gemSecurityPricing: Compute Security Market Equilibria for Some Simple Cases

Description

Compute the equilibrium of a security (i.e. asset) market by the function sdm2 and by computing marginal utility of securities (see Sharpe, 2008).

Usage

gemSecurityPricing(
  S = diag(2),
  UP = diag(nrow(S)),
  uf = NULL,
  muf = NULL,
  ratio_adjust_coef = 0.05,
  numeraire = 1,
  ...
)

Value

A general equilibrium containing a value marginal utility matrix (VMU).

Arguments

S: an n-by-m supply matrix of securities.
UP: a unit (security) payoff k-by-n matrix.
uf: a utility function or a utility function list.
muf: a marginal utility function or a marginal utility function list.
ratio_adjust_coef: a scalar indicating the adjustment velocity of demand structure.
numeraire: the index of the numeraire commodity.
...: arguments to be passed to the function sdm2.

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.

Wang Jiang (2006, ISBN: 9787300073477) Financial Economics. Beijing: China Renmin University Press. (In Chinese)

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

# \donttest{
gemSecurityPricing(muf = function(x) 1 / x)

gemSecurityPricing(
  S = cbind(c(1, 0), c(0, 2)),
  muf = function(x) 1 / x
)

gemSecurityPricing(
  UP = cbind(c(1, 0), c(0, 2)),
  muf = function(x) 1 / x
)

#### an example of Danthine and Donaldson (2005, section 8.3).
ge <- gemSecurityPricing(
  S = matrix(c(
    10, 5,
    1, 4,
    2, 6
  ), 3, 2, TRUE),
  uf = function(x) 0.5 * x[1] + 0.9 * (1 / 3 * log(x[2]) + 2 / 3 * log(x[3]))
)

ge$p

#### an example of Sharpe (2008, chapter 2, case 1)
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
UP <- cbind(secy1, secy2, secy3, secy4)

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

geSharpe1 <- gemSecurityPricing(
  S = matrix(c(
    49, 49,
    30, 30,
    10, 0,
    0, 10
  ), 4, 2, TRUE),
  UP = UP,
  uf = list(
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 1.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 2.5)
  )
)
geSharpe1$p
geSharpe1$p[3:4] + 3 * geSharpe1$p[2]

## an example of Sharpe (2008, chapter 3, case 2)
geSharpe2 <- gemSecurityPricing(
  S = matrix(c(
    49, 49, 98, 98,
    30, 30, 60, 60,
    10, 0, 20, 0,
    0, 10, 0, 20
  ), 4, 4, TRUE),
  UP = UP,
  uf = list(
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 1.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 2.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 1.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 2.5)
  )
)

geSharpe2$p
geSharpe2$p[3:4] + 3 * geSharpe2$p[2]
geSharpe2$D

## an example of Sharpe (2008, chapter 3, case 3)
geSharpe3 <- gemSecurityPricing(UP,
  uf = function(x) (x - x^2 / 400) %*% wt,
  S = matrix(c(
    49, 98,
    30, 60,
    5, 10,
    5, 10
  ), 4, 2, TRUE)
)
geSharpe3$p
geSharpe3$p[3:4] + 3 * geSharpe3$p[2]

# the same as above
geSharpe3b <- gemSecurityPricing(
  S = matrix(c(
    49, 98,
    30, 60,
    5, 10,
    5, 10
  ), 4, 2, TRUE),
  UP = UP,
  muf = function(x) (1 - x / 200) * wt
)

geSharpe3b$p
geSharpe3b$p[3:4] + 3 * geSharpe3b$p[2]

## an example of Sharpe (2008, chapter 3, case 4)
geSharpe4 <- gemSecurityPricing(
  S = matrix(c(
    49, 98,
    30, 60,
    5, 10,
    5, 10
  ), 4, 2, TRUE),
  UP,
  muf = function(x) abs((x - 20)^(-1)) * wt,
  maxIteration = 100,
  numberOfPeriods = 300,
  ts = TRUE
)

geSharpe4$p
geSharpe4$p[3:4] + 3 * geSharpe4$p[2]

