How Underlying Price and Volatility Affect a Vanilla Call's Value

Summary

The document considers how a plain vanilla call’s price can change between two observation times and asks which explanation is inconsistent with a price decline. It notes that a fall in the underlying can reduce the call’s value, and a short call position can profit when the option price falls. A volatility increase tends to raise call value, but changes in the underlying can offset that effect; similarly, the option can become more in the money while other inputs push its price lower.

The final choice links a Value-at-Risk forecast for a long call position to the change in option value. The text implies this is the inconsistent statement: VaR conventionally describes a loss threshold, while the stated inequality describes a sufficiently large gain. The discussion is qualitative and does not specify a pricing model, quantify “significant” volatility changes, or fully distinguish risk conventions for gains and losses.

Key ideas

• A decline in the underlying can contribute to a lower call price.
• A rise in volatility generally supports a higher call value, though underlying moves can offset it.
• A short call position gains when the option price decreases.
• A call can move further in the money even while its market price falls due to other input changes.
• VaR for a long option concerns potential losses, not a threshold for positive gains.

Tags

# Vanilla option pricing at different points in time


# Vanilla option pricing at different points in time












Let $C(t) = C(t; S,K,T)$ the price at time $t$ of a plain vanilla call option with maturity $T$ and strike $K$ on an underlying $S$; if for $t_1<t_2$ we have $C(t_1) > C(t_2)$, it could not be true that (choose the correct answer):

- The value of $S$ has declined ($S_{t_1} > S_{t_2}$);

- The volatility of $S$ has significantly increased;

- The owner of a short position on this contract is making more money than if the position were naked (no long position on $S$ too);

- The contract is now more in-the-money;

- If a Value-at-Risk $X$ with confidence interval $\alpha$ were forecasted in the time horizon from $t_1$ to $t_2$ for a long position on this contract, then a risk manager expected that $C(t_2) - C(t_1) > X$ with probability $\alpha$.

My thoughts:

- Price decreases, stock decreases: it can be true;

- Other things being equal, a significant increase in the volatility would cause a significant increase of the price of the call (going against $C(t_1) > C(t_2)$). However, movements in volatility can always be counterbalanced by movements in the underlying, resulting in a decrease of the price of the stock: it can be true;

- if the price of the call decreases and you went short, you are making money: it can be true;

- as said before, movements in the stock can always be counterbalanced by movements in volatility: it can be true;

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