never exercise American call (no-div), again

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.

BS-E is PDE not SDE

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?

clarifying questions to reduce confusions in BS discussions

–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.

local vol^stoch vol, briefly

Black-Scholes vol – constant, not a function of anything.
** simplest

stoch vol – there’s a dB element in dσ. See

** 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 “stochastic” dB element.
** middle ground. A simplification of stoch vol

Jensen’s inequality – option pricing

See also

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 !

Jensen\'s Inequality
A graphical demonstration of Jensen’s Inequality. The expectations shown are with respect to an arbitrary discrete distribution over the xi

hockey stick – asymptote

(See also post on fwd price ^ PnL/MTM of a fwd position.)

Assume K = 100. As we get very very close to maturity, the “now-if” graph descends very very close to the linear hockey stick, i.e. the “range of (terminal) possibilities” graph.

10 years before maturity, the “range of (terminal) possibilities” graph is still the same hockey stick turning at 100, but the now-if graph is quite a bit higher than the hockey stick. The real asymptote at this time is the (off-market) fwd contract’s now-if graph. This is a straight line crossing X-axis at K * exp(-rT). See

In other words, at time 0, call value >= S – K*exp(-rT)

As maturity nears, not only the now-if smooth curve but also the asymptote both descend to the kinked “terminal” hockey stick.

Towards expiration, how option greek graphs morph

(A veteran would look at other ways the curves respond to other changes, but I feel the most useful thing for a beginner to internalize is how the curves respond to … imminent expiration.)

Each curve is a rang-of-possibility curve since the x-axis is the (possible range of) current underlier prices.

— the forward contract’s price
As expiration approaches, …
the curve moves closer to the (terminal) payout graph — that straight line crossing at K.

— the soft hockey-stick i.e. “option price vs current underlier”

As expiration approaches, …

the curve descends closer to the kinked hockey stick payout diagram

Also the asymptote is the forward contract’s price curve, as described above.

— the delta curve
As expiration approaches, …

the climb (for the call) becomes more abrupt.

See diagram in

— the gamma curve
As expiration approaches, …

the “bell” curve is squeezed towards the center (ATM) so the peak rises, but the 2 tails drop

— the vega curve
As expiration approaches, …

the “bell” curve descends, in a parallel shift

N(d2), GBM, binary call valuation – intuitive

It’s possible to get an intuitive feel for the binary call valuation formula.
For a vanilla European call, C = … – K exp(-Rdisc T)*N(d2)
N(d2) = Risk-Neutral Pr(S_T > K). Therefore,
N(d2) = RN-expected payoff of a binary call
N(d2) exp(-Rdisc T) — If we discount that RN-expected payoff to Present Value, we get the current price of the binary call. Note all prices are measure-independent.
Based on GBM assumption, we can *easily* prove Pr(S_T > K) = N(d2) .
First, notice Pr(S_T > K) = Pr (log S_T > log K).
Now, given S_T is GBM, the random variable (N@T) 
   log S_T ~ N ( mean = log S + T(Rgrow – σ^2)  ,   std = T σ^2 ). 
Let’s standardize it to get
   Z := (log S_T  – mean)/std    ~  N(0,1)
Pr = Pr (Z > (log K  – mean)/std ) = Pr (Z < (mean – log k)/std) = N( (mean – log k)/std)  = N(d2)

option pricing – 5 essential rules n their assumptions

PCP — arb + extremely tight bid/ask spread + European vanilla option only. GBM Not assumed. Any numeraire fine.

Same drift as the numeraire — tradeable + arb + numeraire must be bond or a fixed-interest bank account.

no-drift — tradeable + arb + using the numeraire

Ito — BM or GBM in the dW term. tradable not assumed. Arb allowed.

BS — tradable + arb + GBM + constant vol