expressing the expected (T-t) day stock portfolio as a function of the NIFTY volatility, hence the possibilty of Vega hedge using NIFTY options

one of the formulae i developed during my leisurely days; check it out tried out on one of my porfolios looks like a good estimate

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Assumptions

• Assets, i.e. shortlisted-stocks for investment in the previous section, are correlated and follow the CAPM specification

• Return on NIFTY is defined as the market return

• NIFTY follows the Black-Scholes-Merton

   process

• First order approximation has been used for the portfolio return-transfer function¹

Some stock-portfolio was prepared (my trade secret, used some wicked fundamental analysis to shortlist a few shares). Opportunity for investment is being verified through the use of probability theory. The following is the model based on the above assumptions.

%DELTA Y_{t}=%ALPHA+ diag(%DELTA m_{t}) %BETA^{T}+%EPSILON_{t} where %EPSILON_{t} sim N(0,%SIGMA)  2

dm(t)= r_f m(t)dt+%sigma m(t)dz, where %D

ELTA z(t)= %eta(t) sqrt{%DELTA t}, %eta(t) sim N(0,1)

The above specifications result in the following distribution for the portfolio value.

ln(e^{r_f (T-t)}v_T divides t)sim N(ln(v_t)+(T-t) W^T (A+%BETA (r_f - {%sigma^2} over {2})), (T-t)({(W^T %BETA)}^2 %sigma^2+W^T %SIGMA W))

Thus, risk neutral valuation provides us with the following formula;

v_t (risk neutral)= E(v_T divides t) =v_t e^{(T-t) (W^T (A+ %BETA (r_f - {%sigma^2} over {2})) + ({(W^T %BETA)}^2 %sigma^2+W^T %SIGMA W)over 2-r_f)}

Note that the above notations hold for the following parameters

Y = vector of prices for individual stocks in the portfolio

m = NIFTY price levels

v = value of portfolio

W = contribution vector (weightage) for individual shares in the portfolio

r_f = risk-free rate of return (6.5%)

T = planning horizon (coincides with the expiry date 

of any traded option on NIFTY)

t = time consumed in the planning horizon

We started tracking the market from 3^rd Nov. 2008 to 5^th Dec 2008. The above formula was used to gauge the intrinsic value of the portfolio in the stock-market. Everyday the parameter %sigma  (implied volatility) was estimated from the derivatives market and the risk-neutral valuation of portfolio was carried out for a planning horizon extending upto 25^th Dec 2008. The following is the chart of expected loss of risk-neutral opportunity.

1  {sum w_{i}p_{i,t}-sum w_{i}p_{i,t-1}} over {sum w_{i}p_{i,t-1}} approx {ln(sum w_{i}p_{i,t} / sum w_{i}p_{i,t-1})} approx sum w_{i}ln(p_{i,t}/p_{i,t-1}) where p=price of one stock and w=portfolio weight

2 Maximum likelihood estimation for %ALPHA, %BETA, %SIGMA