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Add first draft of "implied" module

Joseph Weston authored on 03/03/2021 06:42:35
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environment.yml
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@@ -18,5 +18,7 @@ dependencies:
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     - httpx
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     - jsonpatch
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     - msgpack-python
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+    - numpy
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+    - scipy
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     - toolz
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     - tqdm
implied.py
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new file mode 100644
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@@ -0,0 +1,165 @@
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+"""Tools for dealing with bookmaker's odds and implied probabilities"""
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+
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+from math import sqrt
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+from typing import Sequence
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+
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+import scipy.optimize
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+import toolz
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+
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+
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+# -- Adjusting implied probabilities to true probabilities
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+
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+_adjust_methods = {}
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+
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+
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+@toolz.curry
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+def add_method(name: str, f):
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+    _adjust_methods[name] = f
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+    return f
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+
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+
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+def overround(odds: Sequence[float]) -> float:
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+    return sum(odds) - 1
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+
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+
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+def validate_odds(odds: Sequence[float]) -> None:
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+    if len(odds) < 2:
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+        raise ValueError(f"Expected at least 2 outcomes, but got {len(odds)}")
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+    if any(x >= 1 for x in odds):
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+        raise ValueError(f"Expected all outcomes to have an implied probability < 1")
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+    r = overround(odds)
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+    if r < 0:
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+        raise ValueError(f"Odds {odds} have a negative overround ({r})")
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+    elif r > 0.5:
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+        raise ValueError(f"Odds {odds} have excessive overround ({r})")
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+
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+
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+def adjust_odds(odds: Sequence[float], method="power") -> list[float]:
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+    """Return true odds given bookmaker odds.
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+
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+    Parameters
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+    ----------
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+    odds
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+        The odds of each outcome, given as implied probabilites.
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+
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+    The implied probabilities associated with the odds quoted by a bookmaker
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+    sum to a number > 1. This excess is the "overround" of the odds. There
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+    are a number of models
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+
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+    Notes
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+    -----
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+    All implemented methods are based on the descriptions in the paper by
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+    S. Clarke in the references section.
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+
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+    References
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+    ----------
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+    S. Clarke, S. Kovalchik, M. Ingram
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+        Adjusting Bookmaker's odds to Allow for Overround.
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+    """
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+    try:
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+        f = _adjust_methods[method]
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+    except KeyError:
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+        raise ValueError(
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+            f"Unknown method '{method}', expected one of {list(_adjust_methods)}"
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+        )
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+    validate_odds(odds)
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+    return f(odds)
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+
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+
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+@add_method("additive")
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+def adjust_odds_additive(odds: Sequence[float]) -> list[float]:
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+    """Adjust bookmaker odds using the additive method.
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+
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+    In this method the overround is assumed to be split equality between
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+    the different outcomes, regardless of their relative implied probabilities.
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+    """
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+    y = overround(odds) / len(odds)
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+    return [x - y for x in odds]
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+
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+
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+@add_method("multiplicative")
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+def adjust_odds_multiplicative(odds: Sequence[float]) -> list[float]:
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+    """Adjust bookmaker odds using the multiplicative method.
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+
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+    In this method the overround is assumed to be split among the
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+    different outcomes in proportion to their implied probabilities.
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+    """
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+    s = sum(odds)
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+    r = overround(odds)
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+    return [x - (r * x / s) for x in odds]
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+
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+
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+@add_method("power")
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+def adjust_odds_power(odds: Sequence[float]) -> list[float]:
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+    """Adjust bookmaker odds using the power method.
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+
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+    In this method the true probabilities are assumed to be
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+    related to the implied probabilities by a power law,
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+    i.e. p_i = π_i^k, for some fixed k. The k is then
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+    chosen such that the true probabilities sum to 1.
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+
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+    Notes
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+    -----
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+    The 'k' parameter for the power method is computed by
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+    solving the nonlinear equation Σ_i π_i^k = 1 using
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+    a root-finding algorithm.
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+
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+    References
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+    ----------
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+    S.R. Clarke
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+        Adjusting true odds to allow for vigorish
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+    """
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+    # This bracketing interval should be sufficient for reasonable overrounds
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+    # (not > 50%) and reasonable distribution of implied probability between
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+    # the different outcomes.
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+    try:
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+        k = scipy.optimize.root_scalar(
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+            lambda k: sum(x ** k for x in odds) - 1,
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+            bracket=[0.9, 3],
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+            xtol=1e-14,
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+        ).root
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+    except ValueError as e:
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+        if "different signs" in str(e):
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+            raise ValueError(f"Bracketing interval too small for odds {odds}") from None
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+
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+    return [x ** k for x in odds]
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+
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+
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+@add_method("shin")
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+def adjust_odds_shin(odds):
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+    """Adjust bookmaker odds using the Shin method.
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+
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+    In this method we assume a fraction z of sharp bettors
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+    and adjust the true probabilities accordingly.
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+
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+    Notes
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+    -----
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+    The 'z' parameter for the Shin method is computed
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+    by solving a nonlinear equation using a root-finding algorithm.
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+
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+    References
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+    ----------
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+    H.S. Shin
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+        Prices of State Contingent Claims with Insider Traders
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+        and the Favorite-Longshot Bias
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+    """
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+    s = sum(odds)
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+    n = len(odds)
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+
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+    def residual(z):
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+        return (
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+            sum(sqrt(z ** 2 + 4 * (1 - z) * pi ** 2 / s) for pi in odds)
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+            - 2
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+            - (n - 2) * z
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+        )
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+
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+    # 'z' is in the interval [0, 1). We set the upper limit for the
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+    # bracketing interval to just below 1 to exclude a (spurious) solution at z=1.
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+    try:
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+        z = scipy.optimize.root_scalar(residual, bracket=[0, 0.999], xtol=1e-14).root
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+    except ValueError:
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+        raise ValueError(f"Shin method failed to converge for odds {odds}") from None
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+
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+    return [
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+        (sqrt(z ** 2 + 4 * (1 - z) * pi ** 2 / s) - z) / (2 * (1 - z)) for pi in odds
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+    ]