常见“12 种”推导思路（不同书目列法略有差异，核心是等价的无套利/测度变换/方程视角）：

1. delta 对冲 + Itô → BS PDE
2. 风险中性定价（Girsanov）→ 贴现期望
3. Feynman–Kac（PDE ↔ 期望）
4. PDE 变换为热方程并求解析解
5. 二叉树极限（CRR）→ 连续极限
6. 复制组合/自融资组合的无套利论证
7. change of numeraire（换计价资产）推导 $N(d_1),N(d_2)$ 解释
8. 鞅方法：贴现价格为鞅 + 可选停止/表示定理
9. 远期/期货视角（Black-76）再映射回 spot
10. Fourier/特征函数方法对对数正态直接积分
11. 最小方差对冲（均方误差）在完备市场下与复制一致
12. 静态套利界与凸性/对偶（用于验证/约束，与 BS 形式相容）

其中最“标准”的面试推导是 1) 或 2)。

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英文解析

1. Hedging and the Partial Differential Equation

The original derivation of the Black- Scholes partial differential equation was via stochastic calculus, Ito's lemma and a simple hedging argument (Black & Scholes, 1973).

Assume that the underlying follows a lognormal random walk

$$dS = \mu Sdt + \sigma SdX.$$

Use to denote the value of a portfolio of one long option position and a short position in some quantity of the underlying:

$$\Pi = V(S,t) - \Delta S.$$

The first term on the right is the option and the second term is the short asset position.

Ask how the value of the portfolio changes from time t to $\mathrm{t + dt}$ . The change in

the portfolio value is due partly to the change in the option value and partly to the change in the underlying:

$$d\Pi = dV - \Delta dS$$

From Ito's lemma we have

$$d\Pi = \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} dS + \frac{1}{2}\sigma^2 S^2\frac{\partial^2V}{\partial S^2} dt - \Delta dS$$

The right- hand side of this contains two types of terms, the deterministic and the random. The deterministic terms are those with the dt, and the random terms are those with the dS. Pretending for the moment that we know V and itsderivatives then we know everything about the right- handside except for the value of dS, because this is random.

These random terms can be eliminated by choosing

$$\Delta = \frac{\partial V}{\partial S}.$$

After choosing the quantity A, we hold a portfolio whose value changes by the amount

$$d\Pi = \left(\frac{\partial V}{\partial t} +\frac{1}{2}\sigma^2 S^2\frac{\partial^2V}{\partial S^2}\right)dt.$$

This change is completely riskless. If we have a completely risk- free change d in the portfolio value then it must bethe same as the growth we would get if we put the equivalent amount of cash in a risk- free interest- bearing account:

$$d\Pi = r\Pi dt.$$

This is an example of the no- arbitrage principle.

Putting all of the above together to eliminate and $\Delta$ in favour of partial derivatives of V gives

$$\frac{\partial V}{\partial t} +\frac{1}{2}\sigma^2 S^2\frac{\partial^2V}{\partial S^2} +rS\frac{\partial V}{\partial S} -rV = 0,$$

the Black- Scholes equation.

Solve this quite simple linear diffusion equation with the final condition

$$V(S,T) = \max (S - K,0)$$

This derivation of the Black- Scholes equation is perhaps the most useful since it is readily generalizable (if not necessarily still analytically tractable) to different underlyings, more complicated models, and exotic contracts.

2. Martingales

The martingale pricing methodology was formalized by Harrison & Kreps (1979) and Harrison & Pliska (1981).

We start again with

$$dS_{t} = \mu Sdt + \sigma SdW_{t}$$

The W, is Brownian motion with measure P. Now introduce a new equivalent martingale measure Q such that

$$\tilde{W}_{t} = W_{t} + \eta t,$$

where $\eta = (\mu - \tau) / \sigma$ . Under Q we have

$$dS_{t} = rSdt + \sigma Sd\tilde{W}_{t}.$$

Introduce

$$G_{t} = e^{-r(T - t)}E_{t}^{\mathbb{Q}}[\max (S_{T} - K,0)].$$

The quantity $\mathrm{er(T - G}$ is a Q- martingale and so

$$d\left(e^{r(T - t)}G_{l}\right) = a_{l}e^{r(T - t)}G_{l}d\tilde{W}_{l}$$

for some process a. Applying Ito's lemma,

$$dG_{t} = (r + \alpha \eta)G_{t}dt + \alpha G_{t}dW_{t}.$$

This stochastic differential equation can be rewritten as one representing a strategy in which a quantity $\mathrm{aGr / oS}$ of the stock and a quantity (G- aGt/o) $\mathrm{er(T - t)}$ of a zero- coupon bond maturing at time $\mathrm{T}$ are bought:

