fExpressCertificates (version 1.3)

ExpressCertificate.Classic: Analytical and numerical pricing of Classic Express Certificates

Description

Pricing of Classic Express Certificates using the truncated multivariate normal distribution (early stop probabilities) and numerical integration of the one-dimensional marginal return distribution at maturity

Usage

ExpressCertificate.Classic(S, X, T, K, g = function(S_T) {S_T}, 
  r, r_d, sigma, ratio = 1)

Arguments

S

the asset price, a numeric value

X

a vector of early exercise prices ("Bewertungsgrenzen"), , vector of length (n-1)

T

a vector of evaluation times measured in years ("Bewertungstage"), vector of length n

K

vector of fixed early cash rebates in case of early exercise, length (n-1)

g

a payoff function at maturity, by default g(S_T)=S_T

r

the annualized rate of interest, a numeric value; e.g. 0.25 means 25% pa.

r_d

the annualized dividend yield, a numeric value; e.g. 0.25 means 25% pa.

sigma

the annualized volatility of the underlying security, a numeric value; e.g. 0.3 means 30% volatility pa.

ratio

ratio, number of underlyings one certificate refers to, a numeric value; e.g. 0.25 means 4 certificates refer to 1 share of the underlying asset

Value

a vector of length n with certificate prices

Details

The principal feature inherent to all express certificates is the callable feature with pretermined valuation dates (\(t_1< \ldots < t_n\)) prior to final maturity \(t_n\). Express certificates are typically called, if the underlying price on the valuation date is above a strike price (call level): \(S(t_i) > X(t_i)\).

The payoff of an express classic certificate at maturity is the underlying performance itself. So the payoff function at maturity takes the simple form of \(g(S(t_n)) = S(t_n)\).

We compute early redemption probabilities via the truncated multivariate normal distribution and integrate the one-dimensional marginal distribution for the expected payoff \(E[g(S(t_n))] = E[S(t_n)]\).

References

Wilhelm, S. (2009). The Pricing of Derivatives when Underlying Paths Are Truncated: The Case of Express Certificates in Germany. Available at SSRN: http://ssrn.com/abstract=1409322

See Also

MonteCarlo.ExpressCertificate.Classic and MonteCarlo.ExpressCertificate for Monte Carlo evaluation with similar payoff functions

Examples

# NOT RUN {
ExpressCertificate.Classic(S=100, X=c(100), 
  T=c(1, 2), g = function(S) { S }, 
  K=142.5, r=0.01, r_d=0, sigma=0.3, ratio = 1)

ExpressCertificate.Classic(S=100, X=c(100), 
  T=c(1, 2), g = function(S) { max(S, 151) }, 
  K=142.5, r=0.01, r_d=0, sigma=0.3, ratio = 1)    
# }