Rule 1: For a given no-dividend stock, early exercise of American call is never optimal.
Rule 1b: therefore, the price is similar to a European call. In other words, the early exercise feature is worthless.
To simplify (not over-simplify) the explanation, it’s useful to assume zero interest rate.
The key insight is that short-selling stock is always better than exercise. Given strike is $100 but the current price is super high at $150.
* Exercise means “sell at $150 immediately after buying underlier at $100”.
* Short means “sell at $150 but delay the buying till expiry”
Why *delay* the buy? Because we hold a right not an obligation to buy.
– If terminal price is $201 or anything above strike, then the final buy is at $100, same as the Exercise route.
– If terminal price is $89 or anything below strike, then the final buy is BETTER than the Exercise route.
You can also think in terms of a super-replicating portfolio, but I find it less intuitive.
So in real markets when stock is very high and you are tempted to exercise, don’t sit there and risk losing the opportunity. 1) Short sell if you are allowed
2) Exercise if you can’t short sell
When interest rate is present, the argument is only slightly different. Invest the short sell proceeds in a bond.
I believe BS-equation ( a famous PDE) is not a Stoch differential equation, simply because there’s no dW term in it.
A SDE is really about two integrals on the left and right. At least one integral must be a stochastic integral.
Some (not all) of the derivations of BS-E uses stochastic integrals.
There are many ways to derive the BS-E(quation). See [[Crack]]. Roger Lee covered at least two routes.
There are many ways to derive the BS-F(ormula). See P116 [[Crack]]
There are many ways to interpret the BS-F. Roger Lee and [[Crack]] covered them extensively.
Q: BS-F is a solution to the BS-E, but is BS-F based on BS-E?
A: I would say yes, though some BS-F derivations don’t use any PDE (BS-E is PDE) at all.
BS-E is simpler than BS-F IMO. The math operations in the BS-F are non-trivial and not so intuitive.
BS-F only covers European calls and puts.
BS-E covers American and more complex options. See P74 [[Crack]]
BS-E has slightly fewer assumptions:
– Stock is assumed GBM
– no assumption about boundary condition. Can be American or exotic options. – constant vol?
–At every mention of “pricing method”, ask
Q: Analytical (Ana) or Numerical (Num)?
Q: for European or American+exotic options?
Obviously Analytical methods only work for European style
Q: GBM assumption?
I think most numerical methods do. Every single method has severe
assumptions, so GBM is just one of them.
–At every mention of “Option”, ask
Q: European style of Amerian+Exotic style?
–At every mention of Black-Scholes, ask
Q: BS-E(quation) or BS-F(ormula) or BS-M(odel)?
Note numerical methods rely on BS-M or BS-E, not BS-F
–At every mention of Expectation, ask
Q: P-measure of Q-measure?
The other measures, like the T-fwd measure are too advanced, so no
need to worry for now.
Black-Scholes vol – constant, not a function of anything.
stoch vol – there's a dB element in dσ. See http://en.wikipedia.org/wiki/Stochastic_volatility
** most elaborate
** this BM process is correlated to the BM process of the asset price. Correlation ranging from -1 to 0 to 1.
local vol – sigma_t as a deterministic function of S_t and t, without any dB element.
** middle ground. A simplification of stoch vol
This may also explain why a BM cubed isn’t a local martingale.
Q: How practical is JI?
A: practical for interviews.
A: JI is intuitive like ITM/OTM.
A: JI just says one thing is higher than another, without saying by how much, so it’s actually simpler and more useful than the precise math formulae. Wilmott calls JI “very simple mathematics”
JI is consistent with pricing math of vanilla call (or put). Define f(S) := (S-K)+. This hockey-stick is a kind of convex function. Now Under standard RN measure,
E[ f(S_T) ] should exceed f (E[ S_T ])
LHS is the call price today. RHS simplifies to f (S_0) := (S_0 – K)+ which is the intrinsic value today.
How about a binary call? Unfortunately, Not convex or concave !
A graphical demonstration of Jensen’s Inequality. The expectations shown are with respect to an arbitrary discrete distribution over the xi