Monday, July 21, 2008

Optimizing Vertical and Calendar Spreads


Verticals – Manage Delta
Horizontal – Manage Theta
Diagonals – Manage Vega

Vij = option at month i and strike j
Wij = weight of option at month i and strike j

V(S, τ, r, σ) = ∑ ∑ Wij * Vij(Kj, S, τ, r, σ)

Maximize the expected value of the portfolio under lognormal return expectation (If the maximum loss is bounded, lognormal assumption is conservative; if not disastrous)
Minimize the bound of the maximum loss
Maximize Speed

Maximum loss is bounded at L
L < maximum loss entertained
V < 0 (cash out flow –ve i.e. there is an initial cash inflow)
Delta = 0
Vega = 0
Gemma > 0
Volga > 0
Gamma Gamma > 0

Any comments

Best Regards, Suminda

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Open Thoughts on Finance

A long position’s expected returns are path dependent since it can be sold off. A writer’s position is not so in the absence of dynamic hedging.

Implied volatility is a function volatility at the money and vol + vvol + v2vol + … at other strikes.

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Friday, July 18, 2008

Numeric Adjustment to Black–Scholes for the smile.


This is basically a hunch that I have. Needs further investigation and verification.

Only even moments are risk measures of a returns distribution under risk neutral assumptions.

Volatility (vol), Volatility of Volatility (vvol), Volatility of Volatility of Volatility (v2vol), … all are proxy to the even moments of a returns distribution

Vvol means the tails may become fatter and … v2vol they can become even more fatter.

An option price can be decomposed to basket of options having vol determined using moment matching techniques.

Likewise a option can be de composed to a basket of option vol, vvol, …

The vol skew can be calibrated to be series of vivol.

f1(x; μ1, σ1) = 1 / (σ1 * sqrt(2 * π)) * [1 - exp( - (x - μ1) ^ 2 / (2 * σ1 ^ 2) )]

f1(x; μ1, β1) = 1 / (2 * β1) * [1 - exp( - x - μ1 / β1)]

Using this I think you can calibrate Implied Vol (IV)

IV = ∑ ki * fi * vivol

Such that

σ1 or β1 = 0

σ1 or β1 < σ2 or β2 < …

by calibrating for Ki

Any opinions?

Best regards, Suminda Sirinath Salpitikorala Dharmasena

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Asset Prices Cannot be RW: Are they Mean Reverting, Trending (Serially correlated), or one or the other dominates at a given time?


An asset process would be degenerate when it reach zero. For the asset value to be positive, the drift also needs to become zero which means that the the asset price Bachelier process. In the case of a stock, extremely large trading multiples with respect to valuations are not seen. Therefore, my hunch is that stocks mean revert to its valuation. Price also could have a momentum element.

Any opinions?

Best regards, Suminda Sirinath Salpitikorala Dharmasena

NB: through my initial argument seams to have flaws. Some that I realised are corrected. There is more that I can add but I have left them out.

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