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.

\(n(p)\) 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 \(0\).