Why an ATM Collar Backtest Does Not Establish Arbitrage

Summary

The document examines a backtest of a collar formed with shares, a long put, and a short call, with both options struck at the current stock price. Its payoff expression simplifies the option payoff terms at expiration, leaving the initial call-put premium difference grown at the stated risk-free rate. Applying put-call parity, the questioner interprets this as a return tied to the stock price and risk-free rate and reports a high Sharpe ratio in the backtest.

The central lesson is that this algebra and a strong historical Sharpe ratio do not by themselves demonstrate arbitrage. Put-call parity depends on consistent option terms, financing, discounting, and the timing conventions used in the calculation. The document presents the claim as a question and supplies no answer or checks of the backtest, so it does not establish whether the reported performance is valid. It is useful as a prompt to audit payoff signs, strike and maturity alignment, dividend and funding assumptions, and whether the simulated portfolio is self-financing before drawing arbitrage conclusions.

Key ideas

• The proposed collar combines stock ownership with a long put and a short call.
• At-the-money strikes simplify the stated expiration payoff expression.
• The question applies put-call parity to interpret the option premium difference as a financing-related amount.
• A high backtested Sharpe ratio alone does not prove that a strategy is an arbitrage.

Tags

# Arbitrage from ATM option trading?


# Arbitrage from ATM option trading?












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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.