CONTENTS

Summarization of The pricing of options and corporate liabilities

There's a formula used by the most profitable firms in the world, including Jane Street, Citadel, Optiver, and many others. Yes, you know it: it's the Black-Scholes model. They had written the paper in 1973, and it's still the GOAT formula. In this article, we're going to break down the paper and make it easier to understand.

Assumption

First, the Black & Scholes model brought out a hypothesis: If options are correctly priced, you should not be able to have risk-free profit by combining stock + option positions. This is the idea of no arbitrage.

Based on this idea, the paper derives a valuation formula for options that avoids estimating investors' risk preferences and expected stock returns. They try to avoid expected stock returns because they are hard to observe and to estimate reliably. And the most important thing is that they discovered that if no-arbitrage and replication can be priced, there's no need to estimate risk preferences and expected stock returns. So this paper aims to construct a hedged position whose risk can be made almost zero, so its return must match the risk-free rate.

Then it shows that the same logic applies to corporate liabilities such as warrants, equity, and debt. They share the same characteristics as options, which are default matters.

What is an option

An option is the right to buy or sell an asset at a fixed exercise price (or strike price) on the expiration date (or at any time before expiration for an American option).
Options value vs stock price The figure shows the relation between option value and stock price. A few things can be seen:
  • Higher stock price → Higher call value
  • If the stock price is far below the strike near expiration, the call is near worthless
  • If the expiration is far away from now, the call value approaches the stock value when the strike is very low (deep ITM long maturity intuition)

The valuation formula

The assumptions made for derivation are:

  • Stock price follows a continuous-time and log-normal random walk with a constant variance
  • Interest rate is known as a constant
  • Short selling is allowed, which means borrowing or short-selling can be done at the risk-free rate
  • There is no transaction cost and dividends
  • The option is European style exercise only, meaning it can only be exercised at maturity

Under these assumptions, the value of the option depends only on the stock price and time. Therefore, we can let the option value be w(x,t)w(x, t), where xx is the stock price and tt is the time. And the hedged position consists of long 1 stock + short a number (this number is what we now call delta, but it's not yet introduced in this paper) of options. So the problem is: how can we choose the delta so that the stochastic part cancels?

We choose the number of shares to short when the risk of longing the stock is hedged, in other words, the position is riskless, so we can earn the risk-free rate.

To learn more about how hedging makes the risk become zero, here's the calculation: The derivation of the Black-Scholes Formula

An Alternative Derivation

More Complicated Options

Warrant Valuation

Common Stock and Bond Valuation

Empirical Tests

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