Applying the Chain Rule to Basket Option Vega
Summary
The document asks how to differentiate a basket option price with respect to one constituent asset’s volatility. It lays out dependencies among the option price, basket forward, strike, aggregate volatility, and constituent forward, then applies the multivariable chain rule. The proposed result includes paths through both the constituent forward and the aggregate volatility, with the strike and time treated as independent of the volatility being varied.
The author reports that a finite-difference comparison agrees when a term involving the aggregate volatility’s dependence on the constituent forward is omitted, despite believing that dependence may be nonzero. This highlights the need to check the full dependency graph and distinguish direct from indirect effects when computing a sensitivity. The document provides no definition of the basket pricing or volatility functions, so it does not establish whether the disputed term should vanish; its formula and numerical observation remain an unresolved question rather than a verified result.
Key ideas
- A basket option price can depend on a constituent volatility through multiple intermediate quantities.
- The total derivative must account for each dependency path that changes when the constituent volatility changes.
- The proposed sensitivity includes effects through both the basket forward and aggregate volatility.
- A finite-difference match after dropping a term does not by itself explain why that term vanishes.
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# Calculating Greeks for basket option
# Calculating Greeks for basket option
I am stuck with obtaining nice relative error of my analytical derivative of a function. I am comparing values of my function at points $x$ and $x+ \delta$ and the difference does not coincide with derivative value at $x$ multiplied by $\delta$. I want to calculate derivative of $C$ wrt to $\sigma_i$. Maybe I forgot calculus or I am doing silly mistake somewhere and after 10th revision of formula I can not see it. Can you please shed a light on my problem, cause I am crying on it not the first day.
I have such functions, dependencies: $t$ - independent variable, $$C = C(F,K,\sigma,t)$$ $$F = F(w_i,F_i)$$ $$K = K(w_i,K_i)$$ $$ \sigma = \sigma( w_i,\sigma_i,F_i,t)$$ $$F_i = F_i( S_i, R_i, \rho_{xi},\sigma_{x},\sigma_i)$$ $S_i,R_i,\rho_{xi},\sigma_x,\sigma_i,w_i,K_i$ - as well independent variables.
Am I right that total derivative: $$\frac{dC}{d\sigma_i} = \frac{\partial C}{\partial F}\frac{d F}{d \sigma_i} + \frac{\partial C}{\partial K}\frac{d K}{d \sigma_i} +\frac{\partial C}{\partial \sigma}\frac{d \sigma}{d \sigma_i} +\frac{\partial C}{\partial t} \frac{d t}{d \sigma_i}$$ hence: $$\frac{dC}{d\sigma_i} = \frac{\partial C}{\partial F} (\frac{\partial F}{\partial w_i}\frac{d w_i}{d \sigma_i} + \frac{\partial F}{\partial F_i}\frac{d F_i}{d \sigma_i}) + 0 + \frac{\partial C}{\partial \sigma}(\frac{\partial \sigma}{\partial w_i}\frac{d w_i}{d \sigma_i} + \frac{\partial \sigma}{\partial \sigma_i}\frac{d \sigma_i}{d \sigma_i} + \frac{\partial \sigma}{\partial F_i}\frac{d F_i}{d \sigma_i} + \frac{\partial \sigma}{\partial t}\frac{d t}{d \sigma_i} ) + 0 $$ $$\frac{dC}{d\sigma_i} = \frac{\partial C}{\partial F} (0 + \frac{\partial F}{\partial F_i}\frac{d F_i}{d \sigma_i}) + 0 + \frac{\partial C}{\partial \sigma}( 0 + \frac{\partial \sigma}{\partial \sigma_i} + \frac{\partial \sigma}{\partial F_i}\frac{d F_i}{d \sigma_i} + 0) + 0 $$ Have I omitten something? $$\frac{dC}{d\sigma_i} = \frac{\partial C}{\partial F} \frac{\partial F}{\partial F_i}\frac{d F_i}{d \sigma_i} + \frac{\partial C}{\partial \sigma}\frac{\partial \sigma}{\partial \sigma_i} + \frac{\partial C}{\partial \sigma}\frac{\partial \sigma}{\partial F_i}\frac{d F_i}{d \sigma_i} $$
Update:
- Is it a correct way to compute vega wrt to one of the assets in my basket :$\frac{dC}{d\sigma_i} $? I am askying so as I know that partial derivative of $C$ wrt to $\sigma$ is a vega.
- Interesting, but if I put last summand, i.e. $\frac{\partial C}{\partial \sigma}\frac{\partial \sigma}{\partial F_i}\frac{d F_i}{d \sigma_i} $ to $0$ then I get perfect relative error.
But I am lost, because apparently it is not $0$.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.