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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

$$D_n=\frac{\partial F_n(\hat{S}_n)}{\partial \hat{S}_n } $$


Based on these quantities, how can I compute sensitivities using the adjoint method?


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Very interesting question - could you insert a link to what the "adjoint method" or "adjoint MC method" is? –  Richard Jan 20 '14 at 12:11
Furthermore: with "sensitivities" you mean something like the Greeks or more general derivatives w.r.t. to certain parameters. Have you heard of Malliavin-calculus? –  Richard Jan 20 '14 at 12:12

1 Answer 1

We set out a general scheme for doing this sort of thing in our paper


and its sequel


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.

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Could you perhaps give some outline of the method to answer the question. Although it is appreciated when references are given it is generally not enough for a good answer - Thank you. –  vonjd yesterday
that would be rather long.. the essential idea is that you break up the function into very simple operations and then compute each step's sensitivities using the chain rule. –  Mark Joshi 16 hours ago

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