20762 第9讲：股指、外汇与期货期权

第 9 讲

股指、外汇与期货期权

Hull，第 15 章与第 16 章

每周经典文献

衍生品定价的核心原则在于：在不存在套利机会的市场中，任何完全对冲的投资组合必须获得无风险利率。

Lecture 9

Options on Stock Indices, Currencies and Futures

Hull, Chapter 15 & Chapter 16

Tip of the week

The key principle in derivative pricing is that a hedged position should earn the risk-free rate, in the absence of arbitrage opportunities.

学习目标

支付连续红利收益率资产的期权：
- 推导并计算期权价格的理论上下限
- 调整买卖权平价关系（Put-Call Parity）
- 调整二叉树模型以引入连续红利收益率（参见第8讲）
- 将 Black-Scholes 定价框架扩展至支付红利的资产期权
现实市场实例与实务应用：
- 股票指数期权（例如标普 500、标普 100 指数期权）• 投资组合保险策略
- 外汇期权 • Garman-Kohlhagen 定价模型与外国利率
- 期货期权 • Black (1976) 期货期权估值模型
- 深入理解市场为何优先交易期货期权而非现货期权

Learning Objectives

Options on assets paying a continuous dividend yield:
- Calculate the upper and lower bounds
- Adjust the put-call parity relation
- Adjust the binomial model to cater for a continuous dividend yield (see Lecture 8)
- Extend the Black-Scholes framework to price options on assets paying dividends
Real-world market examples & applications:
- Options on stock indices (e.g. S&P 500, S&P 100) • Portfolio Insurance strategies
- Options on currencies • Garman-Kohlhagen model and foreign interest rates
- Options on futures contracts • Black's 1976 model to value futures options
- Why are futures options used instead of options on the physical spot asset?

支付连续复利红利收益率资产的欧式期权

考虑某资产上的欧式看涨与看跌期权：资产当前价值为 $S_{0}$ ，支付连续复利红利率 $q$ ；期权执行价格为 $K$ ，到期期限为 $T$ 。

按年率 $q$ 支付连续红利会导致该资产的预期资本收益率比不支付红利的情况减少 $q$ 。
在以下两种情况下，时刻 $T$ 资产价格的概率分布完全一致：
- 资产当前价格为 $S_{0}$ ，并在存续期内支付连续红利率 $q$
- 资产当前价格设定为 $S_{0} e^{- q T}$ ，且不提供任何红利收益
我们可以通过将基础资产价格下调至 $S_{0} e^{- q T}$ ，随后假设其不支付红利，以此直接对欧式期权进行估值。
基本转换法则： 在所有 Black-Scholes 定价公式和买卖权平价关系中，直接将 $S_{0}$ 替换为 $S_{0} e^{- q T}$ 。

European Options on Assets paying Continuously Compounded Dividend Yield

We consider a European call and European put with strike $K$ and maturity $T$ on an asset with value $S_{0}$ that pays a continuously compounded dividend yield of $q$ .

The payment of a dividend yield at rate $q$ causes the expected return from the asset to be less than it would otherwise be, by the amount $q$ .
The same probability distribution for the asset price at time $T$ is obtained in each of the following cases:
- The asset price starts at price $S_{0}$ and pays a dividend yield $q$
- The asset price starts at price $S_{0} e^{- q T}$ and provides no dividend income
We can value European options by reducing the stock price to $S_{0} e^{- q T}$ and then assuming there is no dividend.
Fundamental Rule: In all Black-Scholes formulas and parity relations, replace $S_{0}$ with $S_{0} e^{- q T}$ .

价格上下限与买卖权平价定理

看涨期权价格上下限

理论下限：
$c \geq max (S_{0} e^{- q T} - K e^{- r T}, 0)$

理论上限：
$c \leq S_{0} e^{- q T}$

看跌期权价格上下限

理论下限：
$p \geq max (K e^{- r T} - S_{0} e^{- q T}, 0)$

理论上限：
$p \leq K e^{- r T}$

连续红利收益率下的买卖权平价定理

c + K e^{- r T} = p + S_{0} e^{- q T}

回顾： 对于支付现值为 $D$ 的离散红利股票，平价关系式为 $c + K e^{- r T} = p + S_{0} - D$ 。在此处，连续红利率将现货价格折现了 $e^{- q T}$ 。

Bounds and Put-Call Parity

Call Bounds

Lower Bound:
$c \geq max (S_{0} e^{- q T} - K e^{- r T}, 0)$

Upper Bound:
$c \leq S_{0} e^{- q T}$

Put Bounds

Lower Bound:
$p \geq max (K e^{- r T} - S_{0} e^{- q T}, 0)$

Upper Bound:
$p \leq K e^{- r T}$

Put-Call Parity with Continuous Dividend Yield

c + K e^{- r T} = p + S_{0} e^{- q T}

Recall: For a stock with discrete dividends of present value $D$ , the relationship was $c + K e^{- r T} = p + S_{0} - D$ . Here, continuous dividend yield discounts the spot asset by $e^{- q T}$ .

