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So I was testing out a collar options strategy (long put, short call, and long shares of the underlying stock) in a backtest for a school finance project, and the profits & losses are given by the formula

$$\text{PnL}_{t+1} = (S_{t+1} - S_t) + [C_t(K_1) - P_t(K_2)] *(1 + R_t) + \max[K_1 - S_{t+1}, 0] - \max[S_{t+1} - K_2, 0]$$.

where $S_t$ is the price of the underlying asset at time $t$, $C_t(K_1)$ is premium of the call option at time $t$ given strike price $K_1$, $P_t(K_2)$ is premium of the put option at time $t$ given strike price $K_2$.

The first strategy I have tested involves using at-the-money (ATM) options, where the strike price equals the current spot price, i.e. $S_t = K_1 = K_2$. So the $\text{PnL}_{t+1}$ is reduced to just $[C_t(K_1) - P_t(K_2)] *(1 + R_t)$. Then using the put-call parity formula we also have $$C_t - P_t = S_t - \frac{S_t}{1+R_t},$$ then $$[C_t(K_1) - P_t(K_2)] *(1 + R_t) = S_t \cdot R_t,$$ where $R_t$ is the risk free rate for time $t$.

So the value of \$1 following this strategy at period $t = 1 + S_0\cdot R_0 + \dots + S_{t-1} \cdot R_{t-1}$

It is noted that the growth of \$1 under this strategy has an extremely high Sharpe ratio of approximately $2.7$ and the period to period returns is more superior than the risk-free rate of return, doesn't this mean that this is an arbitrage opportunity?

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  • $\begingroup$ When $K_1=K_2=S$ the profit from this strategy is going to be essentially zero, in dollar terms. It is a stock position fully hedged with a "syntetic short" position. No investment, no profit, return $=\frac{0}{0}$ $\endgroup$ – noob2 Nov 6 '18 at 19:23
  • $\begingroup$ @noob2 but I still get some form of arbitrage profits from investing in the difference of the call and the put in a risk free asset like t-bills isn't it? So how do we explain the profits from there? $\endgroup$ – Weichen Christopher Xu Nov 6 '18 at 19:44
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    $\begingroup$ its directly offset by the financing cost of buying the stock which u left out of the PnL equation $\endgroup$ – hjw Nov 7 '18 at 8:36

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