AssetPricing (version 0.1-3)

buildS: Build a piecewise linear price sensitivity function

Description

Builds a price sensitivity function which is piecewise linear in price, in an automated manner, with built-in checks for possible infelicities.

Usage

buildS(alpha, beta, kn, tmax)

Value

• A function of two variables x and t, which is a price sensitivity function. The argument x represents price and the argument t represents (residual) time. The value of the function is interpreted as the probability that a customer arriving at time t will purchase an item offered at price x.

Details

The price sensitivity function is assumed to be of the form $$S(x,t) = \alpha_k(t) + \beta_k(t)x$$ for $x_{k-1} \leq x \leq x_k$ where $x_1, x_2, \ldots, x_K$ are the (non-zero) knots of the function. It is assumed that $x_0 = 0$. The variable x represents price and the variable t represents residual time.

The function is defined over the rectangle $[0,x_K] \times [0,t_{\rm max}]$.

Checks are done to make sure that [object Object],[object Object],[object Object],[object Object]

References

Banerjee, P. K. and Turner, T. R. A flexible model for the pricing of perishable assets. Omega, vol. 40, number 5, 2012, pages 533 -- 540, doi: 10.1016/j.omega.2011.10.001.

See Also

xsolve()

Examples

lambda <- function(t) {
      tn <- 1:4
      A <- matrix(c(0,12,12,12,
              0,-16,16,64,
              20,30,30,0),nrow=4)
      B <- matrix(c(12,0,0,0,
               0,16,0,-16,
               0,-10,-10,0),nrow=4)
      s <- cut(t,breaks=c(0,tn),include.lowest=TRUE,labels=tn)
      s <- as.numeric(levels(s)[s])
      M <- matrix(A[s,] + B[s,]*t,ncol=ncol(A))
      M[!is.finite(M)] <- 0
      M
   }

   alpha <- vector("list",4)
   beta  <- vector("list",4)
   alpha[[1]] <- with(list(lambda=lambda),
	function(t) {
	A <- c(1,1,1)
        lll <- lambda(t)
        dnm <- apply(lll,1,sum)
        dnm[dnm==0] <- 1
        lll%*%A/dnm
   })
   beta[[1]] <- with(list(lambda=lambda),
	function(t) {
	B <- c(0,0,0)
        lll <- lambda(t)
        dnm <- apply(lll,1,sum)
        dnm[dnm==0] <- 1
        lll%*%B/dnm
   })
   alpha[[2]] <- with(list(lambda=lambda),
	function(t) {
	A <- c(1.495,1,1)
        lll <- lambda(t)
        dnm <- apply(lll,1,sum)
        dnm[dnm==0] <- 1
        lll%*%A/dnm
   })
   beta[[2]] <- with(list(lambda=lambda),
	function(t) {
	B <- c(-0.2475,0,0)
        lll <- lambda(t)
        dnm <- apply(lll,1,sum)
        dnm[dnm==0] <- 1
        lll%*%B/dnm
   })
   alpha[[3]] <- with(list(lambda=lambda),
	function(t) {
	A <- c(0.01,2.485,1)
        lll <- lambda(t)
        dnm <- apply(lll,1,sum)
        dnm[dnm==0] <- 1
        lll%*%A/dnm
   })
   beta[[3]] <- with(list(lambda=lambda),
	function(t) {
	B <- c(0,-0.2475,0)
        lll <- lambda(t)
        dnm <- apply(lll,1,sum)
        dnm[dnm==0] <- 1
        lll%*%B/dnm
   })
   alpha[[4]] <- with(list(lambda=lambda),
	function(t) {
	A <- c(0.01,0.01,3.475)
        lll <- lambda(t)
        dnm <- apply(lll,1,sum)
        dnm[dnm==0] <- 1
        lll%*%A/dnm
   })
   beta[[4]] <- with(list(lambda=lambda),
	function(t) {
	B <- c(0,0,-0.2475)
        lll <- lambda(t)
        dnm <- apply(lll,1,sum)
        dnm[dnm==0] <- 1
        lll%*%B/dnm
   })
   kn <- c(2,6,10,14)
   S  <- buildS(alpha,beta,kn,4)
   x  <- seq(0,14,length=41)
   t  <- seq(0,4,length=41)
   z  <- S(x,t)
persp(x,t,z,theta=150,phi=40,d=4,xlab="price",ylab="time",
         zlab="probability",ticktype="detailed")