The Black Scholes option pricing model is defined as follows:



OV = option value.
S = spot price of the underlying asset.
X = exercise price (strike).
r = riskfree interest rate, expressed with continuous compounding.
vol = volatility of the relative price change of the underlying asset.
T = time to maturity measured in years (actual/365 basis).
N(.) = cumulative normal distribution of (.).
iPC = 1 for call / 1 for put.





Solution

The continuous equivalent of the actual/365 riskfree interest rate is calculated as follows:
Referring to the equations for d_{1} and d_{2} (see ), if S = 50, X = 60, r= 0.0677, vol = 0.25, and T = 1 (365/365 days), d_{1} = 0.3334 and d_{2} 0.5837.


As iPC = 1 (call), N(d_{1}) is 0.3693 and N(d_{2}) is 0.2797 (see ), the Black Scholes equation becomes:





Greeks


Delta

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



Gamma

As T = 1, the Gamma equation simplifies to:




The n(d_{1}) equation gives 0.3773, and since S = 50 and vol = 0.25, Gamma is 0.0302

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:



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