Gen Black Scholes Process


Gen Black Scholes Process is a direct subtype of Stoch Process 1D
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with functions Gen Black Scholes Process Functions, keys Gen Black Scholes Process keys and example object GenBSProc

TYPE INCLUSION RELATIONSHIPS

Stoch Process 1D

Gen Black Scholes Process

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AVAILABLE FUNCTIONS

Create

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AVAILABLE CREATE FUNCTION KEYS

Discretization

Div Yield Curve

For Int Rate Curve

Force Discretization

Int Rate Curve

Vol Curve

X0

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TYPICAL OBJECTS OF TYPE Gen Black Scholes Process

GenBSProc

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This type represents a Geometric Brownian stochastic process of a quoted stock share price or a forward price or an fx rate, with a deterministic time-dependent drift parameter and a state- and time- dependent volatility parameter.
Denoting with x the quoted price, its diffusion equation is:
dx = [r(t) - y(t)]xdt + σ(t,x)xdw
The drift parameter is expressed as the difference r(t) - d(t) with r(t) and y(t) representing the - assumed deterministic - instantaneous continuously compounded risk free interest rate and an "asset yield" (see below) at time t respectively.
The asset yield y(t) at time t represents the instantaneous continuously compounded yield earned by holding the underlying asset.
The following 3 cases are typical:
1) Underlying is a stock: The yield y(t) represents the stock's dividend yield at time t
2) Underlying is a forward contract: The yield y(t) equals the risk free interest rate r(t) so that the net drift term reduces to zero.
3) Underlying is a foreign currency: The yield y(t) equals the foreign risk free interest rate.

The volatility parameter σ(t,x) is assumed to be a known function of both the time t and the state x.
The above equation is significantly simplified when expressed in terms of ln(x), the logarithm of x:
d(ln(x)) = [r(t) - y(t) - ½(σ(t,x))²]dt + σ(t,x)dw

The parameters r(t), y(t) and σ(t,x) are not specified directly.
The r(t) and y(t) are implied respectively by objects of type Yield Curve
The σ(t,x) is implied respectively by an object of type Vol Curve

When this process is used in simulations, for any given finite time interval Δt = t₂ - t₁, the corresponding change of the quoted price Δx = x₂ - x₁ is calculated by first calculating the Δ(ln(x)) = ln(x₂) - ln(x₁), which can be computed exactly from the formula d(ln(x)) = [r(t) - y(t) - ½(σ(t,x))²]dt + σ(t,x)dw
After Δ(ln(x)) has been calculated, x₂ can be determined by the formula:
x₂ = x₁eΔ(ln(x))