Đo hiệu suất straddle S&P 500 theo chi phí thực thi

# S&P 500 Options: Cost Sensitivity


# S&P 500 Options: Cost Sensitivity

Selling an at-the-money straddle collects a premium of a few percent of the underlying's value,
and the quoted bid-ask spread on those same contracts is a large fraction of that premium. The
strategy's result therefore depends less on the model than on what fraction of the quoted spread
a trader actually pays. This notebook measures that dependence by re-running one strategy per
model family at each of the declared spread fractions and on both option universes.

Two things make option costs different from the equity convention used elsewhere in the book.
They are denominated as a share of the option premium rather than in basis points of notional,
because the premium and the notional differ by more than an order of magnitude. And under the
hold-to-expiry construction almost every position pays the spread only on the way in: a contract
that reaches expiration settles in cash at intrinsic value, and cash settlement is not a trade,
so there is no exit quote to cross. The exception is a contract whose chain ends before its
expiration date, which the engine buys back at the last quoted ask and charges an exit spread and
commission for. That is a small minority of positions, so the entry fill is still where almost
all of the friction on this axis sits, but the curve below is not a pure entry-cost curve and the
grid moves both charges together: `option_spread_fraction` is the same field on the way out.

Cost variants are diagnostics. They never join the candidate set a strategy is selected from,
because varying the cost assumption after the fact and keeping the most favourable answer is a
way of choosing a result rather than measuring one. Their interpretation is in
`18_strategy_analysis`.

**Learning objectives**

- Express a trading cost in the unit the instrument is quoted in, and say what the unit implies
  about which side of the trade pays it.
- Re-run a fixed strategy across a declared cost grid so that the sensitivity is measured on
  paired series rather than inferred from a single run.
- Keep a sensitivity sweep out of the set a selection ranges over.

**Book reference**: Chapter 18

**Prerequisites**: [`14_risk_management`](14_risk_management.ipynb), which is the last stage that
could add a run to the candidate pool and registers none, and through it
[`13_portfolio_management`](13_portfolio_management.ipynb) and
[`12_backtest`](12_backtest.ipynb). This runs last of the four backtest stages, after everything
that selects, so that no later stage can re-rank the strategies whose cost sensitivity it
reports. What it reads is the complete baseline population published by `12_backtest`.

```python
"""Execute the declared S&P 500 options cost-sensitivity population."""

import plotly.express as px
import polars as pl

from case_studies.research import OfficialPopulation, Result, supersedes_for_run
from case_studies.sp500_options.research_workflow import (
    ALL_LABELS,
    open_study,
    preview_baseline_candidates,
    run_official_backtest_requests,
    strategy_request_frame,
)
from case_studies.utils.sweep_config import get_htm_cost_cascade
from utils.style import COLORS, show_plotly_with_alt

CASE_STUDY = "sp500_options"
BASELINE_POPULATION = "sp500-options-baseline-validation-v1"
COST_POPULATION = "sp500-options-cost-sensitivity-validation-v1"
```

```python
EXECUTION_TIER = "canonical"
WORKSPACE: str = ""
PREVIEW_LABELS: list[str] = []
PREVIEW_MAX_BASELINE_CONFIGS = 0
PREVIEW_COST_FRACTIONS: tuple[float, ...] = (0.203,)
PREVIEW_UNIVERSES: tuple[str, ...] = ("liquid",)
# The generation this run retires. A population is immutable under its name, so a re-run
# whose members have moved has to say which one it replaces; the refusal names the current
# hash, and empty is correct only for a name this registry has never held. This notebook
# published its population with no supersedes at all, so the first upstream change to move
# a baseline identity left it unable to register what it had just computed - which is what
# happened when the label buffer was corrected. Stale the moment the run it authorizes
# succeeds, in the same way as the declarations `12_backtest` and `13_portfolio_management`
# carry.
SUPERSEDES_COST_POPULATION: str = ""
```

## One strategy per model family

The sweep runs on one representative from each family rather than on the whole baseline
population, because the question is how the result moves with the cost assumption and not which
model answers it best. Each family contributes its highest-Sharpe baseline, with the backtest
identity breaking exact ties, so the choice is reproducible from the population alone.