二叉树模型：连续红利收益率

令 $p$ 为风险中性世界中的上涨概率。
在“风险中性”世界中：
- 所有资产的总预期收益率均为无风险利率 $r$ 。
- 支付连续红利率 $q$ 的股票，其预期价格增长率（资本利得）为 $r - q$ 。因此：

其中

S_{Δ t}

、

f_{Δ t}

分别为一步

Δ t

末的股价与期权价值。

S_{0} = e^{- (r - q) Δ t} E [S_{Δ t}] = e^{- (r - q) Δ t} [p S_{u} + (1 - p) S_{d}] (1)

f_{0} = e^{- r Δ t} E [f_{Δ t}] = e^{- r Δ t} [p f_{u} + (1 - p) f_{d}] (2)

由式 (1)，两端除以 $S_{0}$ 可得 $e^{(r - q) Δ t} = p u + (1 - p) d$ 。求解 $p$ 得：
$p = \frac{e^{(r - q) Δ t} - d}{u - d}$
由式 (2)，每个树节点处的期权价值均按无风险利率 $r$ 折现： $f_{0} = e^{- r Δ t} [p f_{u} + (1 - p) f_{d}]$

Binomial Model: Continuous Dividend Yield

Let $p$ be the “risk-neutral” probability of an up movement.
In a “risk-neutral” world:
- The expected return on every asset is the risk-free rate $r$ .
- The expected capital return on a stock paying a continuous yield $q$ is $r - q$ . Thus:

where

S_{Δ t}

and

f_{Δ t}

are the stock price and option value at the end of one step

Δ t

S_{0} = e^{- (r - q) Δ t} E [S_{Δ t}] = e^{- (r - q) Δ t} [p S_{u} + (1 - p) S_{d}] (1)

f_{0} = e^{- r Δ t} E [f_{Δ t}] = e^{- r Δ t} [p f_{u} + (1 - p) f_{d}] (2)

From (1), dividing by $S_{0}$ gives $e^{(r - q) Δ t} = p u + (1 - p) d$ . Solving for $p$ :
$p = \frac{e^{(r - q) Δ t} - d}{u - d}$
From (2), the option value at each node is discounted at the risk-free rate $r$ : $f_{0} = e^{- r Δ t} [p f_{u} + (1 - p) f_{d}]$

Black-Scholes 定价公式（默顿扩展 / Merton Extension）

欧式看涨期权价格为：
$c = S_{0} e^{- q T} N (d_{1}) - K e^{- r T} N (d_{2})$
欧式看跌期权价格为：
$p = K e^{- r T} N (- d_{2}) - S_{0} e^{- q T} N (- d_{1})$

其中：（将 $S_{0}$ 替换为 $S_{0} e^{- q T}$ ）

\begin{aligned} d_{1} & = \frac{\ln (S_{0} e^{- q T} / K) + (r + \frac{1}{2} σ^{2}) T}{σ \sqrt{T}} \\ = \frac{\ln (S_{0} / K) + \ln e^{- q T} + (r + \frac{1}{2} σ^{2}) T}{σ \sqrt{T}} \\ = \frac{\ln (S_{0} / K) + (r - q + \frac{1}{2} σ^{2}) T}{σ \sqrt{T}} \end{aligned}

d_{2} = \frac{\ln (S_{0} / K) + (r - q - \frac{1}{2} σ^{2}) T}{σ \sqrt{T}} = d_{1} - σ \sqrt{T}

Black-Scholes Pricing Formula (Merton Extension)

The price of the European call option is:
$c = S_{0} e^{- q T} N (d_{1}) - K e^{- r T} N (d_{2})$
The price of the European put option is:
$p = K e^{- r T} N (- d_{2}) - S_{0} e^{- q T} N (- d_{1})$

where: (replace $S_{0}$ by $S_{0} e^{- q T}$ )

\begin{aligned} d_{1} & = \frac{\ln (S_{0} e^{- q T} / K) + (r + \frac{1}{2} σ^{2}) T}{σ \sqrt{T}} \\ = \frac{\ln (S_{0} / K) + \ln e^{- q T} + (r + \frac{1}{2} σ^{2}) T}{σ \sqrt{T}} \\ = \frac{\ln (S_{0} / K) + (r - q + \frac{1}{2} σ^{2}) T}{σ \sqrt{T}} \end{aligned}

d_{2} = \frac{\ln (S_{0} / K) + (r - q - \frac{1}{2} σ^{2}) T}{σ \sqrt{T}} = d_{1} - σ \sqrt{T}

