Why a Full Option Value Function Helps Compute Greeks
Summary
The document asks whether estimating an option’s value across underlying prices is more useful than estimating value only at the current spot. The accepted answer points to numerical methods such as finite-difference schemes, which can produce the value function over a range of underlying prices in one calculation.
Knowing that function supports deriving delta and gamma without a separate computational pass, making it useful for assessing how option value changes with the underlying and for hedging. The discussion also emphasizes that hedging decisions are made at the present state: future underlying prices are uncertain, and a model can be reapplied as the state changes. The answer is brief and gives no numerical example or comparison of computational cost, accuracy, or calibration approaches, so the practical benefit depends on the pricing method and its implementation.
Key ideas
- A numerical pricing method can compute option values across a range of underlying prices.
- The value function can be used to obtain delta and gamma without a separate computational pass.
- These sensitivities can inform hedging decisions at the current market state.
- Future prices remain stochastic, so the pricing model can be reapplied as the underlying price changes.
- The discussion gives no quantitative comparison of methods or their computational tradeoffs.
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# Importance of full value functions for option pricing
# Importance of full value functions for option pricing
Suppose the value of an option is given by $v(s_0)$ where $s_0$ is the current price of the underlying asset and $v:\mathbb{R}_+\to\mathbb{R}_+$.
It seems that the literature is mostly focused on getting an estimate of $v(s_0)$. My question is whether there is any benefit from estimating the whole function $v$ instead of just a estimate at one point??? Is there any benefit in terms of hedging, etc???
Thank you.
## Answer by Antoine Conze (score 2, accepted)
https://quant.stackexchange.com/a/37334
Some numerical methods, e.g. finite difference schemes, enable you to compute the entire function $s \mapsto v(s)$ at once. This can be useful as no additional pass is required to compute the delta and the gamma.
## Answer by clarkmaio (score 0)
https://quant.stackexchange.com/a/37333
Ciao,
Of course it would be amazing to know the future (i.e. the whole trajectory of the price process) and it would be really usefull for hedging (it makes the hedging itself a trivial problem). However I think you have to think about that everything in the future depends on stochastic variables and that the only place where hedging makes sense is the present (you always work to be covered right now...at every time!).
In general you can work on the future in a very simple way. Infact at each time $t$ you can just use the initial point $s_t$ instead of $s_0$ in your model.
What changes is that $s_t$ is a stochastic variable rather than a real value like $s_0$. In any case, depending on your model, $s_t$ will depend on $s_0$ (see for example Black Scholes model where $s_t$ depends linearly from $s_0$).
The big problem at this point is that you can not do calibration in $t$ since by definition you need the market data (ok..maybe you can do something with futures value) and this situation stop you from having a proper model for anything.
I hope I was helpful!
Ciao ciao,
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