Vega of In-the-Money Calls under Negative Return Skew

Summary

The document raises a question about how volatility affects the value of an in-the-money European call when the underlying return distribution is negatively skewed. The author observes that, under a variance-gamma pricing formula, increasing the volatility parameter appears to reduce call prices for in-the-money strikes when the skewness parameter is negative. The prompt contrasts this pattern with the familiar intuition that greater dispersion can increase a call’s expected payoff because losses are capped at zero.

The post reports model inputs, including spot, risk-free rate, maturity, and at-the-money, in-the-money, and out-of-the-money strikes, and refers to plots of prices relative to category means. It does not include the plots or establish whether the observed effect is valid. The author explicitly allows for a calculation error, so the result should be treated as an unresolved modeling question rather than evidence that call vega is generally negative under negative skew.

Key ideas

• The question examines call price sensitivity to volatility under a variance-gamma return model.
• The author reports declining prices for some in-the-money calls when skewness is negative.
• The usual positive-volatility intuition for calls may not directly settle behavior under a skewed model.
• The post provides parameter context and references plots, but not enough detail to verify the calculations.
• The claimed pattern is tentative and may reflect an implementation error.

Tags

# Can the vega of ITM call-options be negative when the distribution of the underlyings returns is negatively skewed?


# Can the vega of ITM call-options be negative when the distribution of the underlyings returns is negatively skewed?












While calculating european call option prices, using the variance-gamma model formula provided by Madan, Carr & Chang (1998), I noticed that, holding all other things constant, the value of an ITM call option in the VG-model seems to decline with $\sigma$ when the skewness parameter $\theta < 0$.

Below is depicted, for ATM, ITM and OTM-options respectively, the call price reduced by the mean price of its category (by parameters and moneyness category) versus $\sigma$. My concern regards figures 2 and 5.

I understand the argument, as is also presented in answers to many other questions about the vega of ITM-options, that due to the limited downside of call-options, an increase in volatility will also mean a higher expected payoff and therefore value.

My question is, whether this is also true for ITM options when the underlying distribution of returns is negatively skewed?

It is possible that the reason for this pattern in the figures is incorrect calculations on my behalf.

The option prices are for $S_0 =1000$, $r_f = 0.002$, $(T-t)=0.3$,

$K_{ATM}=1000$, $K_{ITM}=900$ and $K_{OTM}=1100$, and the VG-parameters shown in the graphs.

(The titles of the figures use "," instead of "." as decimal points)

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