股指期权 – 定价原理

股票指数可视为提供连续复利红利率 $q$ 的金融资产（平均反映其所有成分股的红利收益）。
二叉树模型：
$f_{0} = e^{- r Δ t} [p f_{u} + (1 - p) f_{d}], 其中 p = \frac{e^{(r - q) Δ t} - d}{u - d}$
其中 $Δ t$ 为每个时间步长的长度。
Black-Scholes 公式：
$c = S_{0} e^{- q T} N (d_{1}) - K e^{- r T} N (d_{2}), p = K e^{- r T} N (- d_{2}) - S_{0} e^{- q T} N (- d_{1})$
其中： $d_{1} = \frac{\ln (S_{0} / K) + (r - q + \frac{1}{2} σ^{2}) T}{σ \sqrt{T}}, d_{2} = d_{1} - σ \sqrt{T}$

Index Options – Pricing

An index is an asset that provides a continuously compounded dividend yield $q$ (on average across its component stocks).
Binomial Model:
$f_{0} = e^{- r Δ t} [p f_{u} + (1 - p) f_{d}], with p = \frac{e^{(r - q) Δ t} - d}{u - d}$
where $Δ t$ is the length of each time step.
Black-Scholes Formula:
$c = S_{0} e^{- q T} N (d_{1}) - K e^{- r T} N (d_{2}), p = K e^{- r T} N (- d_{2}) - S_{0} e^{- q T} N (- d_{1})$
with: $d_{1} = \frac{\ln (S_{0} / K) + (r - q + \frac{1}{2} σ^{2}) T}{σ \sqrt{T}}, d_{2} = d_{1} - σ \sqrt{T}$

股指期权 – 用远期价格定价

也可以用指数的远期价格 $F_{0}$ 给期权定价，这份远期合约与期权同一天到期（期限都是 $T$ ）：
$F_{0} = S_{0} e^{(r - q) T} ⟹ F_{0} e^{- r T} = S_{0} e^{- q T}$
用远期价格写出的 Black-Scholes 公式：
$c = F_{0} e^{- r T} N (d_{1}) - K e^{- r T} N (d_{2}) = e^{- r T} [F_{0} N (d_{1}) - K N (d_{2})]$ $p = K e^{- r T} N (- d_{2}) - F_{0} e^{- r T} N (- d_{1}) = e^{- r T} [K N (- d_{2}) - F_{0} N (- d_{1})]$
其中： $d_{1} = \frac{\ln (F_{0} / K) + \frac{1}{2} σ^{2} T}{σ \sqrt{T}}, d_{2} = \frac{\ln (F_{0} / K) - \frac{1}{2} σ^{2} T}{σ \sqrt{T}} = d_{1} - σ \sqrt{T}$
好处：公式里没有指数的红利收益率 $q$ 。只要市场上能看到同期限的期货（远期）价格 $F_{0}$ ，就不用估计 $q$ 。

Index Options – Pricing using Forward Price

Options can be priced by using the forward price $F_{0}$ of the index for a contract with maturity $T$ (where $T =$ option's maturity):
$F_{0} = S_{0} e^{(r - q) T} ⟹ F_{0} e^{- r T} = S_{0} e^{- q T}$
Black-Scholes Formula in terms of Forward Price:
$c = F_{0} e^{- r T} N (d_{1}) - K e^{- r T} N (d_{2}) = e^{- r T} [F_{0} N (d_{1}) - K N (d_{2})]$ $p = K e^{- r T} N (- d_{2}) - F_{0} e^{- r T} N (- d_{1}) = e^{- r T} [K N (- d_{2}) - F_{0} N (- d_{1})]$
with: $d_{1} = \frac{\ln (F_{0} / K) + \frac{1}{2} σ^{2} T}{σ \sqrt{T}}, d_{2} = \frac{\ln (F_{0} / K) - \frac{1}{2} σ^{2} T}{σ \sqrt{T}} = d_{1} - σ \sqrt{T}$
Key Advantage: The dividend yield $q$ on the index does not appear in these equations, thus there is no need for it to be estimated!

