Video summary

QUANT FINANCE 1 - Why We Never Use the Black Scholes Equation, 1

Main summary

Key takeaways

Finance

Finance-focused summary (from provided subtitles)

This segment provides a historical and intuition-driven explanation of why the Black–Scholes equation is often misunderstood, and why practitioners may prefer alternative foundations—especially a Bachelier / “Bashier”-style approach—depending on assumptions about how prices evolve.

Core claims / corrections to the Black–Scholes narrative

  • The “Black–Scholes formula” is presented as not primarily being the option-pricing formula itself, but rather an argument that made a formula compatible with prevailing economic theories.
  • The speaker argues that more sophisticated option-pricing formulas existed earlier.
  • What is “used today” is described as effectively a version of the (Bachelier) formula (labeled “Bashier” in the subtitles).
  • A key technical point is emphasized:
    • Black–Scholes relies on modeling the underlying in a way that implies log-normal behavior.
    • The speaker argues that a normal (additive) model (Bachelier-type) can be more appropriate under certain dynamics.

Option pricing setup (framework described)

The speaker frames European call valuation as an “expectation of payoff” problem.

  • Consider a call option with:

    • Strike: (K)
    • Expiration: (T) (example given: “three months”)
    • Payoff at expiration: [ \max(S_T - K, 0) ]

  • The option price at valuation time (t_0) is the expected payoff under a chosen stochastic process for the stock: [ \text{Call price} = \mathbb{E}_{t_0}\left[\max(S_T - K, 0)\right] ]

Step-by-step / methodology elements mentioned

1) Choose a stochastic process for the stock price

Two model families are contrasted:

  1. Bachelier-type (normal increments)

    • Price changes include Gaussian noise.
    • Volatility scales like (\sigma \sqrt{\Delta t}).
    • An SDE is described (in words/equations), e.g.:
      • (dS = \mu S\,dt + \sigma S\,dZ)
    • The discussion links to behavior such as (\sigma\sqrt{t-t_0}).

  2. Black–Scholes-type (log-normal / proportional returns)

    • Converts to a returns interpretation where returns are proportional.
    • This implies multiplicative dynamics and thus log-normal behavior.

2) Compute the distribution of (S_T)

  • Under normal, the distribution is Gaussian.
  • Under log-normal, the implied distribution differs accordingly.

3) Integrate the payoff against the distribution

  • The call price is computed as a probability-weighted integral over outcomes where the stock finishes above the strike.
  • The subtitles describe writing the call price as an integral for (S \ge K) using the Gaussian density (in the normal-model discussion).

Two main “modifications/problems” highlighted

(1) Normal vs log-normal modeling

  • Normal modeling claim:
    • The speaker says that with normal modeling, an asset priced at 100 has equal probability around 150 or 50 (as described).
  • Log-normal modeling claim:
    • With log-normal modeling, outcomes like 50 and 200 are described as being “more likely” in a sense that symmetry shifts to multiplicative/log returns, not absolute price moves.
  • Implication/caution:
    • For interest rates, the speaker argues changes are better treated as basis-point moves (small additive changes).
    • Therefore, normal models may be more appropriate than log-normal in that context.

(2) Mean / drift and arbitrage-linked forward rates

  • The speaker argues Black–Scholes implicitly needs a correct drift/mean tied to arbitrage, not merely historical expectations.
  • Arbitrage intuition using carry:
    • Borrow in one currency and lend in another over the same horizon.
    • The difference affects the forward relationship.
  • Example given:
    • Borrow GBP at 10%
    • Invest USD at 5%
    • The implied differential is described as roughly ~5%.
  • Forward price / expected drift:
    • The forward for maturity ((t - t_0)) is said to be determined by an arbitrage relation, not a “free-choice” expected return.

  • Carry definition (explicit): [ \text{carry} = \text{dividend rate} - \text{cost of funds} ]

    • The speaker links arbitrage to the risk-free rate as the marginal provider.

Explicit numbers/timelines/instruments mentioned

  • Example time to maturity: three months
  • Reference to a horizon like “2 months away”
  • Currency/rate example:
    • GBP borrowing rate: 10%
    • USD lending rate: 5%
    • Implied differential: ~5%
  • Mention of “interest rates at 600 basis points” to illustrate basis-point style moves
  • No specific tickers/ETFs/bonds/commodities/macro indices are named.

Performance metrics / portfolio construction

  • None described in these subtitles; the discussion is theoretical/historical and focused on assumptions and interpretation rather than allocation or measurement.

Disclosures / recommendations

  • No explicit “not financial advice” disclaimer appears in the provided subtitles.
  • No direct investing recommendations are made.

Key sources / presenters mentioned (end of segment)

  • Robert Merton (spelled “Merin” in subtitles)
  • Myron Scholes
  • Louis Bachelier (referred to as “Bashier”)
  • Paul Kutner
  • Ed Thorpe / Ed Thorp
  • Kogarov
  • John M. Keynes (referred to as “kanes”)
  • Black–Scholes / “Black trolls”
  • Espen (friend/adviser/trainer; last name not provided)
  • Hog and “(Tal or) talent Hog” (unclear names in subtitles)
  • Mario Durban (mentioned in a put-call parity context)
  • Paul Lévy (referred to via a “Levy process” / “Levy process for”)
  • Brownian motion (“Brow nian motion”)
  • Einstein (via historical mention of Brownian motion preceding him)

Original video