$$dG_{l} = \frac{\alpha G_{t}}{\sigma S} dS + \frac{G - \frac{\alpha G_{t}}{\sigma S}S}{e^{-r(T - t)}} d(e^{-r(T - t)}).$$

Such a strategy is self financing because the values of the stock and bond positions add up to G. Because of the existence of such a self- financing strategy and because

at time $\mathbf{t} = \mathbf{T}$ we have that Gr is the call payoff we must have that G is the value of the call before expiration. The role of the self- financing strategy is to ensure that there are no arbitrage opportunities.

Thus the price of a call option is

$$e^{-r(T - t)}E_t^{\mathbb{Q}}[\max (S_T - K,0)].$$

The interpretation is simply that the option value is thepresent value of the expected payoff under a risk- neutral random walk.

For other options simply put the payoff function inside the expectation. This derivation is most useful for showing the link between option values and expectations, as it is the theoretical foundation for valuation by Monte Carlo simulation.

Now that we have a representation of the option value in terms of an expectation we can formally calculate this quantity and hence the Black- Scholes formulae. Under Q the logarithm of the stock price at expiration is normally distributed with mean $\mathbf{m} =$ $\operatorname {In}(\mathbf{S}) + (\mathbf{r} - \mathbf{o}2)$ (T- t) and variance $\mathrm{v}2 = \sigma 2(\mathrm{T - t})$ . Therefore the call option value is

$$e^{-r(T - t)}\int_{\frac{\ln K - m}{\nu}}^{\infty}(e^{m + \nu x} - K)\frac{e^{-\frac{\nu^2}{2}}}{\sqrt{2\pi}} dx.$$

A simplification of this using the cumulative distribution func- tion for the standardized normal distribution results in the well- known call option formula.

3. Change of Numeraire

The following is a derivation of the Black- Scholes call (or put) formula, not the equation, and is really just a trick for simplifying some of the integration.

It starts from the result that the option value is

$$e^{-r(T - t)}E_t^{\mathbb{Q}}[\max (S_T - K,0)].$$

This can also be written as

$$e^{-r(T - t)}E_t^{\mathbb{Q}}[(S_T - K)H(S - K)],$$

where HCS- K) is the Heaviside function, which is zero for $\mathbf{S}< \mathbf{K}$ and 1 for $\mathbf{S} > \mathbf{K}$

Now define another equivalent martingale measure $\mathbf{Q}^{\prime}$ such that

$$\tilde{W}_{t}^{\prime} = W_{t} + \eta t - \sigma t.$$

The option value can then be written as

$$S_{t}E_{t}^{\mathbb{Q}^{\prime}}\left[\frac{(S_{T} - K)\mathcal{H}(S - K)}{S_{T}}\right].$$

where

$$dS_{t} = (r + \sigma^{2})Sdt + \sigma Sd\tilde{W}_{t}^{\prime}.$$

It can also be written as a combination of the two expressions,

$$S_{t}E_{t}^{\mathbb{Q}^{\prime}}\left[\frac{S_{T}\mathcal{H}(S - K)}{S_{T}}\right] - Ke^{-r(T - t)}E_{t}^{\mathbb{Q}}[\mathcal{H}(S - K)].$$

Notice that the same calculation is to be performed, an expectation of $\mathrm{H}(\mathrm{S - K})$ , but under two different measures. The end result is the Black- Scholes formula for a call option.

This method is most useful for simplifying valuation prob- lems, perhaps even finding closed- form solutions, by using the most suitable traded contract to use for the numeraire.

The relationship between the change of numeraire result and the partial differential equation approach is very simple, and informative.

