陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「The Black-Scholes equation」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

Some time ago, I wrote a short unpublished note (mostly for my own benefit) when I was trying to understand the derivation of the Black-Scholes equation in financial mathematics, which computes the price of various options under some assumptions on the underlying financial model. In order to avoid issues relating to stochastic calculus , Itō’s formula , etc. I only considered a discrete model rather than a continuous one, which makes the mathematics much more elementary. I wa

The basic type of problem that the Black-Scholes equation solves (in particular models) is the following. One has an underlying financial instrument S, which represents some asset which can be bought and sold at various times t, with the per-unit price of the instrument varying with t. (For the mathematical model, it is not relevant what type of asset S actually is, but one could imagine for instance that S is a stock, a commodity, a currency, or a bond.) Given such an underl

已知结果和反例

The problem is this: what is the “correct” price, at time , to assign to an European option (such as a put or call option) at a future expiration time ? Of course, due to the volatility of the underlying instrument S, the future price of this instrument is not known at time . Nevertheless – and this is really quite a remarkable fact – it is still possible to compute deterministically, at time , the price of an option that depends on that unknown price , under certain assumpti

Before we do any mathematics, we must first settle a fundamental financial question – how can one compute the price of some asset A? In most economic situations, such a price would depend on many factors, such as the supply and demand of A, transaction costs in buying or selling A, legal regulations concerning A, or more intangible factors such as the current market sentiment regarding A. Any model that attempted to accurately describe all of these features would be hideously

证明或构造的主线

But the situation is much simpler for purely financial products, such as options, at least when one has a highly deep and liquid market for the underlying instrument S. More precisely, 下面会 make the following (unrealistic) assumptions:

With these assumptions, the supply situation is simplified enormously, because any participant in this market can, in principle, use cash to create an option to sell to others (for instance one can sell a call option for S and cover it by buying a unit of S at any time before the expiration time), in contrast to physical assets (e.g. barrels of oil) which cannot be created purely from market transactions. This freedom of supply leads to upper bounds on the price of a financia

阅读时建议盯住的点

As a simple example of such an upper bound, if a deep and liquid market allows one to repeatedly buy individual units of A at a price of X per unit, then for any integer , the price of k units of A has an upper bound of kX. (The true price may be lower, due for instance to volume discounts, but in general the price of k units of A will be a subadditive function of A. Note though that if the market is not infinitely deep, then each purchase of a unit may increase the price of

As another example, the price at time of a put option for a unit of S at time at strike price P cannot exceed P, because any market participant can create (and then sell) such an option simply by setting aside P units of cash to cover the future expense of buying a unit of S. (This is an extremely crude upper bound, of course, as the option buyer might not exercise the option, in which case the P units of cash are recovered, or the option buyer does exercise in the option, in

值得单独记下的条目

• Infinite depth. Each sale of a unit of S of does not affect the price of futher sales of units of S.
• No transaction costs. The purchase price and sale price of an asset is the same: in other words, the money spent by a buyer in a sale is exactly equal to the money earned by the seller.
• No arbitrage . There do not exist risk-free opportunities for market participants to instantaneously make money.
• Infinite divisibility . Stock can be sold in arbitrary non-integer amounts.
• Short selling . Market participants can borrow arbitrary amounts of stock, at no interest, for arbitrary amounts of time.
• No storage costs. Market participants can hold arbitrary amounts of stock at no cost for arbitrary amounts of time.

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

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