Option valuation for arbitrary distribution using monte carlo simulation

Hi:
    I think this could work.  

    Now, I always find it useful to identify the hard parts up front.  Let's see if I can sketch this in a couple minutes.

1)  You need to estimate variance (vol^2) from the historical prices.  Now 50 days closing prices is a little low and when close to expiration you'd want some intraday data too.  But the model is a little crude anyhow, so perhaps it doesn't matter.  So assume the stock price St follows a log-normal process:  dlog(St) = (r - vol^2/2) dt + vol dBt and take first differences of log(St) for the returns process Rt ~ log(St/S(t-1)).  Rt is normally distributed so vol^2 ~ Sum{(Rt - R(t-1))^2}/48 - (1/49 Sum{Rt})^2}.  
2)  You need the risk-free rate, so set r=treasusry-rate for the same maturity as the stock option time-to-expiration -- this is in calendar days.
3)  Let t = today and T=expiration. 
    The sample paths of the stock price can be assumed to follow: Ss -> exp{r s + vol N(0, 1) sqrt(s)}  // N(0,1) =normal distribution.  
                 Let c_(ns) = exp{-r (T-t)} max{ exp{r (T-t) + vol ns sqrt(T-t)} - X, 0} -- where ns is a random sample from N(0,1).
                 european call(X,S,r,vol,T) = { c_(ns1) + c_(ns2) + ... + c_(ns(num_samples)) } / num_samples.

My math could be off a little (e.g. maybe a ito factor), but at least for me, it really helps to get some brief clean calculations up front, and then code.;-)

Option valuation for arbitrary distribution using monte carlo simulation

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