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I recently came across the Optimal Rebalancing Strategy Using Dynamic Programming for Institutional Portfolios by W. Sun, A. Fan, L. Chen, T. Schouwenaars, M. Albota paper that examines the cost of different rebablancing methods. For example, one might use calendar rebalancing: i.e. rebalance every month / quarter / year. Or one might use threshold rebalancing: i.e. rebalance only if asset weights break out from a certain band around the policy mix, say 3%.

To investigate the cost of the different rebalancing methods, authors run 10,000 simulations. Today, I want to show how to simulate asset price paths given the expected returns and covariances. I will assume that prices follow the Geometric Brownian Motion. Also I will show a simple application of Monte Carlo option pricing. In the next post I will evaluate the cost of different rebalancing methods.

Let’s assume that a stock price can be described by the stochastic differential equation:

where is the expected annual return of the underlying asset, is the annualized volatility, and is a Brownian Motion. This stochastic differential equation has the following solution

which I implemented in the asset.paths() function. The asset.paths() function is based on the Simulating Multiple Asset Paths in MATLAB code in Matlab.

asset.paths <- function(s0, mu, sigma, nsims = 10000, periods = c(0, 1) # time periods at which to simulate prices ) { s0 = as.vector(s0) nsteps = len(periods) dt = c(periods[1], diff(periods)) if( len(s0) == 1 ) { drift = mu - 0.5 * sigma^2 if( nsteps == 1 ) { s0 * exp(drift * dt + sigma * sqrt(dt) * rnorm(nsims)) } else { temp = matrix(exp(drift * dt + sigma * sqrt(dt) * rnorm(nsteps * nsims)), nc=nsims) for(i in 2:nsteps) temp[i,] = temp[i,] * temp[(i-1),] s0 * temp } } else { require(MASS) drift = mu - 0.5 * diag(sigma) n = len(mu) if( nsteps == 1 ) { s0 * exp(drift * dt + sqrt(dt) * t(mvrnorm(nsims, rep(0, n), sigma))) } else { temp = array(exp(as.vector(drift %*% t(dt)) + t(sqrt(dt) * mvrnorm(nsteps * nsims, rep(0, n), sigma))), c(n, nsteps, nsims)) for(i in 2:nsteps) temp[,i,] = temp[,i,] * temp[,(i-1),] s0 * temp } } }

Next let’s visualize some simulation asset paths:

############################################################################### # Load Systematic Investor Toolbox (SIT) # http://systematicinvestor.wordpress.com/systematic-investor-toolbox/ ############################################################################### setInternet2(TRUE) con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb')) source(con) close(con) #***************************************************************** # Plot some price paths #****************************************************************** S = c(100,105) X = 98 Time = 0.5 r = 0.05 sigma = c(0.11,0.16) rho = 0.63 N = 10000 # Single Asset for 10 years periods = 0:10 prices = asset.paths(S[1], r, sigma[1], N, periods = periods) # plot matplot(prices[,1:100], type='l', xlab='Years', ylab='Prices', main='Selected Price Paths') # Multiple Assets for 10 years periods = 0:10 cov.matrix = sigma%*%t(sigma) * matrix(c(1,rho,rho,1),2,2) prices = asset.paths(S, c(r,r), cov.matrix, N, periods = periods) # plot layout(1:2) matplot(prices[1,,1:100], type='l', xlab='Years', ylab='Prices', main='Selected Price Paths for Asset 1') matplot(prices[2,,1:100], type='l', xlab='Years', ylab='Prices', main='Selected Price Paths for Asset 2')

Next, let’s look at examples of Monte Carlo option pricing using asset.paths() function in random.r at github.

#***************************************************************** # Price European Call Option #****************************************************************** load.packages('fOptions') # Black–Scholes GBSOption(TypeFlag = "c", S = S[1], X = X, Time = Time, r = r, b = r, sigma = sigma[1]) # Monte Carlo simulation N = 1000000 prices = asset.paths(S[1], r, sigma[1], N, periods = Time) future.payoff = pmax(0, prices - X) discounted.payoff = future.payoff * exp(-r * Time) # option price mean(discounted.payoff) #***************************************************************** # Price Asian Call Option #****************************************************************** load.packages('fExoticOptions') Time = 1/12 # Approximation TurnbullWakemanAsianApproxOption(TypeFlag = "c", S = S[1], SA = S[1], X = X, Time = Time, time = Time, tau = 0 , r = r, b = r, sigma = sigma[1]) # Monte Carlo simulation N = 100000 periods = seq(0,Time,1/360) n = len(periods) prices = asset.paths(S[1], r, sigma[1], N, periods = periods) future.payoff = pmax(0, colSums(prices)/n - X) discounted.payoff = future.payoff * exp(-r * Time) # option price mean(discounted.payoff) #***************************************************************** # Price Basket Option #****************************************************************** Time = 0.5 # Approximation TwoRiskyAssetsOption(TypeFlag = "cmax", S1 = S[1], S2 = S[2], X = X, Time = Time, r = r, b1 = r, b2 = r, sigma1 = sigma[1], sigma2 = sigma[2], rho = rho) # Monte Carlo simulation N = 100000 cov.matrix = sigma%*%t(sigma) * matrix(c(1,rho,rho,1),2,2) prices = asset.paths(S, c(r,r), sigma = cov.matrix, N, periods = Time) future.payoff = pmax(0, apply(prices,2,max) - X) discounted.payoff = future.payoff * exp(-r * Time) # option price mean(discounted.payoff) #***************************************************************** # Price Asian Basket Option #****************************************************************** Time = 1/12 # Monte Carlo simulation N = 10000 periods = seq(0,Time,1/360) n = len(periods) prices = asset.paths(S, c(r,r), sigma = cov.matrix, N, periods = periods) future.payoff = pmax(0, colSums(apply(prices,c(2,3),max))/n - X) discounted.payoff = future.payoff * exp(-r * Time) # option price mean(discounted.payoff)

Please note that Monte Carlo option pricing requireies many simulations to converge to the option price. It takes longer as we increase number of simulations or number of time periods or number of assets. On the positive side, it provides a viable alternative to simulating difficult problems that might not be solved analytically.

In the next post I will look at the cost of different rebalancing methods.

To view the complete source code for this example, please have a look at the asset.paths.test() function in random.r at github.

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