Keeping one per family rather than a single overall representative is what allows the curves to
be read against each other: if the sensitivity to costs were driven by the model rather than by
the instrument, the four curves would separate.

```python
study = open_study(execution_tier=EXECUTION_TIER, workspace=WORKSPACE or None)
if EXECUTION_TIER == "canonical":
    if PREVIEW_LABELS or PREVIEW_MAX_BASELINE_CONFIGS:
        raise ValueError("canonical execution cannot declare preview reductions")
    population = OfficialPopulation.one(study, name=BASELINE_POPULATION)
    baseline_hashes = population.require_complete()
    baseline = study.backtests.table().filter(pl.col("backtest_hash").is_in(baseline_hashes))
elif EXECUTION_TIER == "preview":
    if not WORKSPACE or not PREVIEW_LABELS or PREVIEW_MAX_BASELINE_CONFIGS < 1:
        raise ValueError(
            "preview execution requires WORKSPACE, PREVIEW_LABELS and PREVIEW_MAX_BASELINE_CONFIGS"
        )
    unknown = sorted(set(PREVIEW_LABELS) - set(ALL_LABELS))
    if unknown:
        raise ValueError(f"preview labels this case study does not declare: {unknown}")
    baseline = preview_baseline_candidates(
        study, labels=PREVIEW_LABELS, limit=PREVIEW_MAX_BASELINE_CONFIGS
    )
else:
    raise ValueError(f"unsupported execution tier: {EXECUTION_TIER!r}")
if baseline.is_empty() or baseline.filter(~pl.col("complete")).height:
    raise RuntimeError("cost sensitivity requires complete baseline results")
if baseline.get_column("sharpe").null_count():
    raise RuntimeError("a baseline result carries no Sharpe ratio")

representatives = (
    baseline.sort("sharpe", "backtest_hash", descending=[True, False])
    .group_by("family", maintain_order=True)
    .head(1)
    .sort("family")
)
```

```python
representatives.select(
    "family",
    "config_name",
    "checkpoint_kind",
    "checkpoint_value",
    "sharpe",
    "backtest_hash",
)
```

## The cost grid

A **spread fraction** is the share of the quoted half-spread the entry trade is assumed to pay.
At the top of the grid the order crosses the full quote. At the bottom it is filled close to the
midpoint, which is roughly what measured effective-to-quoted spread ratios imply for algorithmic
execution in at-the-money equity options; the grid printed below gives both ends and the values
in between. Every other cost - the per-contract
option commission, the per-share equity commission, the half-spread paid whenever the delta
hedge trades the underlying - is held at its configured value, so the fraction is the only field
that moves along a curve.

The **universe** axis is the second rung of the same question. `full` prices the strategy on the
whole at-the-money straddle surface; `liquid` restricts it to the fifth of that surface with the
tightest quoted half-spread on each decision date. The canonical strategy in `12_backtest` is
pinned to `liquid`; this is where the restriction is priced rather than assumed.

The concentration is fixed at the cascade's own value for every request here, so the curves are
comparable to each other. They are not paired with the baselines the representatives came from,
which hold a different number of symbols.

```python
cascade = get_htm_cost_cascade(CASE_STUDY)
cost_fractions = tuple(float(value) for value in cascade["cost_fractions"])
universes = tuple(str(value) for value in cascade["universes"])
cost_top_k = int(cascade["top_k"])
if EXECUTION_TIER == "preview":
    cost_fractions = PREVIEW_COST_FRACTIONS
    universes = PREVIEW_UNIVERSES
if not cost_fractions or not universes:
    raise ValueError("cost-sensitivity request axes cannot be empty")
print(f"Spread fractions of the quoted half-spread: {list(cost_fractions)}")
print(f"Universes: {list(universes)}; symbols held per decision date: {cost_top_k}")
```

## The requests

One request per representative, universe and spread fraction. Each starts from its
representative's signal and overrides three fields: the universe restriction, the concentration,
and the spread fraction. Within one representative and universe only the fraction varies, which
is what makes a curve a curve.