股指期权 – 合约规格

标普 500™ 指数期权（CBOE 交易代码：SPX）

合约参数	规格详情
指数	标普 500 指数是由来自广泛行业的 500 只股票构成的市值加权指数。成分股按总市值（股价 × 已发行股数）加权，因此成分股价格变动的影响与其总市值成正比；对 500 只股票求和后除以预先设定的基期值，基期值会因并购、配股、成分股替换等情况调整。
合约乘数	$100 / 指数点（例如指数处于 1,000 点时 → 合约名义价值为 $100,000）。
权利金报价	以小数形式报出，1 点等于 $100。最小变动价位：低于 3.00 的期权为 0.05（$5.00），其他为 0.10（$10.00）。
执行价格间距	5 点（远月合约为 25 点间距）。
到期月份	三个近月，加上 3 月季度循环（3月、6月、9月、12月）中的三个后续月份。
到期日	到期月份第三个周五之后的周六。
行权方式	欧式 – SPX 期权通常仅在到期日前最后一个营业日方可行权。
最后交易日	SPX 期权通常在计算行权结算价的前一个营业日（通常为周四）停止交易。
行权结算	行权结算价 SET 根据到期日前最后一个营业日（通常为周五）各成分股在主要市场的开盘（第一笔）成交价计算。若某只股票当天未开盘，则使用其最后成交价。结算金额 = (SET − 执行价) × $100。行权后于到期后的下一个营业日交付现金。
保证金	距到期 9 个月或以内的看跌或看涨期权买入，必须全额支付。无保障的看跌或看涨期权卖方必须存入并维持期权收入的 100% 加上合约总价值（当前指数点位 × $100）的 15%，再减去期权虚值的金额（如有），且不低于规定的最低额。

Index Options – Contract Specifications

S&P 500™ Index Options (CBOE Symbol: SPX)

Parameter	Specification Details
Underlying	The Standard & Poor’s 500 Index is a capitalization-weighted index of 500 stocks from a broad range of industries. Component stocks are weighted by total market value (share price × shares outstanding), so the impact of a component’s price change is proportional to its total market value; these are summed over all 500 stocks and divided by a predetermined base value, which is adjusted for mergers, acquisitions, stock rights, substitutions, etc.
Multiplier	$100 per index point (e.g. index at 1,000 → contract notional value = $100,000).
Premium Quote	Stated in decimals. One point equals $100. Minimum tick: 0.05 ($5.00) for options trading below 3.00, and 0.10 ($10.00) for all other series.
Strike Price Intervals	Five points. 25-point intervals for far months.
Expiration Months	Three near-term months followed by three additional months from the March quarterly cycle (March, June, September, December).
Expiration Date	Saturday following the third Friday of the expiration month.
Exercise Style	European – SPX options generally may be exercised only on the last business day before expiration.
Last Trading Day	Trading in SPX options ordinarily ceases on the business day (usually a Thursday) preceding the day on which the exercise-settlement value is calculated.
Settlement of Option Exercise	The exercise-settlement value, SET, is calculated from the opening (first) reported sales price in the primary market of each component stock on the last business day (usually a Friday) before the expiration date. If a stock does not open that day, its last reported price is used. The settlement amount equals (SET − strike) × $100. Exercise results in delivery of cash on the business day following expiration.
Margin	Purchases of puts or calls with 9 months or less until expiration must be paid for in full. Writers of uncovered puts or calls must deposit and maintain 100% of the option proceeds plus 15% of the aggregate contract value (current index level × $100), minus the amount (if any) by which the option is out of the money, subject to a minimum.

