April 2019 - Paper A (Q7 ii)

[Nandan]

Hi,
The report mentions the following formula for the calculation of the changed option price -
dV = delta*dS + 0.5*gamma*(dS)² + theta*dt

I even found a similar formula (with some more terms) in assignment X3 Q1 -
dV = delta*dS + 0.5*gamma*(dS)² + vega*d(sigma) + rho*dr + theta*dt --- (1)

Though, I understand the reason behind the formula and could solve the question, I could not find this mentioned/derived in the course notes. Can anyone please let me know if I am missing something / a short derivation of this (the formula marked (1)) formula if possible?

Thank you in advance!

 

[Joe Hook]

Hi Nandan,

You're right that the result is not mentioned explicitly in the course but it comes from Taylor's formula in 5 variables: share price, time, volatility, risk-free rate and dividend yield. We need only the first partial derivative of all variables apart from the share price as we assume like dt that dx^2=0, as is all the cross terms of dx*dy. So with pricing function f we get (representing partial derivative as pd:

df = pdf/pdS*dS + 0.5*pd^f/pdf^2*(dS)^2 + pdf/pdt*dt + pdf/pdsigma*dsigma+pdf/pdr*dr+pdf/pdq*dq

Replacing the partial derivatives with the greeks then gives the expression you've mentioned above.

So the change in the value of the derivative is the sum of the contributions from each of the changing parameters.

Joe