**Why the four fractions are these four.** They are anchors rather than a linear sweep. `1.0` is
the order crossing the full quoted spread, which is what a marketable order pays if nothing goes
its way. `0.75` is close to the population-average ratio of effective to quoted spread, so it
stands for ordinary execution. `0.203` is the best case reported for algorithmic execution in
at-the-money equity options, and it is the number that decides whether this strategy is viable
at all: if the result only survives there, it survives only for a desk that executes as well as
anyone has been measured to. `0.5` sits between the two middle cases so the curve is not read
from three points.

**The universe axis is not a robustness check, it is a second question.** `full` prices the
strategy on every name it selects; `liquid` restricts to the bottom quintile of quoted
half-spread at each rebalance. A strategy that only clears its costs on the liquid subset is a
different, smaller strategy than the one selected upstream, and reporting the two together is
what keeps that from being presented as the same result at a better cost assumption.

```python
request_rows = []
for row in representatives.iter_rows(named=True):
    baseline_result = Result.open(
        study,
        row["backtest_hash"],
        include_preview=EXECUTION_TIER == "preview",
    )
    base_signal = baseline_result.spec()["strategy"]["signal"]
    for universe in universes:
        if universe not in {"full", "liquid"}:
            raise ValueError(f"unsupported option cost universe {universe!r}")
        for fraction in cost_fractions:
            signal = dict(base_signal)
            if universe == "liquid":
                signal["universe_filter"] = "liquid"
            else:
                signal.pop("universe_filter", None)
            signal["top_k"] = cost_top_k
            signal["option_spread_fraction"] = fraction
            request_rows.append(
                {
                    "request_name": f"{row['family']}-{universe}-spread-{fraction:g}",
                    "prediction_hash": row["prediction_hash"],
                    "label": row["label"],
                    "family": row["family"],
                    "universe": universe,
                    "spread_fraction": fraction,
                    "signal": signal,
                    "allocation": None,
                    "risk": None,
                    "costs": None,
                    "chapter": "ch18",
                }
            )
requests = strategy_request_frame(request_rows)
print(
    f"{requests.height} requests: {representatives.height} families x "
    f"{len(universes)} universes x {len(cost_fractions)} fractions"
)
```

## Execute

Each request resolves its own contracts, because the universe restriction changes which symbols
are eligible and the concentration changes how many are held. The engine validates the paired
option lifecycle, that every selected contract ends either by cash settlement or by liquidation,
the retained hedge, and every cost input before publishing.

**Re-resolving contracts per request is what makes the comparison honest and what makes it
slow.** The alternative - resolving once and re-pricing - would compare one contract set at
several cost assumptions, which answers a narrower question than the one asked here: under a
liquid-universe restriction the strategy does not hold the same options more cheaply, it holds
different options. Sharing a contract set across requests would hide that substitution and
report the cost of a portfolio the strategy would not have held.

```python
execution = run_official_backtest_requests(
    study,
    requests,
    population_name=COST_POPULATION if EXECUTION_TIER == "canonical" else None,
    supersedes=supersedes_for_run(
        study,
        population_name=COST_POPULATION,
        declared=SUPERSEDES_COST_POPULATION or None,
        execution_tier=EXECUTION_TIER,
    ),
)
catalog = execution.catalog_rows.sort("request_name")
if catalog.height != requests.height or catalog.filter(~pl.col("complete")).height:
    raise RuntimeError("cost execution did not publish every declared request")
```