股指期权 – 市场行情报价

2005年12月1日；标普 100 指数期权（OEX）；指数点位 580.22

看涨期权 CALL (2006年1月)							执行价	看跌期权 PUT
收盘价	开盘价	最高价	最低价	最新价	成交量	未平仓量		未平仓量	成交量	最新价	最低价	最高价	开盘价	收盘价
181.8	0	0	0	0	0	0	400	100	0	0	0	0	0	0.25
161.9	0	0	0	0	0	0	420	10	0	0	0	0	0	0.25
142	0	0	0	0	0	0	440	240	0	0	0	0	0	0.25
122.2	0	0	0	0	0	0	460	555	0	0	0	0	0	0.25
112.2	0	0	0	0	0	0	470	12	0	0	0	0	0	0.25
102.3	0	0	0	0	0	0	480	142	0	0	0	0	0	0.4
97.3	0	0	0	0	0	0	485	0	0	0	0	0	0	0.4
92.4	0	0	0	0	0	0	490	477	20	0.15	0.15	0.15	0.15	0.15
87.4	0	0	0	0	0	0	495	0	0	0	0	0	0	0.4
82.5	0	0	0	0	0	0	500	1153	0	0	0	0	0	0.4
77.6	0	0	0	0	0	0	505	0	5	0.25	0.25	0.25	0.25	0.25
72.6	0	0	0	0	0	0	510	2288	0	0	0	0	0	0.5
67.7	0	0	0	0	0	0	515	51	49	0.4	0.4	0.4	0.4	0.4
62.8	0	0	0	0	0	0	520	1823	0	0	0	0	0	0.65
57.9	0	0	0	0	0	0	525	210	0	0	0	0	0	0.75
53.1	0	0	0	0	0	3	530	2111	6	0.7	0.7	1	1	0.7
48.2	0	0	0	0	0	0	535	1180	64	0.8	0.8	0.8	0.8	0.8
43.5	0	0	0	0	0	38	540	2614	99	1.1	1.1	1.25	1.25	1.1
38.6	0	0	0	0	0	0	545	1318	25	1.25	1.25	1.45	1.45	1.25
33.9	0	0	0	0	0	301	550	2578	43	1.5	1.5	1.75	1.75	1.5
29.4	0	0	0	0	0	0	555	535	10	1.9	1.9	2.2	2.2	1.9
25	23.8	25.7	23.8	25.7	37	832	560	2952	252	2.7	2.3	3	3	2.7
20.8	21	21	21	21	8	30	565	532	151	3.5	3.2	3.5	3.5	3.5
16.3	16.2	16.8	16.2	16.3	40	1285	570	1489	46	4.6	3.8	5.7	5.7	4.6
12.5	12.3	12.5	12.3	12.5	13	740	575	668	176	5.7	5.2	7.1	7.1	5.7
9.7	9.3	10.1	8.5	9.8	374	4391	580	743	137	7.8	7.2	9	8.6	7.8
7	6.4	7.1	6.3	7	18	737	585	135	63	9.8	9.7	11.1	11.1	9.9
4.7	4.1	5	3.9	4.7	448	4662	590	96	0	0	0	0	0	14.1
3	2.75	3.1	2.5	3.1	423	3085	595	0	0	0	0	0	0	17.7
1.75	1.5	1.75	1.4	1.75	1332	4815	600	9	0	0	0	0	0	22.1
0.9	0.75	0.9	0.75	0.9	1005	1825	605	0	0	0	0	0	0	26.8
0.45	0.4	0.45	0.4	0.45	17	1236	610	0	0	0	0	0	0	31.7
0.2	0.2	0.2	0.2	0.2	7	842	620	0	0	0	0	0	0	41.8
0.25	0	0	0	0	0	100	630	0	0	0	0	0	0	51.8
0.25	0	0	0	0	0	0	640	0	0	0	0	0	0	61.8
0.25	0	0	0	0	0	0	660	0	0	0	0	0	0	81.8

（数据来源：www.marketdataexpress.com • 2006年1月到期） 市场观察： 指数为 580.22 时，平值 580 看涨期权最新价 9.8、看跌期权最新价 7.8（高亮行）。

Index Options – Quotes

1 Dec 2005; S&P 100 Index Options (OEX); Index Level 580.22

CALL (Jan 2006)							STRIKE	PUT
Close	Open	High	Low	Last	Vol	OI		OI	Vol	Last	Low	High	Open	Close
181.8	0	0	0	0	0	0	400	100	0	0	0	0	0	0.25
161.9	0	0	0	0	0	0	420	10	0	0	0	0	0	0.25
142	0	0	0	0	0	0	440	240	0	0	0	0	0	0.25
122.2	0	0	0	0	0	0	460	555	0	0	0	0	0	0.25
112.2	0	0	0	0	0	0	470	12	0	0	0	0	0	0.25
102.3	0	0	0	0	0	0	480	142	0	0	0	0	0	0.4
97.3	0	0	0	0	0	0	485	0	0	0	0	0	0	0.4
92.4	0	0	0	0	0	0	490	477	20	0.15	0.15	0.15	0.15	0.15
87.4	0	0	0	0	0	0	495	0	0	0	0	0	0	0.4
82.5	0	0	0	0	0	0	500	1153	0	0	0	0	0	0.4
77.6	0	0	0	0	0	0	505	0	5	0.25	0.25	0.25	0.25	0.25
72.6	0	0	0	0	0	0	510	2288	0	0	0	0	0	0.5
67.7	0	0	0	0	0	0	515	51	49	0.4	0.4	0.4	0.4	0.4
62.8	0	0	0	0	0	0	520	1823	0	0	0	0	0	0.65
57.9	0	0	0	0	0	0	525	210	0	0	0	0	0	0.75
53.1	0	0	0	0	0	3	530	2111	6	0.7	0.7	1	1	0.7
48.2	0	0	0	0	0	0	535	1180	64	0.8	0.8	0.8	0.8	0.8
43.5	0	0	0	0	0	38	540	2614	99	1.1	1.1	1.25	1.25	1.1
38.6	0	0	0	0	0	0	545	1318	25	1.25	1.25	1.45	1.45	1.25
33.9	0	0	0	0	0	301	550	2578	43	1.5	1.5	1.75	1.75	1.5
29.4	0	0	0	0	0	0	555	535	10	1.9	1.9	2.2	2.2	1.9
25	23.8	25.7	23.8	25.7	37	832	560	2952	252	2.7	2.3	3	3	2.7
20.8	21	21	21	21	8	30	565	532	151	3.5	3.2	3.5	3.5	3.5
16.3	16.2	16.8	16.2	16.3	40	1285	570	1489	46	4.6	3.8	5.7	5.7	4.6
12.5	12.3	12.5	12.3	12.5	13	740	575	668	176	5.7	5.2	7.1	7.1	5.7
9.7	9.3	10.1	8.5	9.8	374	4391	580	743	137	7.8	7.2	9	8.6	7.8
7	6.4	7.1	6.3	7	18	737	585	135	63	9.8	9.7	11.1	11.1	9.9
4.7	4.1	5	3.9	4.7	448	4662	590	96	0	0	0	0	0	14.1
3	2.75	3.1	2.5	3.1	423	3085	595	0	0	0	0	0	0	17.7
1.75	1.5	1.75	1.4	1.75	1332	4815	600	9	0	0	0	0	0	22.1
0.9	0.75	0.9	0.75	0.9	1005	1825	605	0	0	0	0	0	0	26.8
0.45	0.4	0.45	0.4	0.45	17	1236	610	0	0	0	0	0	0	31.7
0.2	0.2	0.2	0.2	0.2	7	842	620	0	0	0	0	0	0	41.8
0.25	0	0	0	0	0	100	630	0	0	0	0	0	0	51.8
0.25	0	0	0	0	0	0	640	0	0	0	0	0	0	61.8
0.25	0	0	0	0	0	0	660	0	0	0	0	0	0	81.8

