Notes

Additive cases

Two stocks

  • Context: same as in the paper, one risky asset, one cash account with zero interest rate

  • stock price initialized at \(0\)

  • decision variable: number of shares \(n_{t}\)

  • buy and hold portfolio initialized with \(1\) share

  • contrarian trading initialized with one share: \(n_{0}=1\)

  • a contrarian trading rule is a function \(n(p)\), decreasing in \(p\), initialized with \(n(0)=0\), continuous in \(p\)

  • without being entirely rigorous, \(n(p)=N'(p)\) for \(N(\cdot)\) concave, \(N(0)=0\)

  • i.e. a contrarian trading rule is generated by a concave negative potential \(N(\cdot)\)

  • performance atttribution, contrarian trading versus buy-and-hold for price trajectory ending at \(p_{T}\), assuming standard integration works along the trajectory:
    \[N(p_{T})-p_{T}\]

  • discrete trading:
    \[\sum_{t=0}^{T-1}N'(p_{t})(p_{t+1}-p_{t})-p_{T}=\]
    \[\sum_{t=0}^{T-1}\left(N'(p_{t})(p_{t+1}-p_{t})-(N(p_{t+1})-N(p_{t}))\right)+N(p_{T})-p_{T}\]

  • second order approximation:
    \[\sum_{t=0}^{T-1}\left(N'(p_{t})(p_{t+1}-p_{t})-(N(p_{t+1})-N(p_{t}))\right) \approx -\frac{1}{2}N''(p_{t})(p_{t+1}-p_{t})^{2}\]
    \[N(p_{T})-p_{T} \approx \frac{1}{2}N''(0)p_{T}^{2}\]

  • Assuming \(N''(\cdot)\) is constant, this is:
    \[\sum_{t=0}^{T-1}N'(p_{t})(p_{t+1}-p_{t})-p_{T}\approx \frac{1}{2}N''(0)\left(p_{T}^{2}-\sum_{t=0}^{T-1}(p_{t+1}-p_{t})^{2}\right)\]

  • The result depends on:
    \[p_{T}^{2}-\sum_{t=0}^{T-1}(p_{t+1}-p_{t})^{2}\]
    which is a measure of excess volatility.

Multistocks

  • \(n(p)\) and \(p\) are now a vector

  • by assumption, \(n(p)\) is the gradient of the real function \(N(p)\)

  • \(N(p)\) is the value of a portfolio which is continuously adjusted along the smooth trajectory of prices \(0 \rightarrow p\)

  • \(L(p)\) is the value of the buy-and-hold portfolio along the trajectory of prices \(0 \rightarrow p\); it is a linear function of \(p\)

  • the buy-and-hold portfolio \(L\) is based on the number of shares \(n(0)\)

  • \(N(0)=L(0)=0\)

  • \(N(p)-L(p)\) is the pay-off of rebalancing / buy-and-hold, i.e. the trading policy

  • if we rebalance discretely both portfolios, we get the trading pay-off:
    \[\sum_{t=0}^{T-1}N'(p_{t})(p_{t+1}-p_{t})-L(p_{T})\]
    which we can rewrite:
    \[\sum_{t=0}^{T-1}\left(N'(p_{t})(p_{t+1}-p_{t})-N(p_{t+1})-N(p_{t})\right)+N(p_{T})-L(p_{T})\]
    (\(N(0)=0\)).

  • for contrarian trading:
    \[N'(p_{t})(p_{t+1}-p_{t})-N(p_{t+1})-N(p_{t})\]
    is convex positive while:
    \[N(p_{T})-L(p_{T})\]
    is concave negative since \(L(\cdot)\) is tangent to \(N(\cdot)\) at \(0\).

Multiplicative cases

Two stocks

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