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Rutgers University MATH 612 - Computing Greeks using Monte Carlo methods

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IntroductionIntroductionIntroductionIntroductionIntroductionIntroductionPayoff RegularisationPayoff RegularisationPayoff ErrorPayoff ErrorPayoff ErrorPayoff ErrorPayoff ErrorPayoff ErrorPayoff ErrorPayoff ErrorMonte Carlo errorMonte Carlo errorMonte Carlo ErrorMonte Carlo ErrorMonte Carlo ErrorMonte Carlo errorMonte Carlo ErrorMonte Carlo ErrorMonte Carlo ErrorMonte Carlo errorMonte Carlo ErrorMonte Carlo ErrorMonte Carlo ErrorDown-and-out callDown-and-out callPayoff ErrorPayoff ErrorPayoff ErrorTimestep ConvergenceMonte Carlo ErrorMonte Carlo ErrorMonte Carlo Error2D Stratified SamplingConclusionsComputing Greeksusing Monte Carlo methodsMike [email protected] University Computing LaboratoryMonte Carlo Greeks – p. 1/41IntroductionGiven a stochastic o.d.e.dS = µ(S, t) dt + Σ(S, t) dW,subject to initial conditions S(0) = S0, the expected value ofa payoffV ≡ E[f(S(T ))]can be approximated by calculating the average over anumber of discrete paths given by some time-marchingapproximationSn+1= Sn+ µ(Sn, tn) ∆t + Σ(Sn, tn) ∆Wn.Monte Carlo Greeks – p. 2/41IntroductionStrengths:simple;scales well to multiple correlated factors.Weaknesses:American options are challenging (Glasserman,Broadie);methods to reduce Monte Carlo variance compromisethe simplicity (stratified sampling, importance sampling,Latin hypercube sampling, quasi-MC);computing Greeks can be a problem.Monte Carlo Greeks – p. 3/41IntroductionThe pathwise approach to computing∆ =∂∂S0E[f(S(T )]relies on µ, Σ and f being Lipschitz continuous, andpiecewise differentiable, so one can interchange thederivative and the expectation to get∆ = E∂∂S0f(S(T ))= EdfdSeS(T ).whereeS(t) ≡∂S(T )∂S0.Monte Carlo Greeks – p. 4/41IntroductioneS(t) is approximated by linearising the time discretisation togeteSn+1=eSn+∂µ∂SneSn∆t +∂Σ∂SneSn∆W,and the expectation is again approximated by averagingover multiple paths.Monte Carlo Greeks – p. 5/41IntroductionIf µ, Σ and f have Lipschitz-continuous first derivative, andare piecewise twice-differentiable, thenΓ ≡∂2∂S20E[f(S(T )] = E∂2∂S20f(S(T ))= Ed2fdS2eS2(T ) +dfdSeeS(T ).whereeeS(t) ≡∂2S(t)∂S20can be approximated by again differentiating the discretestochastic equations.Monte Carlo Greeks – p. 6/41IntroductionWhat can go wrong?Consider geometric Brownian motiondS = r S dt + σ S dW,and a digital optionf(S(1)) = H(S(1) − K) =(1, S > K0, S < KFor this case, ∆ and Γ are both non-zero, but we get zerovalues forEdfdSeS(1), Ed2fdS2eS2(1) +dfdSeeS(1).Monte Carlo Greeks – p. 7/41Payoff RegularisationKey idea: replace discontinuous payoff H(S−K) by asmooth payoff Hε(S−K) with width ε.