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Finance[BlackScholesProcess] - create new Black-Scholes process

Calling Sequence

BlackScholesProcess(S0, sigma, r, d)

BlackScholesProcess(S0, sigma, r, d, t, S)

Parameters

S0

-

non-negative constant; initial value

r

-

non-negative constant, procedure or yield term structure; risk-free rate

sigma

-

non-negative constant, procedure or a local volatility structure; volatility

d

-

non-negative constant, procedure or yield term structure; dividend yield

t

-

name; time variable

S

-

name; state variable

Description

• 

The BlackScholesProcess command creates a new Black-Scholes process. This is a process St governed by the stochastic differential equation (SDE)

dStSt=rtdtσSt,t22dt+σSt,tdWt

where

– 

r=rt is the risk-free rate,

– 

σ=σS,t is the local volatility,

– 

d=dt is the dividend yield,

and

– 

Wt is the standard Wiener process.

• 

The parameter S0 defines the initial value of the underlying stochastic process. It must be a real constant.

• 

The parameter r is the risk-free rate. The parameter d is the continuous dividend yield. Time-dependent risk-free rate and dividend yield can be given either as an algebraic expression, a Maple procedure, or a yield term structure. If r or d is given as an algebraic expression, then the fifth parameter t must be passed to specify which variable in r should be used as the time variable. Maple procedure defining a time-dependent drift must accept one parameter (the time) and return the corresponding value for the drift.

• 

The sigma parameter is the local volatility. It can be constant or it can be given as a function of time and the value of the state variable. In the second case it can be specified as an algebraic expression, a Maple procedure or a local volatility term structure. If sigma is specified in the algebraic form, the parameters t and S must be given to specify which variable in sigma represents the time variable and which variable represents the value of the underlying.

Examples

withFinance:

First define a Black-Scholes process with constant parameters.

S0:=100.0

S0:=100.0

(1)

σ:=0.3

σ:=0.3

(2)

r:=0.05

r:=0.05

(3)

d:=0.01

d:=0.01

(4)

S:=BlackScholesProcessS0,σ,r,d:

You can compute the expected payoff of a European call option with strike 100 maturing in 1 year.

T:=1.0

T:=1.0

(5)

K:=100

K:=100

(6)

DiscountFactorr,TExpectedValuemaxSTK,0,timesteps=100,replications=105,output=value

13.64080475

(7)

You can then compare the result to the theoretical price.

BlackScholesPriceS0,K,T,σ,r,d,'call'

13.61641736

(8)

This is incorporating local volatility term structure.

σ:=LocalVolatilitySurface0.03+0.0001S,t,S:

X:=BlackScholesProcessS0,σ,r,d:

Again, you can compute the expected payoff of a European call option with strike 100 maturing in 1 year.

V:=DiscountFactorr,TExpectedValuemaxXTK,0,timesteps=100,replications=103,output=value

V:=4.142053200

(9)

Then you can compute the implied volatility.

ImpliedVolatilityV,S0,K,T,r,d

0.03719843687

(10)

In this example we implied volatility surface obtained using a piecewise interpolation of known prices.

r:=0.05

r:=0.05

(11)

d:=0.0

d:=0.

(12)

S0:=87

S0:=87

(13)

K:=85

K:=85

(14)

C:=BlackScholesPriceS0,k,t,0.03+0.0001k,r,d:

U:=seq0.1i,i=1..20:

S:=seq80+i,i=0..20:

V:=Finance:-LocalVolatilityC,S,U,r,d,t,k:

Σ:=LocalVolatilitySurfaceU,S,V:

X:=BlackScholesProcessS0,Σ,r,d:

T:=0.35

T:=0.35

(15)

P:=DiscountFactorr,TExpectedValuemaxXTK,0,timesteps=100,replications=103,output=value

P:=3.501775067

(16)

ImpliedVolatilityP,S0,K,T,r,d

0.03818795821

(17)

See Also

Finance[BlackScholesPrice], Finance[BrownianMotion], Finance[Diffusion], Finance[Drift], Finance[ExpectedValue], Finance[ForwardCurve], Finance[GeometricBrownianMotion], Finance[ImpliedVolatility], Finance[ItoProcess], Finance[LocalVolatility], Finance[LocalVolatilitySurface], Finance[MertonJumpDiffusion], Finance[PathPlot], Finance[SamplePath], Finance[SampleValues], Finance[StochasticProcesses]

References

  

Glasserman, P., Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.

  

Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.


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