippage)
❌ More time and effort
❌ Overhedging risk in volatile markets

Less frequent rehedging:

Pros:
✅ Lower transaction costs
✅ Less time intensive
✅ Avoid overtrading

Cons:
❌ Delta drifts from 0 (directional exposure)
❌ Miss gamma scalping opportunities
❌ Larger rehedge sizes (more slippage)

Rehedging Strategies

1. Fixed-Delta Threshold

Rule: Rehedge when delta exceeds ±5

Example:
Initial: Delta = 0
BTC moves → Delta = +4 (don't rehedge)
BTC moves more → Delta = +6 (REHEDGE! ✅)

2. Fixed-Time Intervals

Rule: Rehedge every 6 hours

9 AM: Check delta, rehedge to 0
3 PM: Check delta, rehedge to 0
9 PM: Check delta, rehedge to 0
3 AM: Check delta, rehedge to 0

Simple, systematic

3. Gamma-Based Rehedging

Rule: Rehedge more frequently when gamma is high

High gamma (near expiry):
- Rehedge every 1-2 hours

Low gamma (long-dated):
- Rehedge daily or less

Adapts to risk level

4. Volatility-Based Rehedging

Rule: Rehedge more when market is moving

High volatility day (±5% moves):
- Rehedge every 2 hours

Low volatility (±1% moves):
- Rehedge once daily

Matches market conditions

Transaction Costs vs Risk Reduction

The P&L Components

Delta-hedged option selling P&L:

Total P&L = Theta Gain + Gamma P&L - Transaction Costs - Financing Costs

Theta Gain: Premium decay (positive if short options)
Gamma P&L: Profit from rehedging (buy low, sell high)
Transaction Costs: Fees from rehedging
Financing Costs: Interest on borrowed funds (if applicable)

Example: 30-Day Delta-Hedged Trade

Position:
Sell 100 ATM straddles on Bitcoin at $50k
Premium collected: $10,000
Delta: 0 (initially)
Gamma: +0.05 (high!)
Theta: +$300/day

Scenario A: Low volatility month (realized vol = 50%)
Bitcoin range: $48k - $52k

Daily rehedging costs:
- Average 20 BTC rehedged per day
- Fees: 0.1% = $100/day
- 30 days × $100 = $3,000 total

P&L:
+ Theta: 30 × $300 = +$9,000 ✅
+ Gamma: Minimal (small moves)
- Transaction costs: -$3,000
- Net: +$6,000

Success! Theta exceeded hedging costs.

Scenario B: High volatility month (realized vol = 120%)
Bitcoin range: $40k - $60k (wild swings!)

Daily rehedging costs:
- Average 60 BTC rehedged per day (volatile!)
- Fees: 0.1% × higher volumes = $300/day
- 30 days × $300 = $9,000 total

P&L:
+ Theta: 30 × $300 = +$9,000
+ Gamma: Minimal (dynamic hedge offsets)
- Transaction costs: -$9,000
- Net: $0

Break-even. Hedging costs ate all theta gains.

Key Insight: Need realized vol < implied vol to profit from short vol + delta hedge.

Dynamic Hedging

Adjusting hedge continuously as market moves.

The Perfect Hedge (Theoretical)

Continuous rehedging:
- Every microsecond, adjust hedge
- Always delta = 0
- Perfect gamma capture

Reality:
- Impossible (infinite transaction costs!)
- Theoretical benchmark only

Practical Dynamic Hedging

Balance perfection with costs.

Dynamic Hedging Algorithm:

1. Set delta tolerance: ±3
2. Set time check interval: Every hour
3. Monitor portfolio delta
4. IF |delta| > 3:
     Rehedge to delta = 0
   ELSE:
     Do nothing

5. Repeat

Result: Stays close to delta-neutral without overtrading

Delta Hedging During Events

Before Event (High IV):
- Delta changes slowly (despite high IV)
- Rehedge less frequently
- Gamma is moderate

During Event (Price spike):
- Delta changes RAPIDLY
- Rehedge immediately
- Critical to stay neutral

After Event (IV crush):
- Delta stabilizes
- Resume normal rehedging schedule

Gamma Scalping in Practice

Profiting from rehedging while delta-neutral.