−5 −4 −3 −2 −1 0 1 2 3 4 5−0.200.20.40.60.811.2xhHHεHow should Hεbe chosen? How big should ε be?Monte Carlo Greeks – p. 8/41Payoff RegularisationThe optimum size for ε comes from balancing two errors:error due to modified payoff:O(εm) for some m > 0;Monte Carlo sampling error:O(M−1/2ε−p) for some p > 0, with M = #paths.Monte Carlo Greeks – p. 9/41Payoff ErrorIf we define p(S) to be the probability density function forthe final state S(1), thenV =Z∞0p(S) H(S−K) dS.Alternatively, if we defineW = Φ−1(U),where Φ−1is the inverse Normal cumulative probabilityfunction, andU is uniformly distributed on (0, 1), thenV =Z10HS0expr −12σ2+ σ Φ−1(U)− KdU.Monte Carlo Greeks – p. 10/41Payoff ErrorSuppose now that we define a new regularised expectationVε=Z∞0p(S) Hε(S−K) dS,where Hε(x) is a regularised Heaviside function defined byHε(x) = h(0)(x/ε),with h(x) being a function such ash(0)(x) =12(1 + tanh(x)) ,which has the properties that h(0)(x) −12is an odd functionwhich approaches±12exponentially as x → ±∞.Monte Carlo Greeks – p. 11/41Payoff ErrorBy making the substitution S = K + εs, it follows thatVε− V =Z∞0p(S) {Hε(S−K) − H(S−K)} dS≈ εZ∞−∞p(K +εs)h(0)(s) − H(s)ds.Performing a Taylor series expansion of p(K+εs), we obtainthe asymptotic expansionVε− V = a2ε2+ a4ε4+ a6ε6+ . . .whereak=1(k−1)!∂k−1p∂Sk−1KZ∞−∞sk−1h(0)(s) − H(s)ds.Monte Carlo Greeks – p. 12/41Payoff ErrorDifferentiating the asymptotic error expansion, we obtainddεVε= 2a2ε + 4a4ε3+ O(ε5),soVε−12εddεVε− V = −a4ε4+ O(ε6).Straightforward manipulations show thatVε−12εddεVε=Z∞0p(S) h(1)S−KεdS,whereh(1)(x) = h(0)(x) +x2ddxh(0)(x).Monte Carlo Greeks – p. 13/41Payoff ErrorThe same argument can be repeated to eliminate thisleading order error by usingh(2)(x) defined byh(2)(x) = h(1)(x) +x4ddxh(1)(x),and more generally, the recursive definitionh(n)(x) = h(n−1)(x) +x2nddxh(n−1)(x),generates a sequence of regularised Heaviside functionswhich give errors of leading orderO(ε2n+2).Monte Carlo Greeks – p. 14/41Payoff Error0.01 0.02 0.05 0.110−1210−1010−810−610−410−2100epserror in valuedigital call: effect of payoff smoothingH(0)H(1)H(2)Monte Carlo Greeks – p. 15/41Payoff Error0.01 0.02 0.05 0.110−1210−1010−810−610−410−2100epserror in deltadigital call: effect of payoff smoothingH(0)H(1)H(2)Monte Carlo Greeks – p. 16/41Payoff Error0.01 0.02 0.05 0.110−1210−1010−810−610−410−2100epserror in gammadigital call: effect of payoff smoothingH(0)H(1)H(2)Monte Carlo Greeks – p. 17/41Monte Carlo errorReminder: given M samples xmfrom a distribution withmeanµxand variance σ2x, the sample meanx =1MXmxmhas mean µxand variance M−1σ2x.Hence,x − µx= O(M−1/2σx).and the 3σ confidence range for µxis[x − 3M−1/2σx, x + 3M−1/2σx].Monte Carlo Greeks – p. 18/41Monte Carlo errorUsing standard Monte Carlo sampling, taking values forS(1) from the appropriate probability distribution, the digitaloption value is O(1) and the variance is also O(1).However, for∆, there is a fraction O(ε) of paths which havevalue O(ε−1), so the mean is O(1) but the variance isO(ε−1). Hence the sampling error is O(N−1/2ε−1/2).ForΓ it is even worse, with a mean O(1) but variance isO(ε−3), so the error is O(N−1/2ε−3/2).Monte Carlo Greeks – p. 19/41Monte Carlo Error0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−1−0.500.511.522.5x 10−3epserror in valuedigital call: M=1000000Monte Carlo Greeks – p. 20/41Monte Carlo Error0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−0.1−0.08−0.06−0.04−0.0200.020.04epserror


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Rutgers University MATH 612 - Computing Greeks using Monte Carlo methods

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