Delta-Hedge Simulator

Write or buy European options at their Black–Scholes–Merton value, delta-hedge them with shares at discrete dates over many simulated stock paths, and see what the hedge leaves behind.

Learning focus: a delta hedge rebalanced at discrete dates replicates the option only on average: the P&L is noisy, the noise falls roughly like 1 ÷ √(rebalances), and hedging at an implied volatility the stock does not deliver leaves a systematic gain or loss.

Option position

Option type

Position

Number of options

Stock price today

Strike

Time to expiry

yrs

Risk-free rate

Dividend yield

Volatility and stock paths

Implied volatility

Realised volatility

Drift

Hedge intervals (rebalances)

Paths

Random seed

Mean hedging P&L

3.22

± 2.83 standard error

Std of P&L

89.51

Std as % of premium

8.56%

5th percentile

−144.17

95th percentile

142.80

BSM premium (per option)

10.4506

Position 1,045.06

Rule-of-thumb std

92.23

Volatility-gap benchmark

0.00

Initial hedge

63.68 shares

With realised volatility equal to implied, the mean P&L (3.22) is within three standard errors of zero: discrete hedging adds noise, not a systematic gain or loss. The spread is about 8.6% of the premium with 52 hedge intervals.

One sample path

A written option's hedge buys shares after the stock rises and sells after it falls, always trading after the move, which is why discrete hedging is imperfect.

Hedging P&L across paths

Each path's final hedged value is discounted to today at the risk-free rate; the width of this distribution is the cost of hedging only at discrete dates.

Spread of P&L against rebalancing frequency

Same seed at each frequency: quadrupling the hedge intervals roughly halves the spread, the 1 ÷ √N shape.

Sample path, rebalance by rebalance

Date	Time (yrs)	Stock	Δ per option	Shares traded	Shares held	Cash
0	0.0000	100.00	0.6368	63.68	63.68	−5,323.25
1	0.0192	102.77	0.6861	4.93	68.61	−5,834.62
2	0.0385	102.88	0.6872	0.12	68.72	−5,852.08
3	0.0577	106.20	0.7420	5.47	74.20	−6,439.10
4	0.0769	105.86	0.7366	−0.54	73.66	−6,387.98
5	0.0962	105.78	0.7351	−0.15	73.51	−6,378.46
6	0.1154	106.57	0.7478	1.27	74.78	−6,520.28
7	0.1346	106.10	0.7402	−0.76	74.02	−6,445.57
8	0.1538	103.34	0.6916	−4.86	69.16	−5,949.52
9	0.1731	107.21	0.7584	6.68	75.84	−6,671.26
10	0.1923	107.15	0.7576	−0.08	75.76	−6,669.18
11	0.2115	103.55	0.6940	−6.36	69.40	−6,017.31
12	0.2308	103.86	0.6996	0.55	69.96	−6,080.71
13	0.2500	99.32	0.6040	−9.56	60.40	−5,137.19
14	0.2692	102.54	0.6721	6.82	67.21	−5,841.19
15	0.2885	105.32	0.7265	5.44	72.65	−6,419.79
16	0.3077	109.49	0.7984	7.19	79.84	−7,212.74
17	0.3269	112.92	0.8479	4.95	84.79	−7,778.40
18	0.3462	114.53	0.8690	2.12	86.90	−8,028.24
19	0.3654	115.06	0.8769	0.78	87.69	−8,125.81
20	0.3846	110.88	0.8245	−5.23	82.45	−7,553.56
21	0.4038	112.73	0.8522	2.77	85.22	−7,872.67
22	0.4231	117.50	0.9078	5.56	90.78	−8,533.59
23	0.4423	110.44	0.8229	−8.49	82.29	−7,603.63
24	0.4615	113.95	0.8742	5.14	87.42	−8,196.17
25	0.4808	115.00	0.8891	1.49	88.91	−8,375.20
26	0.5000	111.98	0.8526	−3.65	85.26	−7,974.41
27	0.5192	113.39	0.8748	2.22	87.48	−8,233.33
28	0.5385	118.29	0.9297	5.50	92.97	−8,891.57
29	0.5577	115.51	0.9061	−2.37	90.61	−8,626.76
30	0.5769	114.74	0.9006	−0.55	90.06	−8,572.22
31	0.5962	115.65	0.9141	1.35	91.41	−8,736.20
32	0.6154	117.97	0.9394	2.53	93.94	−9,043.03
33	0.6346	115.52	0.9200	−1.93	92.00	−8,828.58
34	0.6538	116.33	0.9321	1.20	93.21	−8,977.19
35	0.6731	118.75	0.9557	2.36	95.57	−9,266.56
36	0.6923	118.74	0.9593	0.36	95.93	−9,317.68
37	0.7115	119.05	0.9650	0.57	96.50	−9,394.44
38	0.7308	113.67	0.9216	−4.33	92.16	−8,910.89
39	0.7500	111.05	0.8893	−3.23	88.93	−8,560.95
40	0.7692	116.59	0.9613	7.19	96.13	−9,407.90
41	0.7885	116.68	0.9670	0.57	96.70	−9,483.21
42	0.8077	121.13	0.9903	2.34	99.03	−9,775.48
43	0.8269	120.23	0.9909	0.05	99.09	−9,791.12
44	0.8462	114.06	0.9652	−2.56	96.52	−9,508.07
45	0.8654	116.47	0.9863	2.11	98.63	−9,762.94
46	0.8846	119.00	0.9963	1.00	99.63	−9,891.22
47	0.9038	124.31	0.9999	0.35	99.99	−9,944.80
48	0.9231	126.09	1.0000	0.01	100.00	−9,956.12
49	0.9423	128.45	1.0000	0.00	100.00	−9,965.82
50	0.9615	125.96	1.0000	0.00	100.00	−9,975.41
51	0.9808	132.07	1.0000	0.00	100.00	−9,985.01
52	1.0000	131.74	Expiry	−100.00	0.00	5.39

The first path of the simulation, for the whole position. At expiry the shares are sold and the option settled, so the last cash figure is this path's hedging P&L before discounting.