(Source: www.marketdataexpress.com • Jan 2006 expiry) Market Observation: With the index at 580.22, the ATM 580 call last traded at 9.8 and the ATM 580 put at 7.8 (highlighted row).

利用股指期权构建投资组合保险

投资组合保险策略的核心在于构建保护性看跌期权（Protective Put） – 建立股票投资组合多头的同时，买入股指看跌期权作为下行保险。

每张股指期权合约覆盖的名义价值为 $$ 100 \times S_{0}$ ，其中 $S_{0}$ 为当前指数点位。
价值为 $P$ 、贝塔系数为 $β$ 的股票组合，可通过买入 $N$ 张看跌期权进行完全保险：
$N = β \frac{P}{100 \times S_{0}}$
看跌期权的执行价格 $K$ 可按需选定，用以锁定投资者所要求的最低组合保底价值（floor）。
保险成本即为购买这些看跌期权所支付的权利金。
相比静态套期保值 / 远期抛售的显著优势：
- 期权多头代表权利而非义务 – 在防范下行风险的同时，完整保留组合的上行获利潜力！
- 投资者可自由选择任意执行价格 $K$ ，灵活定制免赔额度与期权成本。

Index Options for Portfolio Insurance

A portfolio insurance strategy is to construct a protective put – a long position in the equity portfolio and a long position in put index options.

One index option contract covers $$ 100 \times S_{0}$ , where $S_{0}$ is the current index level.
A portfolio of value $P$ with beta $β$ is insured by buying $N$ put options:
$N = β \frac{P}{100 \times S_{0}}$
The strike price $K$ of the put options is chosen to establish the desired minimum portfolio value (the floor).
The cost of the insurance is the option premium required to purchase the puts.
Advantages over static hedging / forward sales:
- A long option represents a right, not an obligation – downside protection while retaining full upside potential!
- Flexibility to choose any strike price $K$ to customize the deductible level and premium cost.

简单例子 – 投资组合保险 ( $β = 1$ )

某 $β = 1$ 的股票组合价值为 $500,000。指数点位为 1,000，无风险利率 5%，指数红利率 2%（均为年率），且组合与指数的红利率完全相同。如何通过交易在未来 1 年内防范组合价值跌破 $450,000？

所需期权合约张数： $N = 1 \times \frac{$ 500, 000}{100 \times 1, 000} = 5 张看跌期权合约$
执行价格计算： 要求的保底价值为 $450,000 → 允许的最大回撤幅度为 10%（ $β = 1$ 时 $K$ 与 $r$ 、 $q$ 无关）。 $K = 0.90 \times S_{0} = 0.90 \times 1, 000 = 900$ 操作策略： 买入 5 张执行价格为 900、期限为 1 年的股指看跌期权合约。
假设指数到期下跌至 $S_{T} = 880$ （跌幅 −12%）：
- 到期现货股票组合价值： $(1 - 0.12) \times $ 500, 000 = $ 440, 000$
- 每张期权合约的到期收益： $max (900 - 880, 0) = 20$ 个指数点
- 期权总收益： $20 \times 5 \times $ 100 = $ 10, 000$
- 被保险组合总价值： $$ 440, 000 + $ 10, 000 = $ 450, 000$ （精准达成保底目标！）。

Simple Example – Portfolio Insurance ( $β = 1$ )

A stock portfolio with $β = 1$ is worth $500,000. The index stands at 1,000; $r = 5 %$ and $q = 2 %$ (annual), with identical dividend yields on portfolio and index. What trade protects against the value falling below $450,000 within 1 year?

