If we don't ignore the value of the piggyback options, then under the Black-Scholes (B&S) assumptions, with σ ∈ [70%, 100%], σ_LO=σ_PBO=σ, μ_LO=μ_PBO=r=3.80%, OP=$0.18, and N=280,000:
- If PAR's share price was at $1.50 tomorrow, PAROA's B&S price would be in the range [$1.34, $1.48] (σ_LO=σ_PBO ∈ [70%, 100%]), i.e. a $326K - $364K gain.
- If PAR's share price was at $1.50 in a year's time, PAROA's B&S price would be in the range [$1.25, $1.32] (σ_PBO ∈ [70%, 100%]), i.e. a $299K - $320K gain.
- To break-even, PAR's share price in a year's time would need to be in the range [$0.68, $0.72] (σ_PBO ∈ [100%, 70%])
- For PAROA options (purchased @ $0.18) to match the ROI of PAR (purchased @ 50c), PAR's share price in a year's time would need to be in the range [$0.75, $0.81] (σ_PBO ∈ [100%, 70%])
It's worth noting that the risk of a total loss is higher when purchasing PAROA, as you can't always sell your options at the B&S price and your PAROA options will expire worthless if the share price is under ~$0.55 in a year's time (unless you think σ_PBO is greater than 100%).
However, if instead you only invested half of what you had planned to invest in PAR in PAROA, then under B&S assumptions, PAR's share price would only have to be trading at [$0.89, $0.99] (σ_PBO ∈ [100%, 70%]), in a year's time, for the total return on $10K invested in PAROA @ $0.18 to be higher than the total return on $20K invested in PAR @ $0.50.
So if PAR's share price in a year's time is between $0.25 and [$0.89, $0.99] (σ_PBO ∈ [100%, 70%]), you would have been better off investing $20K in PAR @ 50c per share, but if PAR's share price is less than $0.25 or greater than [$0.89, $0.99] you would have been better off investing $10K in PAROA @ 18c per option.
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