跳跃过程/跳扩散与定价

跳过程：价格存在不连续跳变（如 Poisson 跳），跳扩散：扩散（布朗）+ 跳（Poisson）叠加。

在纯扩散 BS 世界里，1 只股票 + 现金即可连续对冲（市场完备）。

加入跳后：单靠股票 delta 对冲无法对冲跳风险（因为跳是不可预测的离散冲击），市场通常变为不完备，需要：

• 引入额外可交易资产（如不同执行价/期限的期权）或
• 指定风险溢价/选择某个等价鞅测度（如 Merton 跳扩散下可得到显式的“BS 混合”定价式）

因此：

$$\boxed{\text{跳模型下可以定价，但通常不能靠 BS 的完全复制对冲得到唯一价格}}.$$

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英文解析

Introductory courses typically do not say much about jump processes.

The Black and Scholes (1973) model naively assumes that stock prices are continuous. That is, they assume that you can draw the price history without lifting your pencil from the paper. You need only stand on the floor of an exchange, 52 watch a real- time feed (e.g., on a Bloomberg terminal), or read the WSJ headlines after an "event" to see that prices do not move smoothly. Indeed, the fact that stock prices are typically quoted with a minimum tick size (either exchange- imposed or effective) means that stock prices cannot move continuously. You can think of big stock price jumps as being stock price responses to the arrival of information in the market; small stock price jumps might just be due to the random ebb and flow of non- information- based (i.e., liquidity- related) transactions.

A "jump" price process is a price process that has infrequent jumps (i.e., discontinuities) in it. If the jump process is a very simple one, the Black- Scholes/Merton

no- arbitrage technique can still be used to hedge and price options on an asset whose price follows the process. If the jump process is more complicated, the no- arbitrage technique breaks down. See the following discussion, and go to the references if you need more details. I have included some lengthy comments and references. This is because I think it is relevant, and it is often not covered in introductory courses.

A simple jump process example (that is not a diffusion) has $\frac{dS}{S} = \mu dt + (J - 1)d\pi$ (Cox and Ross [1976, p. 147]). In this example, $J = 1$ is the jump amplitude (where $J \geq 0$ ), $d\pi$ takes the value $+1$ with probability $\lambda dt$ and $0$ with probability $1 - \lambda dt$ .⁵³ The percentage stock price change $\frac{dS}{S}$ can thus jump suddenly to $J - 1$ (which may itself be random); such a jump pushes $S$ to $SJ$ .

Figure 8.16: Time Value of a European Call Option

Note: The difference $c(t) - \max [S(t) - X,0]$ is the value of not exercising. When the option is deep out- of- the- money, $\max [S(t) - X,0] = 0$ , and $c(t)$ is approximately zero. When the option is deep in- the- money, you save $X$ by not exercising now, but it costs you the present value of exercising at maturity: $X \times e^{- r(T - t)}$ . The left- hand limit of $X - X \times e^{- r(T - t)} = X \times \left(1 - e^{- r(T - t)}\right)$ is always non- negative. The "kink" in $\max [S(t) - X,0]$ puts the "cusp" in the plot of $c(t) - \max [S(t) - X,0]$ versus $S(t)$ at $S(t) = X$ .

In this simple example, if $J$ is fixed (i.e., non- random), a riskless hedge portfolio

can be formed, and options on an asset whose price follows this simple jump process can be valued using the Black- Scholes/Merton no- arbitrage technique. This should come as no big surprise. The only real difference between this "pure Poisson process" case, and the simple binomial option pricing situation (Sharpe [1978]; Cox, Ross, and Rubinstein [1979]; Rendleman and Bartter [1979]; Cox and Rubinstein [1985]; Crack [2014a]) is that the arrival time of the jump up or jump down is a random variable. You do not need to know when the stock price will jump to hedge the risk in a binomial setting. This "pure Poisson process" is a special case of a more general jump diffusion process discussed next.

