Finite-Difference Delta for Worst-of Options with Normalized Payoffs
Summary
The document explains how to estimate a partial delta for a worst-of option by bumping one underlying’s initial spot, simulating its terminal value, and comparing the resulting payoff with the unbumped payoff. The payoff uses each asset’s terminal price divided by its initial price, so the questioner mistakenly updates the denominator along with the bumped spot. That normalization cancels the intended change and can make the option value appear unchanged.
The answer clarifies that the initial spot in the contract’s payoff definition must stay fixed during the bump. Recalculate the terminal price from the shifted starting spot, but still divide by the original spot when evaluating the payoff. The document illustrates the semantic issue rather than assessing Monte Carlo implementation. It does not discuss bump-size choice, common random numbers, discounting, or convergence, so it is not a full guide to estimating Greeks.
Key ideas
- For a finite-difference partial delta, bump one underlying’s starting spot and recompute its terminal price.
- Keep the contract’s original reference spot in the normalized payoff denominator.
- Updating the denominator with the bumped spot can cancel the effect the bump is meant to measure.
- The discussion addresses payoff semantics rather than Monte Carlo accuracy or variance reduction.
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# Monte carlo delta calculation for Worst/Best Of Option
# Monte carlo delta calculation for Worst/Best Of Option
I try to calculate the Delta for WO by finite difference.
For example, $K = 1.$
$$ S_t = S_0 e^{(r - d_1 - \frac{\sigma_1^2}{2})t + \sigma_1 W_t^1} $$ $$ F_t = F_0 e^{(r - d_2 - \frac{\sigma_2^2}{2})t + \sigma_2 W_t^2} $$
$$ Payoff = (\min\{ \frac{S_t}{S_0}, \frac{F_t}{F_0}\} - K )_{+}$$
For the partial delta calculation I shift the spot and rerun monte carlo, such that, my the bumped forwards is following:
$$ S_t^{up} = (S_0 + S_0 * 0.01) e^{(r - d_1 - \frac{\sigma_1^2}{2})t + \sigma_1 W_t^1} $$ $$ F_t = F_0 e^{(r - d_2 - \frac{\sigma_2^2}{2})t + \sigma_2 W_t^2} $$
$$ Payoff^{up} = (\min\{ \frac{S_t^{up}}{S_0}, \frac{F_t}{F_0}\} - K )_{+}$$ Then I calculate the simple difference: $ \varDelta_{proxy} = Payoff^{up} - Payoff$
As result I get the partial sensitivity, but arises the problem with explanation, when I shift initial spots by the shift size, due to I use the ratio in payoff, my shifted forward is divided into shifted spot and the price of option unchanged.
About the monte carlo engine, please don't care. I have a semantic error related to the payoff.
Can someone explain to me where I`m wrong with my unchanged price?
## Answer by Valometrics.com (score 1)
https://quant.stackexchange.com/a/51067
$S_0$ should remain unchanged as it is defined in the contract terms. In order to compute delta by finite difference, you should shift the price from $S_t$ to $S_t+shift$, generate the price at maturity then compute the payoff using generated price WITHOUT FORGETTING TO KEEP $S_0$ IN THE DENOMINATOR UNCHANGED.
You can find a pricer for BO and WO basket options in my website ValoMetrics.com for test purpose.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.