hi ppl,i am doing a project on option prcing using Monte Carlothe pricing part is quite ok, but got problem in greeksi read Glasserman's book and know how to estimate delta, vegai attempted to compute theta and rho, took derivatives by following the book's methodfor exampletheta = exp(-r*T)*S(T)*T;i ran the simulation but got a theta, which has more than 10% diff as compared to the analytical BlackSchole's my delta, vega results r very close to the analytical value, so i think the MC process is correct anyone has estimated theta using pathwise method?

hi james88,it depends what kind of options do you look at. In general there are three methods 1) Finite Differencing 2) Pathwise Methods and 3) Likelihood Ratio.Which method do you apply? I suppose Pathwise (looking at your formula...) But pathwise has limitations, e.g. for discontinuous payoffs and also for somediffusion processes if the value which you differentiate wrt S(0) does not behave as in Black-Scholes world, see the explanations in Glasserman.Without knowing the problem you encounter in full detail my suggestion would be to combine your Greeks simulation with var reduction techniques. This canimprove your results very very much. Especially, if you consider e.g. barriers or some other exotic stuff (Otherwise MC would not be your method of choice, would it?).If you have some examples of options and diffussions you use one has the chance to analyse the problem further.Best Lapsi

QuoteBut pathwise has limitations, e.g. for discontinuous payoffs and also for somediffusion processes if the value which you differentiate wrt S(0) does not behave as in Black-Scholes worldWhat happens in such cases? Would the situation improve if you mollify the payoff? Lapsi,Can you motivate (mathematically) the following aspect of the pathwise method: take the usual SDE, then apply Euler and then differentiate with respect to the initial condition. So, the idea is: we can differentiate FDM scheme for an SDE on both sides with respect to its initial condition (eq. 7.24), yes?I don't get that, at all. It feels strange, but maybe it's just PO differential calculus; and maybe it's just me

Hi Cuch,I will try and work it out. This can be best grabbed by looking at a problem from ordinary analysis.Sometimes you encounter the situation that you would like to interchange integration and differentiation, e.g.d/dx int f. You might know the derivative d/dx f and you would be able to calculate int d/dx f. Therefore, you would like tohave d/dx int f = int d/dx f. Then, you are done. The main theorem underpinning such results is known as Lebesgue dominated convergence theorem. Under some circumstances interchanging of d/dx and int is feasible. This idenity is reflected in the pathwise problem by writing E[d/dx Payoff] = d/dx E[Payoff].Now, differently to ordinary analysis the Payoff is not a function but a stochastic variable. Here comes some of the trouble from. Different notions of convergence(weak, strong, norm), different notion of solution to a stochastic equation (strong, weak). If we are in a simple model, e.g. Black-Scholes, we may write the second equation d/dx E[Payoff] asd/dx int (Payoff x density) with a simple densityand we are in the setting of our inital examle and can work out the quantities. Especially, if we work in Black-Scholes model we have the exact solution and we are therefore able to compute everything. Suppose now now closed form solution is available but we have a general SDE dX = mu(t.X(t)) dt + sigma(t,X(t)) dW(t). Than our only chance is to fall back to some approximation, e.g. Euler to approximate dX(t)/dX(0) for example. This derivative in fact exists under mild assumptions and what we are actually do is we approx this derivative. Differentiation on both sides mean:Left hand side: Symbol for describing the differentiate process (just notation)Right hand side: A possible fd approximation of the differential -> The actual rule for carring out the approximationFor the big theory you might consult Protter (Stochastic Integration and Differential Equations) or Kunita (Stochastic Flows and Stochastic Differential Equations). This is the theoretical underpinning of the way practioneers do it.Hope this helps...Ciao LapsiUsing this approx we are also able to approximate the differential of

QuoteSuppose now now closed form solution is available but we have a general SDE dX = mu(t.X(t)) dt + sigma(t,X(t)) dW(t). Than our only chance is to fall back to some approximation, e.g. Euler to approximate dX(t)/dX(0) for example. This derivative in fact exists under mild assumptions and what we are actually do is we approx this derivative. Let's take the example of a closed solutionX(t) = X(0) exp((mu - 0.5*sig^2)t + sig*W(t))If we can take the derivative of Euler wrt X(0) then we can do it also for 1) exact solution and 2) SDE (btw is it more correct to write as an integral equation?).Is this too simple?dX(t)/dX(0) = exp((mu - 0.5*sig^2)t + sig*W(t)) ??? It's probably a bit more subtle than that?

Yeah! That is basically it.Suppose you want the delta of simple call, then Payoff is e^(-rT) max(S(T)-K,0).Therefore, dPayoff / dS(0) = dPayOff/dS(T) * dS(T) /dS(0).dPayoff/dS(T) = 0 if S(T) < K and 1 if S(T) > K (since we are taking E[] we do not care that the derivative does not exist for S(T) = K...)dS(T) / dS(0) = S(T) / S(0) = exp((r-sigma^2/2) t + sigma sqrt(t) * z)In fact very easy....

Quotei attempted to compute theta and rho, took derivatives by following the book's methodfor exampletheta = exp(-r*T)*S(T)*T;Your formula for theta looks wrong. Theta is the derivative with respect to time, not r.Suggestion: make your Monte Carlo work with an arbitrary payoff function, and try to price a simple fixed cashflow. Then check your derivatives on pricing a simple discounted cashflow via Monte Carlo. Your theta on a simple cashflow ought to be simply one less day of discounting. Then when you incorporate a more complex payoff function, the change ought to reflect less discounting plus a shorter time to expiry.

Try,Theta = - exp(-rT)S(T) 1{S(T) > K} (r + ln(S(T)/S(0)) / 2T - (r+sigma^2/2)/2)as the estimator this might work... Try to find out that this is the pathwise derivative wrt to to t.You may want to use the identiy 0 = T Theta + r Rho + sigma/2 Vega....Cheers, Lapsi