Number of contracts required: $N = 1 \times \frac{$ 500, 000}{100 \times 1, 000} = 5 put option contracts$
Strike calculation: The required floor is $450,000 → the maximum allowed drawdown is 10% (for $β = 1$ , $K$ is independent of $r, q$ ). $K = 0.90 \times S_{0} = 0.90 \times 1, 000 = 900$ Strategy: Buy 5 index put option contracts with strike 900 and 1-year maturity.
Suppose the index falls to $S_{T} = 880$ at expiry (−12%):
- Spot portfolio value at expiry: $(1 - 0.12) \times $ 500, 000 = $ 440, 000$
- Payoff per option contract: $max (900 - 880, 0) = 20$ index points
- Total option payoff: $20 \times 5 \times $ 100 = $ 10, 000$
- Total insured portfolio value: $$ 440, 000 + $ 10, 000 = $ 450, 000$ (floor met exactly!).

简单例子 – 投资组合保险 ( $β = 2$ )

数字取最简单的：期限 1 年，利率 5%，红利率 2%

已知：

β = 2

的组合价值 $500,000，指数 1,000 点；无风险利率 5%，红利率 2%（均为年率）。目标：1 年内不跌破 $450,000。

第一步，算合约张数： $N = β \frac{P}{100 \times S_{0}} = 2 \times \frac{$ 500, 000}{100 \times 1, 000} = 10 张看跌期权合约$
第二步，用 CAPM 算执行价格：
- 保底 $450,000 → 组合资本收益率最多 $- 10 %$ ；加上 2% 红利，目标总收益率 $E (R_{p}) = - 10 % + 2 % = - 8 %$ 。
- CAPM： $- 8 % = 5 % + 2 (r_{m} - 5 %) ⟹ r_{m} = - 1.5 %$ （市场总收益率）；指数资本收益率 $= - 1.5 % - 2 % = - 3.5 %$ ，故执行价格 $K = 0.965 \times 1, 000 = 965$ 。

交易策略：买入 10 张执行价格 965、期限 1 年的股指看跌期权。下一页检验：指数跌 10% 时能否守住保底？

Simple Example – Portfolio Insurance ( $β = 2$ )

Simplest round numbers: 1-year term, 5% rate, 2% yield

Given: a

β = 2

portfolio worth $500,000; index at 1,000; risk-free rate 5%, dividend yield 2% (both annual). Goal: keep the value from falling below $450,000 over 1 year.

Step one, number of contracts: $N = β \frac{P}{100 \times S_{0}} = 2 \times \frac{$ 500, 000}{100 \times 1, 000} = 10 put contracts$
Step two, strike from CAPM:
- Floor $450,000 → capital return at most $- 10 %$ ; plus the 2% dividend, target total return $E (R_{p}) = - 10 % + 2 % = - 8 %$ .
- CAPM: $- 8 % = 5 % + 2 (r_{m} - 5 %) ⟹ r_{m} = - 1.5 %$ ; index capital return $= - 1.5 % - 2 % = - 3.5 %$ , so $K = 0.965 \times 1, 000 = 965$ .

Strategy: buy 10 index puts with strike 965 and 1-year maturity. Next page: if the index falls 10%, is the floor preserved?

简单例子 – 投资组合保险效果检验

接上一页：10 张

K = 965

的看跌期权，假设到期指数

S_{T} = 900

第一步，指数的总收益率： $\frac{900 - 1, 000}{1, 000} + 2 % = - 10 % + 2 % = - 8 %$
第二步，CAPM 给出组合的总收益率： $E (R_{p}) = r + β (r_{m} - r) = 5 % + 2 (- 8 % - 5 %) = - 21 %$
第三步，扣掉红利得资本收益率，算现货价值： $- 21 % - 2 % = - 23 %$ ，所以组合价值 $= (1 - 0.23) \times $ 500, 000 = $ 385, 000$ 。
第四步，看跌期权收益：每张合约 $max (965 - 900, 0) = 65$ 个指数点，共 $65 \times 10 \times $ 100 = $ 65, 000$ 。

结论：被保险组合总价值

= $ 385, 000 + $ 65, 000 = $ 450, 000

，精准守住保底！课本原题（3 个月、12% 利率）步骤完全一样，完整解答见《Hull 习题与解答》第 9 讲。

Simple Example – Portfolio Insurance Outcome Check

Continued: 10 puts with

K = 965

; suppose the index at expiry is

S_{T} = 900

Step one, total return on the index: $\frac{900 - 1, 000}{1, 000} + 2 % = - 10 % + 2 % = - 8 %$
Step two, CAPM gives the portfolio's total return: $E (R_{p}) = r + β (r_{m} - r) = 5 % + 2 (- 8 % - 5 %) = - 21 %$
Step three, subtract the dividend to get the capital return, then compute the spot value: $- 21 % - 2 % = - 23 %$ , so the portfolio value is $(1 - 0.23) \times $ 500, 000 = $ 385, 000$ .
Step four, put payoff: per contract $max (965 - 900, 0) = 65$ index points, totalling $65 \times 10 \times $ 100 = $ 65, 000$ .