First let us make the comparison between the risk- neutral expectation and the Black- Scholes equation as transparent as possible. When we write

$$e^{-r(T - t)}E_{t}^{\mathbb{Q}}[\max (S_{T} - K,0)]$$

we are saying that the option value is the present value of the expected payoff under the risk- neutral random walk

$$dS = rSdt + \sigma Sd\tilde{W}_{t}.$$

The partial differential equation

$$\frac{\partial V}{\partial t} +\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}} +rS\frac{\partial V}{\partial S} -rV = 0$$

means exactly the same because of the relationship between it and the Fokker- Planck equation. In this equation the diffusion coefficient is always just one half of the square of the randomness in dS. The coefficient of aV/as is always the risk- neutral drift rS and the coefficient of V is always minus the interest rate, - r, and represents the present valuing from expiration to now.

If we write the option value as $\mathbf{V} = \mathbf{S}\mathbf{V}$ then we can think of Vas the number of shares the option is equivalent to, in value terms. It is like using the stock as the unit of currency. But if we rewrite the Black- Scholes equation in terms of $\mathbf{V}$ using

$$\frac{\partial V}{\partial t} = S\frac{\partial\overline{V}}{\partial t},\quad \frac{\partial V}{\partial S} = S\frac{\partial\overline{V}}{\partial S} +\overline{V},\quad \mathrm{and}\quad \frac{\partial^{2}V}{\partial S^{2}} = S\frac{\partial^{2}\overline{V}}{\partial S^{2}} +2S\frac{\partial\overline{V}}{\partial S},$$

then we have

$$\frac{\partial\overline{V}}{\partial t} +\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}\overline{V}}{\partial S^{2}} +(r + \sigma^{2})S\frac{\partial\overline{V}}{\partial S} = 0.$$

The function $\mathbf{V}$ can now be interpreted, using the same comparison with the Fokker- Planck equation, as an expectation, but this time with respect to the random walk

$$dS = (r + \sigma^{2})Sdt + \sigma Sd\tilde{W}_{t}^{\prime}.$$

And there is no present valuing to be done. Since at expiration we have for the call option

$$\frac{\max (S_{T} - K,0)}{S_{T}}$$

we can write the option value as

$$S_{t}E_{t}^{\mathbb{Q}^{\prime}}\left[\frac{(S_{T} - K)\mathcal{H}(S - K)}{S_{T}}\right].$$

where

$$dS_{t} = (r + \sigma^{2})Sdt + \sigma Sd\tilde{W}_{t}^{\prime}.$$

Change of numeraire is no more than a change of dependent variable.

4. Local Time

The most obscure of the derivations is the one involving the concept from stochastic calculus known as ' local time.' Local time is a very technical idea involving the time a random walk spends in the vicinity of a point.

The derivation is based on the analysis of a stop- loss strategy in which one attempts to hedge a call by selling one share short if the stock is above the present value of the strike, and holding nothing if the stock is below the present value of the strike. Although at expiration the call payoff and the stock position will cancel each other exactly, this is not a strategy that eliminates risk. Naively you might think that this strategy would work, after all when you sell short one of the stock as it passes through the present value of the strike you will neither make nor lose money (assuming there are no transaction costs). But if that were the case then an option initially with strike above the forward stock price should have zero value. So clearly something is wrong here.

To see what goes wrong you have to look more closely at what happens as the stock goes through the present value of the strike. In particular, look at discrete moves in the stock price.

As the forward stock price goes from K to $\mathbf{K} + \mathbf{e}$ sell one share and buy K bonds. And then every time the stock falls below the present value of the strike you reverse this. Even in the absence of transaction costs, there will be a slippage in this process. And the total slippage will depend on how often the stock crosses this point. Herein lies the rub. This happens an infinite number of times in continuous Brownian motion.

If U(e) is the number of times the forward price moves from K to $\mathbf{K} + \mathbf{e}$ , which will be finite since $\epsilon$ is finite, then the financing cost of this strategy is

$$\epsilon U(\epsilon).$$

Now take the limit as $\rightarrow 0$ and this becomes the quantity known as local time. This local- time term is what explains the apparent paradox with the above example of the call with zero value.

Now we go over to the risk- neutral world to value the local- time term, ending up, eventually, with the Black- Scholes formula.

It is well worth simulating this strategy on a spreadsheet, using a finite time step and let this time step get smaller and smaller.

