Greeks for a European Option on a Two-Asset Basket
Summary
The document considers delta, vega, and gamma for a European call whose underlying is a weighted sum of two stock prices. It questions whether the basket’s vega can be obtained by weighting the individual asset vegas, and proposes shifting both volatilities to create a chosen change in the basket’s standard deviation before measuring the option-price response.
The response cautions that the option depends on the two assets separately, so a single basket Greek is not generally well-defined from the current basket value alone. In some cases, the individual deltas may be similar enough that a weight-adjusted combination is an acceptable approximation. The discussion does not provide a general pricing model or formulas for calculating the full set of sensitivities; the approximation’s suitability depends on the assets and model.
Key ideas
- A basket option’s value depends on each underlying asset, not solely on the basket’s current value.
- A single basket delta or vega may not be meaningful without specifying a model that treats the basket as an underlying.
- A weighted combination of individual sensitivities can be an approximation when the asset sensitivities are sufficiently close.
- Shifting multiple volatilities together defines a particular scenario sensitivity rather than a universal basket vega.
Tags
Full text
# Greeks of a Basket Option
# Greeks of a Basket Option
I want to estimate delta, vega and gamma for a basket option. This option is a European Call option. The underlying is $S=\omega_1 S_1 +\omega_2 S_2$
Where:
$S1$ = stock price of asset 1
$S2$ = stock price of asset 2
$\omega_1 $ and $\omega_2$ are the weights
It is easy to compute $\nu_{S_1} $ and $\nu_{S_2}$ but my question is about the definition of $\nu_{S}$
I started computing $\nu_{S}$ as $\omega_1 \nu_{S_1} +\omega_2 \nu_{S_2}$ but then I realized that maybe this is not the correct way to compute it.
My second option was to estimate a number $\theta$ with the following property: "A variation on $\theta$ of each of the stock's volatilities produces a variation on the SD of S of magnitude $\epsilon$", then, compute vega as $\nu= (V(\sigma_1+\theta, \sigma_2 +\theta)-V(\sigma_1, \sigma_2))/ \epsilon$ for a small $\epsilon$ value
Where V represent the option price and $\sigma_i$ the volatlity of the ith asset.
I want to know if one of these methods is correct.
I am new at pricing options so every information or recomendation will be useful for me.
## Answer by Andrea (score 2)
https://quant.stackexchange.com/a/81175
None of the is exactly correct.
The current value of the basket is not an input to the formula, so you cannot differentiate it, and no delta can be computed.
The option really depends on the 2 underlyings separately, so it does not make sense to think in terms of the basket delta.
You can change the model to have the model as underlying (with some approximation).
And in many cases the 2 deltas will be close enough, so $\omega_1 \nu_{S_1} +\omega_2 \nu_{S_2}$ is acceptable.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.