Description

Consider a European call option on a stock that has a current spot price of $100, a volatility of 30%, and pays two dividends of $4.00 on 1 September 2002 and 1 February 2003. The option has a strike price of $100 and matures on 1 March 2003. The zero curve is flat at 5% (on an actual/365 basis). What is the value of this option as at 1 April 2002?

Function Specification

=oBSdd(1, "1/3/02", "1/3/03", 100, 100, 0.3, D4:F4, D5:E5, 0)

It is assumed the cell references for the zero curve and dividend schedule contain the appropriate input values.

Solution

The continuous equivalent of the flat actual/365 zero curve is calculated as follows:

S*, the spot price less the present value of the future dividends, is calculated as follows:

S* = 100 - 7.73 = 92.27

Referring to the equations for d₁ and d₂ (see model definition), if S* = 92.27, X = 100, r = 0.0488, vol = 0.3 and T = 1 (365/365 days), d₁ = 0.0446 and d₂ -0.2554.

As iPC = 1 (call), N(d₁) is 0.5178 and N(d₂) is 0.3992 (see oCumNorm( ) function), the Black Scholes equation becomes:

Greeks

The following Greeks are computed using the formulas specified in oBSdd() Model Greeks:

Delta

As iPC = 1 (call), the delta equation simplifies to:

Gamma

As T = 1, the Gamma equation simplifies to:

The n(d₁) equation gives 0.398546, and since S* = 92.27 and vol = 0.3, Gamma is 0.014397

Theta

As iPC = T = 1, the equation for Theta becomes:

Vega

As T = 1, the equation for Vega becomes:

Rho

As iPC = T = 1, the equation for Rho becomes: