Delta-hedging simulator

Sell an option, hedge it with the stock, and see what you are left with at expiry. The Greeks stop being a table and become money: hedge at the volatility you sold, and the book earns ½(σ²ᵢₘₚ − σ²ᵣₑₐₗ) S² Γ per unit of time, plus noise from hedging only now and then, minus what the trading costs.

Contract

Spot price (S)

Strike (K)

Implied volatility (sold at)

The volatility the option is priced and hedged at

Time to maturity (T)

Years

Risk-free rate (r)

Annualised, 0.05 = 5%

Dividend yield (q)

Annualised, continuous

Call

Put

Try this

0 of 3 done

Break even

Sell at 20% with realised volatility 20%, hedging every step, no costs.

What to look for: The average result is zero within its standard error and the histogram is narrow. The premium was exactly the cost of the hedge.

Harvest the difference

Keep selling at 20% but set realised volatility to 15%, then 25%.

What to look for: The average turns positive, then negative, and the theory figure moves with it. A short option hedged at the implied volatility is a bet that realised volatility comes in lower.

The price of laziness

Set realised back to 20% and rebalance every 21 steps instead of every step. Then add 10 basis points of cost with daily rebalancing.

What to look for: Infrequent hedging keeps the mean near zero but widens the histogram by roughly the square root of 21. Costs narrow nothing and shift the whole histogram left.

The world

Realised volatility

What the spot actually does

Drift

Real-world expected return

The hedge

Rebalance

Cost (bps of traded value)

Steps

252 = daily for a year

Paths

Seed

Premium received

$10.45

Sold at 20.0%

Average result at expiry

-$0.007

± 0.015 over 1,000 paths

Theory says

+$0.000

½(σ²ᵢₘₚ − σ²ᵣₑₐₗ) S² Γ summed over the life

Spread of results

0.468

50% of paths profitable; costs 0.000 per path

Sold at exactly the realised volatility, the book breaks even on average: everything left is the noise of hedging only now and then.

The book is the short option, the shares and the cash, marked to market at the implied volatility. It drifts with the theoretical line and wobbles around it: the wobble is the cost of not hedging continuously, largest when Gamma is large.

A tighter histogram means a better hedge. Hedge more often and it narrows; add costs and it shifts left.

Watch the hedge

The path in the charts above plays in time, 252 steps from today to expiry. On each rebalancing date the hedge trades back to the Black-Scholes delta; in between it holds still while the delta moves on. Below, the book's profit builds up against what theory says a continuous hedge earns, and the gap between them is the hedging error. The thin line is the same path hedged 20 times as often, without costs, which is close to continuous hedging: it wobbles far less, because the error shrinks with the square root of the time between trades.

Playback speed

Spot Strike

Shares held (discrete hedge) Black-Scholes delta (continuous hedge)

Hedged book (discrete) Theory (continuous hedge) Hedging error Hedged near-continuously

On this path the hedged book ends at +$0.169, against +$0.000 from theory: a hedging error of +$0.169. Hedged near-continuously it ends at +$0.099.