Relationship between two multi-valued child tables
Before I attempt to answer your questions, just a couple of hints that may help
you to create the perfect model :-) Bear with me.


When you construct an ERD, try and avoid "database" terminology such as: 
primary key, foreign key, table, row.  Instead, use the terms "entity", "relationship", "unique identifier", "instance".
Use singular nouns as entity names.  When creating an entity, think about a SINGLE instance (of an entity).  This may make it easier to find the correct cardinalities.
You can draw as many relationships as you like.  However, only keep the relationships that make sense in your system (eg if there are 5 entities, and the relationships in your diagram make it look a bit like a pentagram inscribed into a 5-sided polygon, then there are - in all probability - too many relationships).  The MANY side of a relationship will eventually be transformed into a foreign key (which we should not worry about when working at the ERD level).


Now to your questions:

Q0) However, is it possible to create a Many:Many relationship between ElibibleEquity and EligibleIndices if they are children to the TradeBasket?

Answer 0)  Yes.  Just make sure that this really makes sense (see answer 1).

Q1) To facilitate a relationship between EligibleEquity and EligibleIndices, what should the cardinalities should i adopt (red or purple - in relation to the ERD)?

Answer 1)  Find out, and use, whatever is correct!
   Does EACH EligibleEquity (potentially) use/contain ONE OR MORE EquitableIndices?
   AND: Is EACH EquitableIndex used/contained within ONE OR MORE EquitableEquities?
   -> If the answer to both of these questions is: YES, then model a many-to-many relationship. (This will require an "intersection" or "junction" later on -> see the HYPOTHETICAL... entity in the attached ERD.)



Q2) If the EligibleEquity and EligibleIndices should be mapped as multivalued entities of the TradingBasket, what would the primary key of the junction table between EligibleEquity and EligibleIndices be?

Answer 2) -> The key for this could be: the combination of the UNIQUE IDENTIFIERs (UIDs) of TradingBasked and EligibleEquity (or EligibleIndex, respectively).  

Eg - situation as discussed in "Answer 1": if the UIDs are the attributes(EligibleEquity)TYPE, and (EligibleIndex)INDEX_NAME, respectively, then a combination of these (unique) values can serve as a UID of the intersection/junction. (-> See the keys of the HYPOTHETICAL... table in the attached Relational model.) 



There is more to be considered, and I'm sure that there will be more questions.  Let us know how you get on!