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by: Domingo Parker


Marketplace > Oklahoma State University > Finance > FIN 5883 > QUANT FINANCIAL APPL
Domingo Parker
OK State
GPA 3.5


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Class Notes
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This 0 page Class Notes was uploaded by Domingo Parker on Sunday November 1, 2015. The Class Notes belongs to FIN 5883 at Oklahoma State University taught by Staff in Fall. Since its upload, it has received 7 views. For similar materials see /class/232778/fin-5883-oklahoma-state-university in Finance at Oklahoma State University.




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Date Created: 11/01/15
Monte Carlo Derivative valuation Things we learned from the Arbitrage Theorem 1 With no payouts the arbitrage theorem states that the risk neutral probability measure produces expected return 1r for all asset prices 2 If an asset has a known positive payout the expected return under the risk neutral measure is reduced by the size of the payout 3 For futures contracts the risk neutral measure produces an expected return of zero Further the noarbitrage value of the derivative is its expected payoff discounted to the present at the risk free rate Ct11r E N CH1 Given a geometric Brownian motion for the underlying dSI uSIdtoSIdWI Recognize that a geometric Brownian motion is only one possibility for the assumed continuous time stochastic process The risk neutralized stochastic processes 1 dSI rSldt 06391th 2 dSI r qSIdt 06391th 3 dSI 06391th Monte Carlo valuation exploits these results to provide a exible riskneutral valuation methodology for European derivatives with fixed expiration dates T Time frame t T l Specify the risk neturalized process 2 Use a random number generator to produce the first path terminal value see exact method below of the underlying on the expiration date of the derivative S 1T 3 Compute the payoff on the derivative given pathterminal value SlT of the underlying ClT 4 Repeat steps 2 and 3 for ntrials n 5 Average derivative payoff Z 11 7 n 6 Discount to present using risk free rate r Ct ef T X 2 171 n Simulating geometric Browninan motion continuous process with discrete process dSI uSIdtaSIdWI GBM dW E iid Weiner process ie independent draws from normal distribution with mean zero and variance dt Where dt is the continuous time limit of the time index The continuous time increment dt is de ned such that alt 0 for at gt 1 In the limit as dt gt 0 dS S S 4 w MSW 1nSt SI SI N u gtlt dt 0392 X dt where d3 is the normal probability lnction If GBM then SHdI SI eXp039dVI7 u 0392dt Sldl N LN 2 Elstdll Stewioja M In SM ln St u 0502 dt 0251 For multiple time steps N H jdl Wdel l dWs 13 I WHJdt N N M X jdt 0392 X jdt N 2 Sljdl SI eXP039Wrjdt u 039 NC For a discrete representation of GBM De ne dt calendar time over interval tl t As in preVious notes let X lnSt 7 lnSt1 x udt UghE or dS S SH SH gtlt yda E Note the iid Weiner process dW has been replaced with its discrete counter part Where is now a iid draw from the standard normal N0l Simulating 7 Stochastic Modeling 7 Monte Carlo methods Write dX WE then dSI SI SI1 S171 gtlt udt039dX Euler method of simulating series S from starting value So for a series of length T 0 Generate T random draws 11 12 T from standard normal 0 Given estimates uhat sigmahat update S at each time step using these random draws by simply putting the latest value for S into the righthand side to calculate dS When using this method to simulate the evolution of a stochastic differential equation this method produces discretization error of Odt Which means that in the limit as dt gt0 the ratio of the discretization error to dt tends to a fixed constant This implies the discretization error is asymptotically proportional to dt in the continuous time limit There are other methods with smaller discretization errors The Milstein method has a discretization error of Odtz An exact method with no discretization error exists for the process developed above Use Ito s lemma to de ne the process lnS dlnSlnSI lnSt1u 02dt odX implying ygazmm ldX SIS1718 71 Where possible for Monte Carlo simulations the exact formulation is preferred to methods involving discretization error Precision of the Monte Carlo derivative value To control the absolute error 501 6quot S with confidence level loc The confidence interval for an estimate of the mean n 5i 217094 7 the