Greeks of Basket Products with Linear and Nonlinear Payoffs

Summary

The question asks whether the delta, gamma, vega, rho, and theta of a multi-asset product can be calculated by adding the corresponding Greeks of its underlying assets. The product described pays annual coupons when the performance of an equally weighted basket reaches a barrier, making its payoff potentially dependent on a threshold rather than a simple weighted sum.

The answer states that Greeks add linearly when the basket payoff itself is a constant-weight sum of the underlying prices or payoffs. For a nonlinear payoff, such as one involving a barrier, the relevant Greek must instead be obtained by differentiating the full basket payoff with respect to the risk factor in question. The response gives the linear delta principle but does not work through the coupon structure, model assumptions, asset correlations, or numerical valuation. Its conclusion is therefore a general rule, not a complete pricing treatment of the described product.

Key ideas

• Greeks of a constant-weight linear basket payoff are the same weighted sums of the component Greeks.
• A barrier or other nonlinear payoff generally prevents direct addition of the underlying Greeks.
• Compute each basket Greek by differentiating the complete payoff with respect to the relevant risk factor.
• The discussion does not provide a model or numerical valuation for the coupon-bearing product.

Tags

# Greeks of Basket


# Greeks of Basket












I am considering a product composed of 10 underlying assets. The maturity is 5 year. Each year if the performance of the equi-weighted portfolio reach a barrier, it pays a coupon.

My question concern the computation of the greeks. For example, is it true to compute delta as the sum of the delta of each underlying assets ? Same question for the gamma, vega, rho and theta.

## Answer by ash (score 3)

https://quant.stackexchange.com/a/7741

Freddy has already answered it and my answer had an assumption in it so clarifying -

If payoff of basket with underlined securities A,B and C are $$ P_b = C_1*P_A + C_2*P_B + C_3*P_C $$ Where $$C_1 , C_2 ,C_3 $$ are contants then portfolio delta is $$ \delta_b = C_1*\delta_a+C_2*\delta_b+C_3*\delta_c $$

In short as Freddy Said , and I assumed if the potfolio payoff is merely a sum of all underlined then yes the delta will be sum of deltas of underlined. If not and then you have to apply differentiation on the payoff fuction of basket

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