How It Works

You're delta-neutral with positive gamma:

Bitcoin at $50,000
Portfolio delta = 0

Bitcoin rises to $52,000:
Gamma effect → Portfolio delta now +4 (positive)

Action: Sell 4 BTC at $52,000 to rehedge
You sold high! ✅

Bitcoin falls to $50,000:
Gamma effect → Portfolio delta now -4 (negative)

Action: Buy 4 BTC at $50,000 to rehedge
You bought low! ✅

Net result:
Sold 4 BTC at $52,000 = $208,000
Bought 4 BTC at $50,000 = $200,000
Gamma scalping profit: $8,000 ✅

Requirements for Profitable Gamma Scalping

1. Positive gamma (long options)
✅ Gives you the convexity to profit from rehedging

2. Sufficient volatility
✅ Need price moves to scalp

3. Realized vol > Implied vol
✅ Movement exceeds what you paid (theta)

4. Low transaction costs
✅ Fees don't eat profits

When it works:

Realized volatility exceeds implied volatility
You have positive gamma
Can rehedge efficiently

When it fails:

Realized vol < implied vol (theta exceeds gamma gains)
Transaction costs too high
Whipsaw markets (constant reversals)

Hedging Errors & Risks

Error 1: Incorrect Delta Calculation

❌ Wrong:
Used stale delta from yesterday
Bitcoin moved significantly
Your hedge is way off

✅ Right:
Recalculate Greeks in real-time
Use current prices
Adjust for intraday moves

Error 2: Ignoring Other Greeks

❌ Wrong:
Hedged delta perfectly
Ignored massive vega exposure
IV spiked, lost money anyway!

✅ Right:
Monitor ALL Greeks
Delta hedge removes directional risk
But vega, gamma, theta still affect P&L

Error 3: Over-Hedging

❌ Wrong:
Bitcoin at $50k, delta = +2
Rehedge
Bitcoin at $50.1k, delta = +2.5
Rehedge again (too soon!)
Bitcoin at $50.2k, delta = +3
Rehedge again (overtrading!)

Result: Death by transaction costs

✅ Right:
Set minimum thresholds
Only rehedge when meaningful drift occurs
Balance precision vs costs

Error 4: Slippage in Illiquid Markets

❌ Wrong:
Need to buy 100 BTC immediately
Market is thin
Slippage: 1% = $50,000 loss ❌

✅ Right:
Scale into hedge gradually
Use limit orders
Consider derivatives (futures) for size

Advanced Delta Hedging Strategies

1. Partial Delta Hedging

Hedge only percentage of delta.

Strategy: Hedge 80% of delta, keep 20% unhedged

Position delta: +40
Hedge: Buy 32 BTC (80% of 40)
Remaining delta: +8

Why?
- Reduce hedging costs (20% fewer trades)
- Accept small directional exposure
- Still mostly neutral

2. Delta Band Hedging

Rehedge only when outside range.

Target delta: 0
Band: ±5

Current delta: +3 → DON'T hedge (within band)
Current delta: +6 → HEDGE back to 0 (outside band)

Reduces unnecessary rehedging

3. Cross-Hedging with Correlated Assets

Use correlated asset instead of underlying.

Example: Hedge Bitcoin options with Bitcoin futures

Advantages:
- Lower transaction costs (futures fees < spot)
- Better liquidity
- Leverage (less capital required)

Disadvantages:
- Basis risk (futures ≠ spot exactly)
- Rollover costs

4. Ratio Hedging

Hedge with options instead of underlying.

Position: Short 100 calls, delta = +40

Instead of buying 40 BTC:
Buy 200 calls at higher strike (delta = 0.20 each)
Hedge delta: 200 × 0.20 = -40 ✅

Effect:
- Delta neutral ✅
- But changes gamma, vega, theta exposures
- More complex to manage

Real-World Case Study: Market Maker Delta Hedging

Scenario

Market Maker: Making markets in Bitcoin options
Daily volume: 1,000 options traded
Goal: Profit from bid-ask spread, not direction