sample variance of C and n is the number of trials where 2105 is the quantile from the standard normal s2n is Perform reasonable number of trials say 100 calculate the sample variance for the derivative s payoff in these 100 trials Use this estimate for s2 to determine the number of trials n required such that 2 S 21705o4 7 3 3 Monte Carlo pricing is computationally expensive Variance reduction techniques seek to reduce the variance of the sampling distribution of C hence the required number of trials to achieve the desired accuracy Antithetic sampling can be applied whenever the simulated value price of the underlying is a monotonic transformation of the random increment Antithetic sampling creates a second series from the original 41 4 4T such that the resulting series is negatively correlated with the rst At its simplest this is accomplished by drawing 41 4 4T and creating 7141 4 4T The paths ofthe underlying price constructed from these two sets of standard normal increments will be negatively correlated Calculate derivative value f1 from n trials 41 4 4T Calculate derivative value f from n trials 7141 4 4T The MonteCarlo simulated value using antithetic variance reduction is simply f 05f1 f Parameterizing a model estimating u 6 from sample M of observations dS N u gtlt dt 0392 X dt where d3 is the normal probability function A 1M A d 1 M A2 Milt H3195 039 t WEIOCI Annualized estimated values u 6 A 1 M A 1 M A 2 M X 6125 a M71de E109 0 Estimating the annualized mean from M observations with small dt is imprecise 0 Mean over time interval dt scales directly with dt 0 Central Limit Theorem applied to the sample mean indicates that for samples drawn from a normally distributed random variable or as the sample size increases the sample mean is itself distributed normal and the variance of the distribution of the sample mean decreases with sample size M 2 A A 039 A 039 Ni gt sei M xM In contrast 0 Standard deviation over time interval dt scales with dt 0 The sampling distribution of the sample variance amp2 Naza4i gt seampamp1 i M l 2M Notice that for a given sample size M the estimate of standard deviation is more precise than the estimate of the mean Additionally as the sample size is increased the precision of the estimate of standard deviation improves at a faster rate than the precision of the estimate of the mean Alternative estimates annualized volatility of return For dt small and LPG A 1 A24 2 a M71xdt 110 Exponential weighting From daily High 7 Low prices 1 M 2 039 thaogHI 10gt Preliminaries S price of underlying asset V value of a position that contains options written on S q vector of quantities of the position V fS 6 T r K q Step 1 Use a Taylor series expansion to describe the changes in position value in terms of the price of the underlying asset 2 61V dS 2S7 alS2 additional terms in expansion The deltagamma approach uses only terms of second order or lower Substituting for dS SdSSSX and the option s delta and gamma X logarithmic change in price of underlying assumed normal distributed dV qASX qGS2X2 or Y aX bx2 a qAS l 2 7 E qGS To apply the DelatGamma Cornish Fisher approximation to measure the risk of a hedge portfolio such as question 2 on the nal it is necessary to de ne the hedge portfolio s netdelta and the hedge portfolio s netgamma and substitute these net value into the de nition of a and b Remember that Delta for long underlying position is 1 Delta for short underlying position is 71 Gamma for long or short underlying position is 00 Delta for a long position in call option is positive Delta for a long position in a put option is negative Delta for a short position in a call option is negative Delta for a short position in a put option is positive Gamma for a long position in either call or put options is positive Gamma for a short position in either call or put option is negative An example the collar position explored in class Collar is long q units of underlying long qquot Hp units of put option and short qquot HC units of call option q1000 long position underlying option values and Greeks from BlackScholes atmoneyput option kl100 outmoneycall option k2110 k100 110 00 p0 blspriceS k r T s Dc Dp blsdeltaS k r T s Gamma blsgammaS k r T s HpabslDpl hedge ratio deltaneutral put hedge offset25 portion of put cost to offset Hc 0ffS IHpp0lCO2 hedge ratio call option aq SlHpDplHcDc2 bq0 5SAZ0HpGammal HcGamma2


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