Non-LinearStatistical1 min read
◆ The PatternProtective puts, collars, and delta-neutral overlays
Options provide non-linear hedging: a protective put caps downside while preserving upside. A collar finances the put by selling an upside call, reducing net cost.
Collar payoff = Stock + Put(K1) − Call(K2)
Delta hedging continuously adjusts the hedge ratio. The cost is realized volatility — if realized vol < implied vol, the hedge is profitable; otherwise it is a drag.
// Interactive — put strike and payoff profile
Put strike %
| Strategy | Max Loss | Max Gain | Net Cost |
|---|---|---|---|
| Protective put | Premium | Unlimited | Premium paid |
| Collar | Floored | Capped | Low / zero |
| Delta hedge | Slippage | Vol spread | Variable |
Pattern bridge: A protective put pays max(K − S, 0) — the same hinge as the ReLU in activation functions, flat on one side and linear on the other. Loss aversion is why investors pay for that floor. The same idea for exchange-rate exposure, usually with forwards, is currency hedging.
Worked example
A stock at 100, a 90 put for 2.5 and a 115 call sold for 2.0. With the put alone, the worst case is −12.5 whether the stock ends at 90 or 70, and at 130 you keep +27.5. The collar uses the call premium to cut the floor to −10.5, but caps the gain at +14.5. Protection is paid for either in premium or in upside.
Where it misleads
- Puts are usually priced above the volatility that follows, so rolling them continuously costs more over time than the losses they prevent for most investors (Israelov 2019).
- The hedge only covers the period you bought; crashes that start just after an expiry are not covered.
- Delta hedging needs continuous trading; in a gap it fails the way portfolio insurance did in 1987.
In code
S0, put_k, put_cost, call_k, call_premium = 100, 90, 2.5, 115, 2.0 # premiums made up ST = np.array([70, 90, 100, 115, 130]) # where the stock ends stock = ST - S0 protective = stock + np.maximum(put_k - ST, 0) - put_cost collar = protective - np.maximum(ST - call_k, 0) + call_premium
Assumes import numpy as np and import pandas as pd. Each snippet carries its own example numbers; the comments say which are made up or simulated.
Sources
- Options, Futures, and Other Derivatives, J. C. Hull, Pearson (any recent edition)
- F. Black & M. Scholes, “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy 81(3), 1973
- R. Israelov, “Pathetic Protection: The Elusive Benefits of Protective Puts”, Journal of Alternative Investments, 2019
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