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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 (1+r) 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 no-arbitrage value of the derivative is its expected payoff discounted to the present at the risk free rate. ~ Ct = (1/(1+r))* E P [Ct+1] Given a geometric Brownian motion for the underlying dS t uS t dt S t dWt Recognize that a geometric Brownian motion is only one possibility for the assumed continuous time stochastic process. The risk neutralized stochastic processes: 1. dS t rS t dt S t dWt 2. dS t (r q)S t dt S t dWt 3. dS t S t dWt Monte Carlo valuation exploits these results to provide a flexible “risk-neutral” valuation methodology for European derivatives with fixed expiration dates, T. Time frame: t .. .. .. .. .. T 1. 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(1,T). 3. Compute the payoff on the derivative given path/terminal value, S(1,T) of the underlying, C(1,T) 4. Repeat steps 2 and 3 for n-trials. n C (i , T ) 5. Average derivative payoff, . i 1 n n C (i, T ) . i 1 n 6. Discount to present using risk free rate, r, C (t ) e r*T 1 Simulating geometric Browninan motion (continuous process) with discrete process: dS t uS t dt S t dWt (GBM) dW iid Weiner process, i.e. 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 defined such that dt 0 for > 1. In the limit as dt 0: dS t S t dt S t ln( S t dt ) ln( S t ) St St dS ~ (u dt , 2 dt ) where is the normal probability function S ~ If GBM then: S t dt S t exp(dWt (u 12 2 )dt ) S t dt ~ LN 2 E[ S t dt ] S t e (u 0.5 )dt ln S t dt ~ (ln S t (u 0.5 2 )dt , 2 dt ) For multiple time steps: t jdt ~ Wt jdt dWs ds t ~ Wt jdt ~ N (u jdt, 2 jdt) ~ S t jdt S t exp(Wt jdt (u 12 2 ) jdt) For a discrete representation of GBM: Define dt = calendar time over interval [t-1 …. t] As in previous notes let xt = ln(St) – ln(St-1) xt dt dt , or dSt St St 1 St 1 (dt dt ) Note the iid Weiner process dW has been replaced with its discrete counter part. Where is now a iid draw from the standard normal, N(0,1). 2 Simulating – Stochastic Modeling – Monte Carlo methods: Write dX dt then: dSt St St 1 St 1 (dt dX ) Euler method of simulating series S, from starting value S0, for a series of length T Generate T random draws, (1, 2, ….T) from standard normal Given estimates u-hat, sigma-hat, update S at each time step using these random draws by simply putting the latest value for S into the right-hand side to calculate dS. When using this method to simulate the evolution of a stochastic differential equation, this method produces discretization error of O(dt). Which means that in the limit as dt0, 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 O(dt2). An exact method with no discretization error exists for the process developed above. Use Ito’s lemma to define the process ln(S) d ln( S ) ln( S t ) ln( S t 1 ) ( 1 2 )dt dX 2 implying: t St St 1e (( 12 2 ) dt dX ) t 1 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, C (n) with confidence level (1-): The confidence interval for an estimate of the mean: 3 s 2 ( n) where z1-0.5* is the quantile from the standard normal, s2(n) is n the sample variance of C and n is the number of trials. C ( z10.5* ) 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; z1 0.5* s2 n 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 (1, 2, ….T) such that the resulting series is negatively correlated with the first. At its simplest this is accomplished by drawing (1, 2, ….T) and creating –1*(1, 2, ….T). The paths of the underlying price constructed from these two sets of standard normal increments will be negatively correlated. Calculate derivative value f1 from n trials (1, 2, ….T). Calculate derivative value f2 from n trials –1*(1, 2, ….T). The MonteCarlo simulated value using antithetic variance reduction is simply: f = 0.5*(f1 + f2). Parameterizing a model (estimating , ) from sample M = # of observations dS ~ (u dt , 2 dt ) where is the normal probability function S ˆdt 1 M xt M t 1 ˆ dt M 2 1 ( xt ˆ ) ( M 1) t 1 Annualized estimated values ( , ) 4 ˆ 1 M xt M dt t 1 ˆ M 2 1 ( xt ˆ ) ( M 1)dt t 1 Estimating the annualized mean from M observations with small dt is imprecise. Mean over time interval dt scales directly with dt. 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. ˆ ~ N ( , 2 M ) se( ˆ ) ˆ M In contrast, Standard deviation over time interval dt scales with The sampling distribution of the sample variance: ˆ 2 ~ N ( 2 , 4 dt 2 1 ) se(ˆ ) ˆ M 1 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, =0 ˆ M 2 1 (xt ) ( M 1)dt t 1 Exponential weighting: ˆ t 1 dt i i j 2 xj j 5 From daily High – Low prices ˆ M 2 1 (log( H t ) log( Lt )) ( M 1) dt 4 log 2 t 1 6