5. Parameters as Variables

The next derivation is rather novel in that it involves dif- .ferentiating the option value with respect to the parameters strike, K, and expiration, T, instead of the more usualdifferentiation with respect to the variables S and t. This will lead to a partial differential equation that can be solved for the Black- Scholes formulæ. But more importantly, this tech- nique can be used to deduce the dependence of volatility on stock price and time, given the market prices of options as functions of strike and expiration. This is an idea due to Dupire (1994) (also see Derman & Kani, 1994, and Rubinstein, 1994, for related work done in a discrete setting) and is the basis for deterministic volatility models and calibration.

We begin with the call option result from above

$$V = e^{-r(T - t)}E_t^\mathbb{Q}[\max (S_T - K,0)],$$

that the option value is the present value of the risk- neutral expected payoff. This can be written as

$$\begin{array}{l}{V(K,T) = e^{-r(T - t^{*})}\int_{0}^{\infty}\max (S - K,0)p(S^{*},t^{*};S,T)dS}\\ {= e^{-r(T - t^{*})}\int_{K}^{\infty}(S - K)p(S^{*},t^{*};S,T)dS,} \end{array}$$

where $\mathfrak{p}(\mathbf{S}^{*},\ast ;\mathbf{S},\mathrm{T})$ is the transition probability density func- tion for the risk- neutral random walk with $\mathbf{S}^{*}$ being today's asset price and $\mathbf{t}^{*}$ today's date. Note that here the arguments of $\mathrm{v}$ are the ' variables' strike, K, and expiration, T.

If we differentiate this with respect to $\mathbf{K}$ we get

$$\frac{\partial V}{\partial K} = -e^{-r(T - t^k)}\int_K^\infty p(S^*,t^*;S,T)dS.$$

After another differentiation, we arrive at this equation for the probability density

function in terms of the option prices

$$p(S^{*},t^{*};K,T) = e^{r(T - t^{*})}\frac{\partial^{2}V}{\partial K^{2}}$$

We also know that the forward equation for the transition probability density function, the Fokker- Planck equation, is

$$\frac{\partial p}{\partial T} = \frac{1}{2}\frac{\partial^{2}}{\partial S^{2}} (\sigma^{2}S^{2}p) - \frac{\partial}{\partial S} (rS p).$$

Here $\sigma (\mathbf{S},\mathbf{t})$ is evaluated at $\mathbf{t} = \mathbf{T}$ . We also have

$$\frac{\partial V}{\partial T} = -rV + e^{-r(T - l^{3})}\int_{K}^{\infty}(S - K)\frac{\partial p}{\partial T} dS.$$

This can be written as

$$\frac{\partial V}{\partial T} = -rV + e^{-r(T - t^{*})}\int_{K}^{\infty}\left(\frac{1}{2}\frac{\partial^{2}(\sigma^{2}S^{2}p)}{\partial S^{2}} -\frac{\partial(rS p)}{\partial S}\right)(S - K)dS.$$

using the forward equation. Integrating this by parts twice we get

$$\frac{\partial V}{\partial T} = -rV + \frac{1}{2} e^{-r(T - t^{*})}\sigma^{2}K^{2}p + r e^{-r(T - t^{*})}\int_{K}^{\infty}S p dS.$$

In this expression $\sigma (\mathbf{S},\mathbf{t})$ has $\mathbf{S} = \mathbf{K}$ and $\mathbf{t} = \mathbf{T}$ . After some simple manipulations we get

$$\frac{\partial V}{\partial T} = \frac{1}{2}\sigma^{2}K^{2}\frac{\partial^{2}V}{\partial K^{2}} -rK\frac{\partial V}{\partial K}.$$

This partial differential equation can now be solved for the Black- Scholes formulae.

This method is not used in practice for finding these formulae, but rather, knowing the traded prices of vanillas as a function of $\mathbf{K}$ and $\mathbf{T}$ we can turn this equation around to find $\sigma$ , since the above analysis is still valid even if volatility is stock and time dependent.

6. Continuous-Time Limit of the Binomial Model

Some of our twelve derivations lead to the Black- Scholes partial differential equation, and some to the idea of the option value as the present value of the option payoff under a risk- neutral random walk. The following simple model (Figure 7.1) does

both.

