gemSecurityPricingExample: Some Examples of Security Pricing

Description

These examples illustrate how to find the equilibrium of a security market by the function sdm2 and by computing marginal utility of securities (see Sharpe, 2008).

Usage

gemSecurityPricingExample(...)

Arguments

...

arguments to be passed to the function sdm2.

Value

A general equilibrium.

References

Danthine, J. P., Donaldson, J. (2005, ISBN: 9780123693808) Intermediate Financial Theory. Elsevier Academic Press.

Sharpe, William F. (2008, ISBN: 9780691138503) Investors and Markets: Portfolio Choices, Asset Prices, and Investment Advice. Princeton University Press.

Xu Gao (2018, ISBN: 9787300258232) Twenty-five Lectures on Financial Economics. Beijing: China Renmin University Press. (In Chinese)

https://web.stanford.edu/~wfsharpe/apsim/index.html

Examples

Run this code

# NOT RUN {
#### an example of Danthine and Donaldson (2005, section 8.3).
uf <- function(x) 0.5 * x[1] + 0.9 * (1 / 3 * log(x[2]) + 2 / 3 * log(x[3]))

ge <- sdm2(
  A = function(state) {
    VMU <- marginal_utility(state$last.A %*% dg(state$last.z), diag(3), uf, state$p)
    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")

    A <- state$last.A * Ratio
    prop.table(A, 2)
  },
  B = matrix(0, 3, 2),
  S0Exg = matrix(c(
    10, 5,
    1, 4,
    2, 6
  ), 3, 2, TRUE),
  names.commodity = c("secy1", "secy2", "secy3"),
  names.agent = c("agt1", "agt2"),
  numeraire = "secy1",
  ts = TRUE
)

ge$p

#### an example of Sharpe (2008, chapter 2)
secy1 <- c(1, 0, 0, 0, 0)
secy2 <- c(0, 1, 1, 1, 1)
secy3 <- c(0, 5, 3, 8, 4) - 3 * secy2
secy4 <- c(0, 3, 5, 4, 8) - 3 * secy2
# unit security payoff matrix
USP <- cbind(secy1, secy2, secy3, secy4)

prob <- c(0.15, 0.25, 0.25, 0.35)
wt <- prop.table(c(1, 0.96 * prob)) # weights

gamma.agt1 <- 1.5
gamma.agt2 <- 2.5

ge <- sdm2(
  A = function(state) {
    Payoff <- USP %*% (state$last.A %*% dg(state$last.z))

    VMU <- marginal_utility(Payoff, USP, list(
      function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / gamma.agt1),
      function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / gamma.agt2)
    ), price = state$p)
    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")

    A <- state$last.A * ratio_adjust(Ratio, coef = 0.05, method = "linear")

    A <- prop.table(A, 2)
  },
  B = matrix(0, 4, 2),
  S0Exg = matrix(c(
    49, 49,
    30, 30,
    10, 0,
    0, 10
  ), 4, 2, TRUE),
  names.commodity = c("secy1", "secy2", "secy3", "secy4"),
  names.agent = c("agt1", "agt2"),
  numeraire = "secy1"
)

ge$p
ge$p[3:4] + 3 * ge$p[2]

#### an example of Xu (2018, section 10.4, P151)
secy1 <- c(1, 0, 0)
secy2 <- c(0, 1, 0)
secy3 <- c(0, 0, 1)
prob <- c(0.5, 0.5)
wt <- c(1, prob)
USP <- cbind(secy1, secy2, secy3)

gamma.agt1 <- 1
gamma.agt2 <- 0.5

ge <- sdm2(
  A = function(state) {
    Payoff <- USP %*% (state$last.A %*% dg(state$last.z))

    VMU <- marginal_utility(Payoff, USP, list(
      # Here CRRA(...)$u, CRRA(...)$CE and CES functions are interexchangeable.
      function(x) CRRA(x, gamma = gamma.agt1, p = wt)$u,
      function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / gamma.agt2)
    ), state$p)
    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")

    A <- state$last.A * Ratio
    prop.table(A, 2)
  },
  B = matrix(0, 3, 2),
  S0Exg = matrix(c(
    1, 0,
    0, 0.5,
    0, 2
  ), 3, 2, TRUE),
  names.commodity = c("secy1", "secy2", "secy3"),
  names.agent = c("agt1", "agt2"),
  numeraire = "secy1",
  maxIteration = 1,
  ts = TRUE
)

ge$p #c(1, (1 + sqrt(5)) / 4, (1 + sqrt(17)) / 16)

## the same as above.
dst.agt1 <- node_new("util",
  type = "CD", alpha = 1, beta = c(0.5, 0.25, 0.25),
  "secy1", "secy2", "secy3"
)

dst.agt2 <- node_new("util",
  type = "CES", alpha = 1, beta = c(2, 1, 1), sigma = 0.5,
  "secy1", "secy2", "secy3"
)

ge <- sdm2(
  A = list(dst.agt1, dst.agt2),
  B = matrix(0, 3, 2),
  S0Exg = matrix(c(
    1, 0,
    0, 0.5,
    0, 2
  ), 3, 2, TRUE),
  names.commodity = c("secy1", "secy2", "secy3"),
  names.agent = c("agt1", "agt2"),
  numeraire = "secy1",
  maxIteration = 1,
  ts = TRUE
)

ge$p
# }