## an example of Sharpe (2008, chapter 6, case 14)
prob1 <- c(0.15, 0.26, 0.31, 0.28)
wt1 <- prop.table(c(1, 0.96 * prob1))
prob2 <- c(0.08, 0.23, 0.28, 0.41)
wt2 <- prop.table(c(1, 0.96 * prob2))

uf1 <- function(x) CES(alpha = 1, beta = wt1, x = x, es = 1 / 1.5)
uf2 <- function(x) CES(alpha = 1, beta = wt2, x = x, es = 1 / 2.5)
geSharpe14 <- gemSecurityPricing(
  S = matrix(c(
    49, 49,
    30, 30,
    10, 0,
    0, 10
  ), 4, 2, TRUE),
  UP = UP,
  uf = list(uf1,uf2)
)

geSharpe14$D
geSharpe14$p
geSharpe14$p[3:4] + 3 * geSharpe14$p[2]
mu <- marginal_utility(geSharpe14$Payoff, diag(5),uf=list(uf1,uf2))
mu[,1]/mu[1,1]
mu[,2]/mu[1,2]

#### an example of Wang (2006, example 10.1, P146)
geWang <- gemSecurityPricing(
  S = matrix(c(
    1, 0,
    0, 2,
    0, 1
  ), 3, 2, TRUE),
  muf = list(
    function(x) 1 / x * c(0.5, 0.25, 0.25),
    function(x) 1 / sqrt(x) * c(0.5, 0.25, 0.25)
  )
)

geWang$p # c(1, (1 + sqrt(17)) / 16)

# the same as above
geWang.b <- gemSecurityPricing(
  S = matrix(c(
    1, 0,
    0, 2,
    0, 1
  ), 3, 2, TRUE),
  uf = list(
    function(x) log(x) %*% c(0.5, 0.25, 0.25),
    function(x) 2 * sqrt(x) %*% c(0.5, 0.25, 0.25)
  )
)

geWang.b$p

#### an example of Xu (2018, section 10.4, P151)
wt <- c(1, 0.5, 0.5)
ge <- gemSecurityPricing(
  S = matrix(c(
    1, 0,
    0, 0.5,
    0, 2
  ), 3, 2, TRUE),
  uf = list(
    function(x) CRRA(x, gamma = 1, prob = wt)$u,
    function(x) CRRA(x, gamma = 0.5, prob = wt)$u
  )
)

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

#### an example of incomplete market
ge <- gemSecurityPricing(
  UP = cbind(c(1, 1), c(2, 1)),
  uf = list(
    function(x) sum(log(x)) / 2,
    function(x) sum(sqrt(x))
  ),
  ratio_adjust_coef = 0.1,
  priceAdjustmentVelocity = 0.05,
  policy = makePolicyMeanValue(span = 100),
  maxIteration = 1,
  numberOfPeriods = 2000,
)

ge$p

## the same as above
ge.b <- gemSecurityPricing(
  UP = cbind(c(1, 1), c(2, 1)),
  muf = list(
    function(x) 1 / x * c(0.5, 0.5),
    function(x) 1 / sqrt(x) * c(0.5, 0.5)
  ),
  ratio_adjust_coef = 0.1,
  priceAdjustmentVelocity = 0.05,
  policy = makePolicyMeanValue(span = 100),
  maxIteration = 1,
  numberOfPeriods = 2000,
  ts = TRUE
)

ge.b$p
matplot(ge.b$ts.p, type = "l")

#### an example with outside position.
asset1 <- c(1, 0, 0)
asset2 <- c(0, 1, 1)

# unit (asset) payoff matrix
UP <- cbind(asset1, asset2)
wt <- c(0.5, 0.25, 0.25) # weights

uf1 <- function(x) prod((x + c(0, 0, 2))^wt)
uf2 <- function(x) prod(x^wt)

ge <- gemSecurityPricing(
  S = matrix(c(
    1, 1,
    0, 2
  ), 2, 2, TRUE),
  UP = UP,
  uf = list(uf1, uf2),
  numeraire = 1
)

ge$p
ge$z
uf1(ge$Payoff[,1])
uf2(ge$Payoff[,2])
# }