Figure 8.5: The model.

In the binomial model the asset starts at S and over a time step St either rises to a value u x S or falls to a value vx S, with $0< \mathrm{u}< 1< \mathrm{u}$ . The probability of a rise is p and so the probability of a fall is 1 - p.

We choose the three constants u, v and p to give the bino- mial walk the same drift, $\mu$ , and volatility, o, as the asset we are modelling. This choice is far from unique and here we use the choices that result in the simplest formula:

$$\begin{array}{c}{u = 1 + \sigma \sqrt{\delta t},}\\ {\nu = 1 - \sigma \sqrt{\delta t}}\\ {p = \frac{1}{2} +\frac{\mu\sqrt{\delta t}}{2\sigma}.} \end{array}$$

Having defined the behaviour of the asset we are ready to price options.

Suppose that we know the value of the option at the time t+ St. For example, this time may be the expiration of the option. Now construct a portfolio at time t consisting of one option and a short position in a quantity of the underlying. At time t this portfolio has value

$$\Pi = V - \Delta S,$$

where the option value $V$ is for the moment unknown. At time $t + 8t$ the option takes

one of two values, depending on whether the asset rises or falls

$$V^{+}\quad \mathrm{or}\quad V^{-}.$$

At the same time the portfolio of option and stock becomes either

$$V^{+} - \Delta uS\quad \mathrm{or}\quad V^{-} - \Delta \nu S$$

Having the freedom to choose, we can make the value of this portfolio the same whether the asset rises or falls. This is ensured if we make

$$V^{+} - \Delta uS = V^{-} - \Delta \nu S.$$

This means that we should choose

$$\Delta = \frac{V^{+} - V^{-}}{(u - \nu)S}$$

or hedging. The portfolio value is then

$$V^{+} - \Delta uS = V^{+} - \frac{u(V^{+} - V^{-})}{(u - \nu)} = V^{-} - \Delta \nu S = V^{-} - \frac{\nu(V^{+} - V^{-})}{(u - \nu)}.$$

Let's denote this portfolio value by

$$\Pi +\delta \Pi .$$

This just means the original portfolio value plus the change in value, But we must also have $8\mathrm{TI} = \mathrm{rII}$ &t to avoid arbitrage opportunities. Bringing all of these expressions together to eliminate, and after some rearranging, we get

$$V = \frac{1}{1 + r\delta t}\left(\rho^{\prime}V^{+} + (1 - \rho^{\prime})V^{-}\right),$$

where

$$p^{\prime} = \frac{1}{2} +\frac{r\sqrt{\delta t}}{2\sigma}.$$

This is an equation for $\mathrm{V}$ given $\mathrm{V}+$ and $\mathrm{V}^{- }$ , the option values at the next time step, and the parameters $\mathrm{r}$ and $\mathrm{o}$ .

The right- hand side of the equation for V can be interpreted, rather clearly, as the present value of the expected future option value using the probabilities p' for an up move and 1- p' for a down.

Again this is the idea of the option value as the present value of the expected payoff under a risk- neutral random walk. The quantity p' is the risk- neutral probability, and it is this that determines the value of the option not the real probability. By comparing the expressions for p and p' we see that this is equivalent to replacing the real asset drift $\mu$ with the risk- free rate of return r.

We can examine the equation for $\mathrm{v}$ in the limit as $\delta t\rightarrow 0$ .We write

$$V = V(S,t),\quad V^{+} = V(uS,t + \delta t)\quad \mathrm{and}\quad V^{-} = V(\nu S,t + \delta t).$$

Expanding these expressions in Taylor series for small 8t we find that

$$\Delta \sim \frac{\partial V}{\partial S}\quad \mathrm{as}\quad \delta t\rightarrow 0,$$

and the binomial pricing equation for $\mathrm{v}$ becomes

$$\frac{\partial V}{\partial t} +\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}} +rS\frac{\partial V}{\partial S} -rV = 0.$$

This is the Black- Scholes equation

7. CAPM

This derivation, originally due to Cox& Rubinstein (1985) starts from the Capital Asset Pricing Model in continuous time. In particular it uses the result that there is a linear relationship between the expected return on a financial instrument and the covariance of the asset with the market. The latter term can be thought of as compensation for taking risk. But the asset and its option are perfectly correlated, so the compensation in excess of the risk- free rate for taking unit amount of risk must be the same for each.

