Interpreting Forward Delta and Vega Ratios for Options
Summary
The document asks how forward delta relates to the vega ratio of out-of-the-money options and to the vega of in-the-money options. It describes a proposed interpretation in which the underlying forward price is used to calculate delta for an option with the relevant maturity, and delta selects a row in a reference table that scales at-the-money vega to estimate another option’s vega.
For in-the-money options, the author proposes using put-call parity and the complementary delta of the corresponding out-of-the-money option. The post gives several example forward-delta and vega-ratio pairs, and suggests that an out-of-the-money put may better reproduce the table values. It is a question with a tentative explanation, not a verified derivation: it does not establish the table’s assumptions, define the precise delta convention, or resolve whether the proposed interpretation is correct.
Key ideas
- Forward delta is proposed as the lookup variable for estimating an out-of-the-money option’s vega relative to at-the-money vega.
- Using the forward price can incorporate the underlying’s carry into the delta calculation for the option’s maturity.
- The author suggests put-call parity links an in-the-money option’s vega ratio to that of its corresponding out-of-the-money option.
- The post offers tentative interpretations and examples rather than a complete derivation.
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Full text
# Vega for ITM and OTM options against the ATM one # Vega for ITM and OTM options against the ATM one In Dynamic Hedging by Taleb, at pag. 182, is presented the concept of "Vega ratio". If I understand correctly, the author in Table 10.3 confronts the Vega of an OTM option with the one of an ATM with same strike and maturity 3 months "by the forward". What I do not understand is: - What does it means "by the forward"? He then states: > The reader can thus derive the Vega for an OTM option provided he knows the delta. - What is the forward delta? What is its role in this Vega ratio? Then, for a ITM option, > the Vega ratio is 100 minus the delta of the OTM corresponding instrument with same strike and maturity thanks to put/call parity. - How and why for in the money option this Vega ratio is linked to the forward delta in this way? Please, let me know if more details are needed. Thanks for the help. Note Some numbers reported in the Table 10.3 for OTM option: Forward delta = 50, Vega Ratio = 100%; Forward delta = 25, Vega Ratio = 77%; Forward delta = 5, Vega Ratio = 25%. Edit I report here a possible answer: - Instead of estimating the dividend pay-out and risk free return of the asset it is possible to use the Forward price for the underlying with the same 3-month maturity. Thus, one can calculate the Forward Delta of the option. - The role of the Forward Delta is only to choose the right row of the mentioned Table. In fact, if you know the FD you can use the Vega Ratio and the Vega for the ATM 3-month option to get the Vega of interest. Numerically, one can find the strike prices reported in the table by seeing the Delta as a function of the strike price. It seems to me that the numbers are better recovered if one uses the delta of a OTM put. - Again, the role of the Forward Delta is only to choose the right row. For the ITM instrument, by leveraging the Put/Call parity, one can find the correct ratio to compute the Vega of interest by lookin at the "100-FD of the ITM" row. Eg. For FD=25 the implicit strike is K=95 and the put is OTM. This Vega Ratio could be used for the ITM call with FD=75. What do you think?
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