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

cost modeling of a firm

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Here we have assumed that both, HUL and ITC, are using the same technology¹. The price of the goods-basket has been normalized to unity for HUL in the year 1996. The average total cost curve for the industry has been fitted using the following specification, which allows for a learning model of capacity expansion and utilization;

Equation Set 2.1 starts here

ATC(Q_t)=%beta %alpha {Q^e_t}^{2} - 2 %alpha Q^e_t Q_t+ %alpha Q_t^2

; where %alpha geslant 0, %beta > 1 

%DELTA lnQ^e_t = %lambda_0 +%lambda_1 %DELTA lnQ^e_{t-1}+%mu %DELTA lnQ_{t-1}+ ε_t ; where 0< %lambda_1 < 1, ε_t sim N(0,%sigma^2)

Equation set 2.1 ends here

Where;

ATC(Q_t) = {Total cost at time 't'} over { Q_t}

Q_t = quantity of good basket produced²

MLE obtained from constrained³ LLF maximization were used to fit the data on HUL and ITC. From the estimators we obtained the maximum capacity corresponding to the minimum average cost for a firm in the industry. From the above we constructed a time-series for capacity utilizations for the two companies, HUL and ITC.

1Equation set: 2.1 has been estimated for data on HUL and ITC using identical set of parameters.

2Calculated as variable cost deflated by prevalent CPI.

3As constrained by parameters.

some high school math

today one of my friends prodded me to do some high school math. after a gap of many years i was again operating in my domain, my own turf; a long needed respite from the everyday grillings of a pgdm course. 

the problem as my friend framed it : "prove that a conic section of the form ax^2+2hxy+by^2=1 is a circle, if any line drawn from a point P(x0,y0) intersects the conic section at two distinct points; A, B; such that PA.PB is constant"

ahem.. my proof 

Consider a change of axis such that the point P becomes origin. The conic section w.r.t. the new co-ord sys can be represented as

ax^2+2hxy+by^2+2gx+2fy+c=0; the line can be represented as y=mx, where

PA.PB=(1+m^2)|x1*x2|; where x1 and x2 are roots of the quad:

(a+2hm+bm^2)x^2+2(g+fm)x+c=0

PA.PB=|c|(1+m^2)/|a+2hm+bm^2|=k... a constant >0

Now the expression given above should trivially hold good for all real-m

|c|(1+m^2)=k|a+2hm+bm^2|

So,

a=b=+/-|c|/k , h=0

So the conic is a circle 

Ya, i agree a small one but nevertheless it was atleast worth a start, as far as my first blog is concerned