Adjoint Monte Carlo Sensitivities for Discretized Diffusions
Summary
The document sets up an option valuation problem using Monte Carlo paths generated from an Euler discretization of a stochastic differential equation. It asks how to compute sensitivities, or Greeks, with respect to model parameters such as the initial value, interest rate, and volatility. The setup identifies the payoff at the terminal state and the derivative of each time-step update with respect to the current state, quantities that an adjoint calculation can use to propagate sensitivity information backward through the simulation.
The responses point to published treatments and suggest a spreadsheet-friendly example that compares finite differences, complex step, tangent linear, and adjoint methods. One response says the adjoint approach can be preferable to likelihood ratio and Malliavin methods when applicable. However, neither response gives the requested recurrence or a worked derivation, and the referenced material is not reproduced here. The document introduces the computational context and methods to consult, but is insufficient on its own to implement or assess an estimator's accuracy or efficiency.
Key ideas
- Monte Carlo option valuation can be built from paths of an Euler-discretized diffusion.
- Adjoint sensitivities propagate derivative information through the simulation steps to the payoff.
- The question identifies state-transition derivatives and model parameters relevant to computing Greeks.
- Finite differences, complex step, tangent linear, and adjoint approaches are named as comparison methods.
- The responses refer to external treatments but do not provide an implementation or numerical evidence.
Tags
Full text
# How to compute greeks using the adjoint Monte Carlo approach?
# How to compute greeks using the adjoint Monte Carlo approach?
Assume I have a stochastic ODE $$dS = a(S)dt + b(S)dW,$$ with Euler approximation $$\hat{S}_{n+1}=F_n(\hat{S}_n)=\hat{S}_n+a(\hat{S}_n)h+b(\hat{S}_n)Z_n\sqrt{h}.$$ This allows me to create sample paths based on drawing normally distributed random numbers $Z_n$ from $N(0,1)$.
Now the estimated value of my option is $$\hat{V}=\frac{1}{N}\sum_i f(S^i_T)$$ where $f$ is the payoff function and $S^i_T$ is the i-th sample path of the process at time $T$.
Assume the ODE and $f$ have various parameters, for example starting value $S_0$, risk-free interest rate $r$ and volatility $\sigma$. Furthermore, f is sufficiently continous such that the derivatives
$$D_n=\frac{\partial F_n(\hat{S}_n)}{\partial \hat{S}_n } $$
exist.
Based on these quantities, how can I compute sensitivities using the adjoint method?
Links:
## Answer by Mark Joshi (score 2)
https://quant.stackexchange.com/a/16750
We set out a general scheme for doing this sort of thing in our paper
http://ssrn.com/abstract=1401094
and its sequel
http://ssrn.com/abstract=1437847
Whilst the case studied is different, the techniques are the same. I also discuss in detail the whole process in a chapter of More Mathematical Finance.
The adjoint method when it applies is generally better than alternatives such as likelihood ratio and Malliavin calculus.
## Answer by CPT (score 1)
https://quant.stackexchange.com/a/21216
If you want a simple example which you can easily reproduce in a spreadsheet, look at section 3 of the paper "Adjoints and automatic (algorithmic) differentiation in computational finance by Christian Homescu. Table 1 is wrong though but you should be able to generate the same numbers using all 4 methods
1) Finite Difference 2) Complex Step 3) Tangent Linear 4) Adjoint
Good Luck !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.