For the stock, the expected return (dividing by dt) is $\mu$ . Its risk is o.

From Ito we have

$$dV = \frac{\partial V}{\partial t} dt + \frac{1}{2}\sigma^2 S^2\frac{\partial^2V}{\partial S^2} dt + \frac{\partial V}{\partial S} dS.$$

Therefore the expected return on the option in excess of the risk- free rate is

$$\frac{1}{V}\left(\frac{\partial V}{\partial t} +\frac{1}{2}\sigma^2 S^2\frac{\partial^2V}{\partial S^2} +\mu S\frac{\partial V}{\partial S} -rV\right)$$

and the risk is

$$\frac{1}{V}\sigma S\frac{\partial V}{\partial S}.$$

Since both the underlying and the option must have the same compensation, in excess of the risk- free rate, for unit risk

$$\frac{\mu - r}{\sigma} = \frac{\frac{1}{V}\left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2V}{\partial S^2} + \mu S\frac{\partial V}{\partial S} - rV\right)}{\frac{1}{V}\sigma S\frac{\partial V}{\partial S}}.$$

Now rearrange this. The $\mu$ drops out and we are left with the BlackScholes equation.

8. Utility Theory

The utility theory approach is not exactly the most useful of the twelve derivation methods, requiring that we value from the perspective of a particularly unrepresentative investor, an investor with a utility function that is a power law. This idea was introduced by Rubinstein (1976). Even though not the best way to derive the famous formulae utility theory is something which deserves better press than it has received so far.

The steps along the way to finding the Black- Scholes formulae are as follows. We work within a single- period framework, so that the concept of continuous hedging, or indeed anything continuous at all, is not needed. We assume that the stock price at the terminal time (which will shortly also be an option's expiration) and the consumption are both lognormally distributed with some correlation. We choose a utility function that is a power of the consumption. A valuation expression results. For the market to be in equilibrium requires a relationship between the stock's and consumption's expected growths and volatilities, the above- mentioned correlation and the degree of risk aversion in the utility function. Finally, we use the valuation expression for an

option, with the expiration being the terminal date. This valuation expression can be interpreted as an expectation, with the usual and oft- repeated interpretation.

9. Taylor Series

Taylor series is just a discrete- time version of Itô's lemma. So you should find this derivation of the Black- Scholes partial differential equation very simple.

$V(S, t)$ is the option value as a function of asset $S$ and time $t$ . Set up a portfolio long the option and short $b f$ the stock:

$$\Pi = V(S, t) - \Delta S.$$

Now look at the change in this portfolio from time $t$ to $t + \delta t$ , with $\delta t$ being a small time step:

$$\delta \Pi = V(S + \delta S, t + \delta t) - V(S, t) - \Delta \delta S,$$

where

$$\delta S = u S \delta t + \sigma S \sqrt{\delta t} \phi$$

is a discrete- time model for the stock and is a random variable drawn from a normal distribution with zero mean and unit standard deviation. (Aside: Does it matter that is normally distributed? It's a nice little exercise to see what difference it makes if comes from another distribution.)

Now expand 8\P in Taylor series for small St to get

$$\begin{array}{r l r} & {} & {\delta \Pi \& \approx \frac{\partial V}{\partial t}\delta t + \frac{\partial V}{\partial S}\left(\mu S\delta t + \sigma S\sqrt{\delta t}\phi\right) + \frac{1}{2}\sigma^{2}S^{2}\delta t\frac{\partial^{2}V}{\partial S^{2}}\phi^{2}}\\ & {} & {\& \quad -\Delta \left(\mu S\delta t + \sigma S\sqrt{\delta t}\phi\right) + O(\delta t^{3 / 2}).} \end{array}$$

where all terms are now evaluated at $S$ and $t$

The variance of this expression is

$$\sigma^{2}S^{2} \delta t \left(\frac{\partial V}{\partial S} - \Delta\right)^{2} + O(\delta t^{3 / 2})$$

which is minimized by the choice

$$\Delta = \frac{\partial V}{\partial S}.$$

Now put this choice for $\Delta$ into the expression for $\mathrm{SI}$ , and set

$$\delta \Pi = r\Pi \delta t.$$

This is a bit naughty, but I'll come back to it in a second. Take the resulting equation, divide by 8t so the leading terms are $\mathrm{O}(1)$ and let $\mathrm{St} \to 0$ . Bingo, you have the Black- Scholes partial differential equation, honest.

