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Interpreting a Short Put’s Marked-to-Market Profit and Loss

Article Quant Q&A · Author: Strictly_increasing
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Summary

The document asks whether the current option premium less the discounted future option price represents the present value of a short European put position. The response clarifies that this expression is better understood as the position’s profit or loss, valued from the initial date, with the future option price still stochastic until the later observation time. A short seller receives the option’s current value and would incur a cost if buying it back at its later market price.

The discussion is a brief conceptual clarification rather than a pricing derivation. It does not establish a valuation framework, explain discounting conventions in detail, or address early exercise, transaction costs, financing, or the cash flows at expiration. Its key distinction is between a position’s future marked-to-market P&L expressed in present-value terms and a deterministic present value known at the outset.

Key ideas

  • A short option position gains value from the premium received and loses value as its repurchase price rises.
  • Discounting a future option price does not make that price known at the initial date.
  • The expression is a stochastic profit-or-loss measure viewed from the initial date.
  • The exchange does not develop a full option valuation or financing treatment.

Tags

OptionsDerivatives pricing
Full text
# Is this the present value of a short position on an option?


# Is this the present value of a short position on an option?












Consider a European put option, whose price at time $0$ is $\Pi_0$. Set: $$\mathcal{L}_0=\Pi_0 - P(0,t_M)\Pi_{t_M}$$ where 0 < $t_M$ and $P(0, t_M)$ is the discount factor from time $0$ to time $t_M$. Is that correct to state that $\mathcal{L}_0$ represents the present value of a short position on the put option? That is, at time $0$ you have a positive value equal to $\Pi_0$ and at time $t_M$ you will have to rebuy the put option at its current price (i.e. $\Pi_{t_M}$), which enters negatively (and discounted, since you have to evaluate everything at time 0) in the value of your position.

## Answer by Kermittfrog (score 0, accepted)

https://quant.stackexchange.com/a/52938

Close to that. Your formula represents the profit (or loss) of that position as seen from $t_0$, which is so far a stochastic term observable at expiry.

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