Delta Hedging: Simulating a Gamma Feedback Loop

A market maker writes options and delta-hedges in an underlying it moves when it trades.

Scenarios
Option written by dealer
Strike \(K/S_0\) 1.05
Time to expiry 30 days
Volatility \(\sigma\) 60%/yr
Drift \(\mu\) +100%/yr
Net exogenous buying pressure (retail demand, news flow).
Dealer hedging
Dealer's option position: 40% of float. Hedge (% of underlying): \(0.40 \times \Delta_t\).
Price impact \(\lambda\) 1.0
Dealer hedging moves prices by \(\lambda\times\)(hedge flow). So \(\lambda=1\) means dealers move the price by 1% for every 1% of float they buy.
Day 0 of 30
Price path, daily hedging flow, and total dealer hedge
━━ Baseline (no feedback)  ·  ━━ With feedback  ·  ‒ ‒ Strike
Day 0 \(S_t = \) $100.00 \(\Delta_t = \) 0.500 \(H_t = \) 0.0% of float
Baseline return
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With-feedback return
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Extra move from feedback
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Peak hedge held
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