The naughty step in this was setting the return on the port- folio equal to the risk- free rate. This is fine as long as the portfolio is itself risk free. Here it is not, not exactly; there is still a little bit of risk. The variance, after choosing the best, is $\mathrm{O}(8t3 / 2)$ . Since there are $\mathrm{O}(T / 8t)$ rehedges between the start of the option's life and expiration, where $\mathrm{T}$ is the time to expiration, the total variance does decay to zero as $\mathrm{St} \to \mathrm{O}$ , thank goodness. And that's the a posteriori justification for ignoring risk.

(In Wilmott, 1994, this analysis goes to higher order in $\mathrm{St}$ to find an even better hedge than the classic Black- Scholes - one that is relevant if $\mathrm{St}$ is not so small, or if gamma is large, or if you are close to expiration. In that paper there is a small typo, corrected in Wilmott, 2006. )

In our final derivation you will see a less mathsy version of this same argument.

10. Mellin Transform

This derivation (see Yakovlev & Zhabin, 2003) is another one that I am only going to do in spirit rather than in detail. Again it is in discrete time but with continuous asset price. In that sense it's rather like the previous derivation and also our final

derivation, Black- Scholes for accountants, only far, far more complicated.

Vk(S) is the option value when the stock price is S at the kth time step. You set up a portfolio, , long one option and short a quantity, , of the underlying asset. The delta is chosen to minimize the variance of this portfolio at the next time step; the resulting expression for $\Delta$ involves the covariance between the stock and the option.

The pricing equation is then

$$e^{r\delta t}\Pi_{k = 1} = E\left[\Pi_k\right],$$

with the obvious notation. There is a slight problem with this, in that there is really no justification for equating value and expectation, at least not until you look at (or check a posteriori) the total variance at expiration and show that it is small enough to ignore (if the time steps are small enough). Anyway..

This equation can be rewritten just in terms of $\mathrm{v}$ as

$$V_{k + 1}(S) = \int_{-\infty}^{\infty}f_k(x)V_k(Se^x)dx.$$

So far none of this has required the stock to be lognormally distributed, it is more general than that. But if it is lognormal then the above iteration will result in the class formulae for calls and puts.

11. A Diffusion Equation

The penultimate derivation of the Black- Scholes partial differential equation is rather unusual in that it uses just pure thought about the nature of Brownian motion and a couple of trivial observations. It also has a very neat punchline that makes the derivation helpful in other modelling situations.

It goes something like this.

Stock prices can be modelled as Brownian motion, the stock price plays the role of the position of the ' pollen particle' and time is time. In mathematical terms Brownian motion is just an example of a diffusion equation. So let's write down a diffusion equation for the value of an option as a function of space and time, i.e. stock price

and time, that's V(S,t). What's the general linear diffusion equation? It is

$$\frac{\partial V}{\partial t} +a\frac{\partial^2V}{\partial S^2} +b\frac{\partial V}{\partial S} +cV = 0.$$

Note the coefficients u . band C .At the moment these could be anythisg. Now for the two trivial observations.

First, cash in the bank must be a solution of this equation.Financial contracts don't come any simpler than this. So plug $V = e^{\pi}$ into this diffusion equation to get

$$r e^{r t} + 0 + 0 + c e^{r t} = 0.$$

So $c = - r$

Second, surely the stock price itself must also be a solution?After all, you could think of it as being a call option with zero strike. So plug $V = \mathrm{S}$ into the general diffusion equation. We find

$$0 + 0 + b + cS = 0.$$

So $b = - cS = rS$

Putting b and C back into the general diffusion equation we find

$$\frac{\partial V}{\partial t} +a\frac{\partial^2V}{\partial S^2} +rS\frac{\partial V}{\partial S} -rV = 0.$$

This is the risk- neutral Black- Scholes equation. Two of the coefficients (those of V and av/as) have been pinned down exactly without any modelling at all. Ok, so it doesn't tell uswhat the coefficient of the second derivative term is, but even that has a nice interpretation. It means at least a couple of interesting things.

