---

FinMath.h File Reference

---

Detailed Description

Standalone methods to compute functions or solve eqations useful in basic financial mathematics.

Author:

Michael J. Meyer

Definition in file FinMath.h.

#include "TypedefsMacros.h"

Go to the source code of this file.

Functions
Real	multiNormalDensity (int dim, Real *z)
Real	N (Real x)
Real	d_plus (Real Q, Real k, Real Sigma)
Real	d_minus (Real Q, Real k, Real Sigma)
Real	blackScholesFunction (Real Q, Real k, Real Sigma)
Real	dBSF (Real Q, Real k, Real Sigma)
Real	bsDiscountedCallPrice (Real S, Real K, Real tau, Real sigma, Real B)
Real	bsDiscountedPutPrice (Real S, Real K, Real tau, Real sigma, Real B)
Real	blackImpliedAggregateCallVolatility (Real Q, Real k, Real y)

---

Function Documentation

Real multiNormalDensity (  int    dim, Real *    z )

The density of the standard multinormal distribution N(0,I).

Real N (  Real    x )

The cumulative distribution function N(x)=Prob(X<=x) of a standard normal variable X derived from the error function Parameters: x  Any real number. Referenced by Pricing::betaCoefficient(), EuclideanRegion::boundaryIntersection(), RandomObject< RangeType >::correlation(), Pricing::correlationWithControlVariate(), RandomObject< RangeType >::covariance(), RandomObject< RangeType >::covarianceMatrix(), RandomObject< RangeType >::expectation(), RandomObject< RangeType >::meanAndVariance(), Pricing::monteCarloForwardPrice(), Vector< Real >::operator/=(), and RandomObject< RangeType >::variance().

Real d_plus (  Real    Q, Real    k, Real    Sigma )

The quantity in Margrabe's formula for the option to exchange assets S_1, S_2 (receive S_1 for kS_2). In this context: where and is the aggregate volatility of log(Q) from current time t to the horizon T. If Q has constant annual volatility this becomes In the case of the European call on S_1 with strike k the asset S_2 is the zero coupon bond maturing at call expiration. Parameters: Q  The quotient S_1(t)/S_2(t) k  Ratio of exchange between assets (receive S_1 for kS_2). Sigma  aggregate volatility of Q on [t,T].

Real d_minus (  Real    Q, Real    k, Real    Sigma )

The quantity in Margrabe's formula for the option to exchange assets S_1, S_2 (receive S_1 for kS_2). In this context: where and is the aggregate volatility of log(Q) from current time t to the horizon T. If Q has constant annual volatility this becomes In the case of the European call on S_1 with strike k the asset S_2 is the zero coupon bond maturing at call expiration. Parameters: Q  The quotient S_1(t)/S_2(t) k  Ratio of exchange between assets (receive S_1 for kS_2). Sigma  aggregate volatility of Q on [t,T].

Real blackScholesFunction (  Real    Q, Real    k, Real    Sigma )

Computes the function with as in d_plus, d_minus. Useful in application of the Black-Scholes formula or Margrabe's formula to price calls or options to exchange assets. For the option to exchange assets S_1,S_2 we have Q=S_1(t)/S_2(t) and In the special case of a call on S we have S_1=S, S_2=B is the zero coupon bond maturing at call expiry and the forward price of the call can be written as where Q=S/B is the forward price of S at expiry. Parameters: Q  The quotient S_1(t)/S_2(t) k  Ratio of exchange between assets (receive S_1 for kS_2). Sigma  aggregate volatility of Q on [t,T].

Real dBSF (  Real    Q, Real    k, Real    Sigma )

Derivative of the Black-Scholes function Q*N(d_+)-k*N(d_-) with respect to Sigma. Parameters: Q  The quotient S_1(t)/S_2(t) k  Ratio of exchange between assets (receive S_1 for kS_2). Sigma  aggregate volatility of Q on [t,T].

Real bsDiscountedCallPrice (  Real    S, Real    K, Real    tau, Real    sigma, Real    B )

Discounted Black-Scholes call price (note that current time may not be zero). Both the current time t and the time T to expiry (from time zero) are needed since the formula will be applied at time other than time zero and S Parameters: S  the forward asset price at expiry T. K  strike price tau  time to expiration sigma  annual asset volatility B  risk free bond B_T at expiry

Real bsDiscountedPutPrice (  Real    S, Real    K, Real    tau, Real    sigma, Real    B )

Discounted Black-Scholes put price (note that current time may not be zero). Here we need the interest rate, the current time t and the time T of expiration explicitly. The formula will be applied at times other than time zero. Parameters: S  the forward asset price at expiry T. K  strike price tau  time to expiry sigma  annual asset volatility B  risk free bond B_T at expiry

Real blackImpliedAggregateCallVolatility (  Real    Q, Real    k, Real    y )

Given solves the equation for using continued bisection. A solution exists if and only if . Used to compute implied volatilities from call forward prices y. Note that the solution is not the implied annual volatility , indeed these are related as , where is time to expiry.

---