Consider the jump diffusion process $\frac{dS}{S} = (\alpha - \lambda k)dt + \sigma dZ + dq$ (described in detail in my Footnote 8 to Question 2.16 on p. 26). When $\sigma = 0$ and $Y \equiv dq + 1$ is non- random, you get Cox and Ross's simple jump process above, and the no- arbitrage technique can be used to hedge and price options on the jump process. Otherwise, when $\sigma > 0$ and $\operatorname {var}(Y) \geq 0$ it is not possible to form a riskless hedge portfolio or use the no- arbitrage technique (Cox and Ross [1976, p. 147]; Cox and Rubinstein [1985, pp. 361- 371]; Merton [1992, p. 316]). Both the (non- jump) diffusion process and the (non- diffusion) simple jump process are the continuous limits of discrete binomial models. However, the jump- diffusion is not. It is for this reason that a riskless hedge cannot be formed in the jump- diffusion case (Cox and Rubinstein [1985, pp. 361- 371]).

The fundamental reason that the no- arbitrage technique can be used to hedge and price options in the standard Black- Scholes world is linearity. In continuous time, the Black- Scholes option price is an instantaneously linear function of the stock price. Portfolio building is a linear operation, and it follows that payoffs to the option can be perfectly replicated by building and continuously rebalancing a portfolio of the stock and the bond. Linearity breaks down when the jump term has positive variance - the call price becomes a nonlinear function of the stock price and perfect hedging is not possible (Merton [1992, p. 316]).

rebalancing a portfolio of the stock and the bond. Linearity breaks down when the jump term has positive variance - the call price becomes a nonlinear function of the stock price and perfect hedging is not possible (Merton [1992, p. 316]).

Although the no- arbitrage technique fails to price the option on the jump diffusion process, you can price the option using an equilibrium argument. An instantaneous

CAPM (capital asset pricing model) approach may be used as it was in the original Black and Scholes (1973) paper. The information that causes jumps may be assumed to be firm- specific (i.e., unsystematic and diversifiable).⁵⁵ You can hedge out the non- jump part of the option and deduce that the remainder (the jump) must have zero beta and, therefore, a riskless rate of return. This yields a partial differential equation that can be solved to give the call option price as an infinite summation:

$$C(S(t),T - t) = \sum_{n = 0}^{\infty}\left\{\frac{\exp[-\lambda(T - t)][\lambda(T - t)]^n}{n!}\times \right.$$ $$E_{n}\left\{W\left[S(t)X_{n}\exp (-\lambda k(T - t)),(T - t);X,\sigma^{2},r\right]\right\}$$

Here $X_{n}$ is a random variable with the same distribution as the product of $n$ independent and identically distributed random variables each identically distributed to the random variable $Y$ (recall that $Y - 1$ is the random percentage change in stock price when a jump occurs), $X_{0} \equiv 1, E_{n}$ is the expectation operator over the distribution of $X_{n}$ , and $W\left[S, (T - t); X, \sigma^{2}, r\right]$ is the standard Black- Scholes pricing formula (see Merton (1992, pp. 318- 320) for a full discussion of the foregoing and Haug [2007, Section 6.9.1] for practical issues). You cannot perfectly hedge the call when the underlying follows the general jump diffusion $\left[\sigma > 0, \operatorname {var}(Y) \geq 0\right]$ . However, you can hedge out the continuous parts of the stock and option price movements. This leaves a risky hedge portfolio following a pure jump process (with stochastic jump size). If you follow the Black- Scholes hedge when you are short the option, then most of the time you earn more than the expected rate of return on the risky hedge portfolio. However, if one of those occasional jumps occurs (i.e., news arrives), you suffer a reasonably large loss. In the non- diversifiable jump case, the return to the hedge portfolio when there is a jump balances the return during normal time to some extent, but not well enough to make the equilibrium return on the hedge equal to the riskless rate; as mentioned above, the hedge is risky.

In general, there is no way to adjust the parameters of the hedge technique $\left(\sigma^{2}, \text{for example}\right)$ to get a better hedge (see Merton [1992, pp. 316- 317] for a full discussion

of the issues).⁵⁶

Finally, if the underlying asset price is modeled as a jump process, the standard Black- Scholes call option formula mis- prices the option. Both the magnitude and the direction of the mis- pricing of the Black- Scholes model relative to the jump model vary with the distributional assumption for the size of the jump component (Trippi et al. [1992]).