Intraday Delta Management

Time  | Trade           | Position Δ | Action       | Hedge Δ | Net Δ
------|-----------------|------------|--------------|---------|------
9:00  | Start of day    | 0          | -            | 0       | 0
9:15  | Sold 50 calls   | +18        | Buy 18 BTC   | -18     | 0
10:00 | Bought 30 puts  | +27        | Buy 9 BTC    | -27     | 0
11:30 | BTC $50k→$51k   | +32        | Buy 5 BTC    | -32     | 0
13:00 | Sold 40 puts    | +20        | Sell 12 BTC  | -20     | 0
15:00 | Bought 25 calls | +11        | Sell 9 BTC   | -11     | 0
16:00 | End of day      | +8         | Buy 3 BTC    | -11     | -3

End of day: Delta ≈ 0 (close enough)

P&L Breakdown

Revenue:
Bid-ask spread capture: $50,000
Theta collected: $15,000
Total revenue: $65,000

Costs:
Hedging transaction fees: $8,000
Gamma losses (fast markets): $5,000
Total costs: $13,000

Net profit: $65,000 - $13,000 = $52,000

Daily success!

Practice Exercise: Delta Hedge a Position

Given

Your position:
- Sold 50 Bitcoin straddles
- Strike: $50,000
- Call delta: 0.52 each
- Put delta: -0.48 each
- Current BTC price: $50,000

Questions:
1. What's your current net delta?
2. How do you hedge to delta-neutral?
3. BTC rises to $51,000 - new call delta = 0.58, put delta = -0.42. What's your new delta?
4. How do you rehedge?

Click for solution

Answer 1: Current net delta

Sold straddles = sold calls + sold puts

Calls: 50 × -0.52 = -26 (short calls = negative delta)
Puts: 50 × -(-0.48) = +24 (short puts = positive delta)

Net delta = -26 + 24 = -2

Answer 2: Initial hedge

You have -2 delta (slightly short)
To neutralize: Sell 2 BTC (gives you +2 delta)

Net: -2 + 2 = 0 ✅ Delta-neutral

Answer 3: New delta after BTC rises to $51k

Calls: 50 × -0.58 = -29
Puts: 50 × -(-0.42) = +21
Options delta = -29 + 21 = -8

Hedge (2 BTC short) = +2

Net delta = -8 + 2 = -6

Answer 4: Rehedge action

Currently -6 delta
Need to sell 4 more BTC (4 × 1 = -4 delta from hedge)

New total hedge: Short 6 BTC
Net delta: -8 + 6 = -2 (close to neutral, good enough!)

Note: You sold high at $51k (good gamma scalping!)

Key Takeaways

1. Delta hedging removes directional risk

Take offsetting position in underlying
Achieve delta-neutral portfolio
Isolate other Greeks (gamma, theta, vega)

2. Who uses it: Market makers, vol traders, issuers

Market makers: MUST hedge (core business)
Vol traders: To isolate volatility bets
Issuers: Hedge structured products

3. Rehedging frequency is a trade-off

More frequent: Lower risk, higher costs
Less frequent: Higher risk, lower costs
Balance based on gamma, volatility, costs

4. Gamma scalping profits from rehedging

Buy low, sell high automatically
Requires positive gamma
Works when realized vol > implied vol

5. Transaction costs can eat profits

Every rehedge incurs fees
Must exceed theta collection
Critical to monitor

6. Dynamic hedging requires discipline

Set clear rules
Automate when possible
Monitor continuously
Avoid overtrading

What’s Next?

You’ve completed all Level 2 Intermediate modules! You now understand:

✅ Black-Scholes pricing
✅ Monte Carlo simulation
✅ Volatility analysis
✅ The Greeks
✅ Delta hedging

Ready for advanced professional strategies?

Continue to: Multi-Asset Structures →

Enter Level 3 and learn correlation trading, basket options, and complex multi-underlying strategies.

Tools & Resources

Hedging Tools:

Delta Calculator - Calculate hedge ratios
Rehedge Optimizer - Optimize frequency
Cost Analyzer - Transaction cost estimation

Simulations:

Gamma Scalping Simulator - Practice rehedging
Dynamic Hedging Backtest - Test strategies

Next Module: Multi-Asset Structures →

Related Topics:

The Greeks - Foundation for hedging
Gamma Scalping - Advanced techniques
Market Making - Professional application