Decompose option PnL into its greeks

Summary

– Decomposition option PnLs into its greeks by using Taylor expansion
– Major greeks attribute to daily PnL are delta, gamma, theta and vega

Taylor expansion

An arbitrary function \( f \left( x \right) \) can be written by below formula:

$$
f \left( x \right) = f \left( a \right) + f^{\left( 1 \right)} \left( a \right) \left( x – a \right) + \frac{f^{\left( 2 \right)} \left( a \right)}{2} \left( x – a \right)^2 + {\rm higher}.
$$

The option present value \( V_t \left( S_t, T, \sigma_t, r_t \right) \) is expressed with its derivatives(option greeks) in a similar manner:

$$
V_t \left( S_t, \tau_t, \sigma_t, r_t \right) = V_{t-1} \left( S_{t-1}, \tau_{t-1}, \sigma_{t-1}, r_{t-1} \right) + \\
\frac{{\partial}V}{{\partial}S_{t-1}}{\rm d}S + \frac{{\partial}V}{{\partial}\tau_{t-1}}{\rm d}\tau + \frac{{\partial}V}{{\partial}\sigma_{t-1}}{\rm d}\sigma + \frac{1}{2} \frac{{\partial}^2V}{{\partial}S_{t-1}^2}{\rm d}S^2 + {\rm residuals},
$$

where \( S_t \) is underlying, \( \tau \) is time to expiry, \( \sigma \) is volatility and \( r \) is interest rate respectively. Usually the effect of the interest rate sensitivity(rho, PV01 sensitivity for swaptions) is limited and included to the “residual” term as well as other higher derivatives.
Daily PnL \( R_t = V_t – V_{t-1} \) is hence written by below formula:

$$
R_t \approx \Delta_{t-1}{\rm d}S + \theta_{t-1}{\rm d}\tau + \nu_{t-1}{\rm d}\sigma + \frac{1}{2} \Gamma{\rm d}S^2.
$$

If you consider stochastic volatility model such as SABR, above equation also holds sensitivity terms to that parameters(volvol, beta, rho). Furthermore, other related greeks including forward-delta, forward-gamma, normal-vega can be applied above interchangeably.

Reference

Taylor series – Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Taylor_series

Riding the Swaption Curve by Johan G. Duyvesteyn, Gerben J. De Zwart :: SSRN
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2008841