First, if we do start to move outside the Black- Scholes worldthen chances are it will be the diffusion coefficient that we must change from its usual 1o2s2 to accommodate new models.

Second, if we want to fudge our option prices, to massage them into line with traded prices for example, we can only do so by fiddling with this diffusion coefficient,

i.e. what we now know to be the volatility. This derivation tells us that our only valid fudge factor is the volatility.

12. Black-Scholes for Accountants

The final derivation of the Black- Scholes equation requires very little complicated mathematics, and doesn't even need assumptions about Gaussian returns, all we need is for the variance of returns to be finite.

The Black- Scholes analysis requires continuous hedging, which is possible in theory but impossible, and even undesirable, in practice. Hence one hedges in some discrete way. Let's assume that we hedge at equal time periods, 8t. And consider the value changes associated with a delta- hedged option.

We start with zero cash We buy an option We sell some stock short Any cash left (positive or negative) is put into a risk- free account.

We start by borrowing some money to buy the option. This option has a delta, and so we sell delta of the underlying

Figure 8.6: How our portfolio depends on S.

stock in order to hedge. This brings in some money. The cash from these transactions is put in the bank. At this point in time our net worth is zero.

Our portfolio has a dependence on S as shown in Figure 7.2.

We are only concerned with small movements in the stock over a small time period, so zoom in on the current stock position. Locally the curve is approximately a parabola, see Figure 7.3.

Now think about how our net worth will change from now to a time St later. There are three reasons for our total wealth to change over that period.

1. The option price curve changes.
2. There is an interest payment on the money in the bank.
3. The stock moves

Figure 8.7: The curve is approximately quadratic.

The option curve falls by the time value, the theta multiplied by the time step:

$$\Theta \times \delta t.$$

To calculate how much interest we received we need to know how much money we put in the bank. This was

$$\Delta \times S$$

from the stock sale and

• v

from the option purchase. Therefore the interest we receive is

$$r(S\Delta -V) \delta t.$$

Finally, look at the money made from the stock move. Since gamma is positive, any stock price move is good for us. The larger the move the better.

The curve in Figure 7.3 is locally quadratic, a parabola with coefficient . The stock move over a time period St is proportional to three things:

the volatility o

the stock price S

the square root of the time step

Multiply these three together, square the result because the curve is parabolic and multiply that by T and you get the profit made from the stock move as

$$\frac{1}{2}\sigma^2 S^2\Gamma \delta t.$$

Put these three value changes together Gignoring the $\delta t$ term which multiplies all of them) and set the resulting expression equal to zero, to represent no arbitrage, and you get

$$\Theta + \frac{1}{2}\sigma^2 S^2\Gamma + r(S\Delta - V) = 0,$$

the Black- Scholes equation

Now there was a bit of cheating here, since the stock price move is really random, What we should have said is that

$$\frac{1}{2}\sigma^2 S^2\Gamma \delta t$$

is the profit made from the stock move on average. Crucially all we need to know is that the variance of returns is

$$\sigma^2 S^2\delta t,$$

we don't even need the stock returns to be normally distributed. There is a difference between the square of the stock prices moves and its average value and this

gives riseto hedging error, something that is always seen in practice. If you hedge discretely, as you must, then Black- Scholes only works on average. But as you hedge more and more frequently, going to the limit $\mathrm{St} = 0$ , then the total hedging error tends to zero, so justifying the Black- Scholes model.

13. Other Derivations

There are other ways of deriving the Black- Scholes equation or formulae but I am only going to give the references (see Gerber & Shiu, 1994, and Hamada & Sherris, 2003). One of the reasons why I have drawn a line by not including them is summed up very nicely by a reader (who will remain anonymous for reasons which will be apparent) who submitted a couple of possible new derivations, in particular one using ' distortion risk theory.' In an email to me he wrote: 'Unfortunately distortion risk theory is completely unknown to quants.. maybe because this theory originated in insurance mathematics, but more probably because is useless, except research paper writing. I wrote master thesis on this topic, from time perspective, completely waste of time.'