Quick Notes on Haug's Article (Know Your Weapon I)

Concerning vanilla options and good-O' Black- Scholes:

1. Delta (the absolute value of which) can be greater than unity
2. As a corollary, ITM put can have absolute Delta lower than 0.5 and OTM call can have Delta greater than 0.5
3. Gamma and Vega attain their max NOT at-the-money-forward
4. Ordinary way of computing Gamma has a saddle point; there's a peak at the long expiry zero underlying boundary
5. As a consequence, GammaP might be more useful than Gamma in describing gamma risk

Ref: Haug, Know Your Weapon Part I

Black-Scholes Pricing in F#

First, get Math.NET Library (Open source).

F# code:

\\ begin
#light

open MathNet.Numerics.FSharp
open MathNet.Numerics.Distributions

let d12 (S0,r,y,sigma,K,T,d1_d2) = (log (S0/K)+(r-y+(if d1_d2 = 1.0 then 1.0 else -1.0))*(sigma*sigma / 2.0)*(T))/(sigma * (sqrt(T)))

let normal = new Normal(0.0, 1.0)

let bsPrice (S0, r, y, sigma, K, T) = exp (-y*T) * S0 * normal.CumulativeDistribution(d12 (S0,r,y,sigma,K,T,1.0))-exp (-r*T) *K * normal.CumulativeDistribution(d12 (S0,r,y,sigma,K,T,2.0))

printfn "Price = %F" (bsPrice (1.0, 0.0, 0.0, 0.2, 1.0, 1.0))

open System
Console.ReadKey(true)
\\ end

Vega hedging (and good modeling practice in general)

Recently re-read two very good papers (see the bottom), and it's interesting to find that they have related themes.

Synopsis for Jarrow: A model can either be theoretical or statistical (think regression). A theoretical model can be rejected (or not) by inspecting the assumptions and implications (note, though, that a rejected model can still be used). If a theoretical model is rejected, then calibration to market data would effectively turn it into an ad hoc statistical model.

For the sake of discussion consider B-S. It sets out to be a theoretical model, but volatility smile renders it rejected. By calibrating it to the volatility smile (local volatility), we are turning it into a statistical model. It can then be used to price other options (although, now being a statistical model, the econometric relationship would change over time and re-calibration is required). What we should not do is to use it for Delta/Gamma hedging, because it is exactly the failure of Delta hedging assumption that breaks its theoretical model status - there exist more than one sources of risk, which is why a single constant sigma does not suffice.

What is worse than using it for Delta hedging is to use it for Vega hedging, because under B-S the stock volatility is assumed to be a constant. Taking partial derivative with respect to sigma is mathematically sound, but it makes no sense other than spitting out an expression that lure you into believing its legitimacy. Which brings us to the uncertain parameters model Wilmott discussed. In his paper (and his book too) he explains why Vega is a terrible measure of volatility sensitivity when Gamma changes sign. Besides cliquet option, barrier option also falls into this category.

He proposes an uncertain parameter model that still uses Delta hedging argument. A non-linear PDE is produced. In a related chapter he discusses a similar technique for interest rate modeling, although in that case Delta hedging is dropped and the result of calibration is a yield envelop.

Jarrow 2011 - Risk Management Models: Construction, Testing, Usage
Wilmott - Cliquet Option Pricing
Also, PWOQF Ch. 52

Quick Notes on IR/Credit/Volatility Comparison

Modeling Approaches
Stochastic instantaneous spot
Short rate - Hazard rate - Stochastic volatility

Stochastic instantaneous forward
Instantaneous forward rate - ??? - xi in the Bergomi model

Market traded
LIBOR/Forward swap rate - Forward CDS rate - Forward variance swap rate

Others (Not exactly modeling)
Yield - ??? - Implied volatility

Question: where does local volatility fit in?