# 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 $$0$$.