## What the run produced

Read each line for its slope rather than its level. The slope is how much of the result is a
claim about execution quality; the gap between the two universes at the same fraction is what
the liquidity restriction buys. Each point carries the block-bootstrap Sharpe interval the
engine registers with every backtest, reported below as `ci95_lo` and `ci95_hi`, so whether a
point clears zero can be read off the bar rather than assumed from the marker. The intervals share a return series across neighbouring
spread fractions, so they are not independent along a line: they say how firm a single point
is, not whether two points on the same line differ.

```python
cost_curve = (
    study.backtests.table(include_preview=EXECUTION_TIER == "preview")
    .select("backtest_hash", "sharpe", "sharpe_ci95_lo", "sharpe_ci95_hi")
    .join(catalog.select("request_name", "backtest_hash"), on="backtest_hash", how="inner")
    .join(
        requests.select("request_name", "family", "universe", "spread_fraction"),
        on="request_name",
        how="inner",
    )
    .sort("family", "universe", "spread_fraction")
)
interval_columns = ["sharpe", "sharpe_ci95_lo", "sharpe_ci95_hi"]
if (
    cost_curve.height != catalog.height
    or cost_curve.select(interval_columns).null_count().sum_horizontal().sum()
):
    raise RuntimeError("the published cost population is missing rows or Sharpe metrics")
```

```python
cost_curve.select(
    "family",
    "universe",
    "spread_fraction",
    "sharpe",
    pl.col("sharpe_ci95_lo").alias("ci95_lo"),
    pl.col("sharpe_ci95_hi").alias("ci95_hi"),
    (pl.col("sharpe_ci95_lo") > 0).alias("clears_zero"),
).sort("family", "universe", "spread_fraction")
```

```python
cost_figure = px.line(
    cost_curve.with_columns(
        ci95_upper=pl.col("sharpe_ci95_hi") - pl.col("sharpe"),
        ci95_lower=pl.col("sharpe") - pl.col("sharpe_ci95_lo"),
    ),
    x="spread_fraction",
    y="sharpe",
    color="family",
    line_dash="universe",
    markers=True,
    error_y="ci95_upper",
    error_y_minus="ci95_lower",
    hover_data=["backtest_hash", "sharpe_ci95_lo", "sharpe_ci95_hi"],
)
cost_figure.add_hline(y=0, line_width=1, line_dash="dash", line_color=COLORS["neutral"])
cost_figure.update_layout(
    title="Validation Sharpe against the share of the quoted spread paid on entry",
    height=520,
    width=1000,
    margin=dict(t=70),
    legend_title_text="family / universe",
)
cost_figure.update_xaxes(title_text="Fraction of the quoted option half-spread paid on entry")
cost_figure.update_yaxes(title_text="Validation Sharpe")
show_plotly_with_alt(
    cost_figure,
    "Line chart of validation Sharpe against the fraction of the quoted option half-spread paid "
    "on entry, one line per model family and option universe, each point carrying its "
    "block-bootstrap Sharpe interval as an error bar.",
)
```

## Key takeaways

- A cost is stated in the unit its instrument is quoted in. A basis-point-of-notional convention
  applied to an option premium misstates the friction by the ratio between premium and notional,
  which for an at-the-money straddle is large.
- Holding to expiration removes the exit-side spread for every position that reaches it, so the
  entry fill is where nearly all execution quality shows up. That is a property of the
  construction, not of the model - and it is a property of reaching expiration, not of intending
  to: a contract whose chain ends first is bought back and pays the exit side like any other.
- A sensitivity grid is evidence about robustness only while it stays outside the set a strategy
  is selected from. Once a cost assumption can be chosen after seeing the result, the grid has
  become a search.

**Known limitations**: the spread fractions are declared assumptions calibrated from published
execution studies, not fills this strategy achieved, so the curve says what the result would be
under each assumption and not which assumption holds. The hedge spread and both commissions
are held fixed, so their contribution is inside every point rather than resolved along an axis.
![notebook output](figures/p1_1.png)