Conclusion: total insured portfolio value

= $ 385, 000 + $ 65, 000 = $ 450, 000

– the floor is held exactly! The original Hull problem (3 months, 12% interest rate) follows exactly the same steps; full solution: see Hull Exercises & Solutions, Lecture 9.

再加一点：合约张数取整与 $β$ 调整

只能买整数张合约、执行价按交易所间距取整，怎么办？

参数设定： 投资组合价值 $P = $ 540, 000$ ，指数 $S_{0} = 1, 000$ ，1 年保底目标 $= $ 486, 000$ ，合约乘数 $= $ 100$ ； $r = 5 %$ ， $q = 2 %$ （仅 (c) 需要）；1 年期为远月合约，执行价间距 25 点。

(a) 基准情形（ $β = 1$ ）： 看跌期权张数： $N = \frac{$ 540, 000}{100 \times 1, 000} = 5.4 \to 6 张合约$ （只能买整数张，向上取整）。
允许的最大资本跌幅 $= - 10 %$ ，执行价格 $K = 0.90 \times 1, 000 = 900$ 。买入 6 张 1 年到期、 $K = 900$ 的看跌期权。
(b) 当指数下跌至 $S_{T} = 800$ （跌幅 −20%）时： 现货股票价值： $(1 - 0.20) \times $ 540, 000 = $ 432, 000$ 。
看跌期权收益： $6 \times (900 - 800) \times $ 100 = 6 \times $ 10, 000 = + $ 60, 000$ 。
组合总价值： $$ 432, 000 + $ 60, 000 = $ 492, 000$ （张数向上取整到 6 张，比 $486,000 保底多出 $6,000）。
(c) 非单位贝塔调整（ $β = 2$ ； $N^{*}$ ： $β$ 调整后未取整的看跌期权张数）： $N^{*} = 2 \times \frac{$ 540, 000}{100 \times 1, 000} = 10.8 \to 11 张合约$ 看跌期权依据现货价格变动行权，而 CAPM 决定的是总收益率！
组合目标总收益率： $E (R_{p}) = - 10 % + 2 % (红利) = - 8 %$ ；
$- 8 % = 5 % + 2 (R_{m} - 5 %) ⟹ R_{m} = - 1.5 %$ （市场总收益率）→ 资本跌幅 $= - 1.5 % - 2 % = - 3.5 %$ 。
所需执行价格： $K = 1, 000 \times (1 - 0.035) = 965$ ，按交易所 25 点间距向上取整得 $K = 975$ 。
更高的 $β$ 不仅需要更多合约张数（11 张 vs 6 张），而且需要更高的执行价格（975 vs 900）！

One Step Further – Rounding the Contract Count & the $β$ Adjustment

Only whole contracts, and strikes only at exchange intervals – what now?

Setup: portfolio value $P = $ 540, 000$ , index $S_{0} = 1, 000$ , 1-year floor target $= $ 486, 000$ , multiplier $= $ 100$ ; $r = 5 %$ , $q = 2 %$ (needed only for (c)); 1-year options are far-month, strikes 25 points apart.

(a) Base case ( $β = 1$ ): Number of puts: $N = \frac{$ 540, 000}{100 \times 1, 000} = 5.4 \to 6 contracts$ (whole contracts only – round up).
Maximum allowed capital drop $= - 10 %$ ; strike $K = 0.90 \times 1, 000 = 900$ . Buy 6 one-year puts with $K = 900$ .
(b) When the index falls to $S_{T} = 800$ (−20%): Spot equity value: $(1 - 0.20) \times $ 540, 000 = $ 432, 000$ .
Put payoff: $6 \times (900 - 800) \times $ 100 = 6 \times $ 10, 000 = + $ 60, 000$ .
Total portfolio value: $$ 432, 000 + $ 60, 000 = $ 492, 000$ (rounding up to 6 puts leaves $6,000 above the $486,000 floor).
(c) Non-unit beta adjustment ( $β = 2$ ; $N^{*}$ = $β$ -adjusted, unrounded put count): $N^{*} = 2 \times \frac{$ 540, 000}{100 \times 1, 000} = 10.8 \to 11 contracts$ Puts pay off on the spot price move, but CAPM pins down the total return!
Portfolio target total return: $E (R_{p}) = - 10 % + 2 % (dividend) = - 8 %$ ;
$- 8 % = 5 % + 2 (R_{m} - 5 %) ⟹ R_{m} = - 1.5 %$ (market total return) → capital drop $= - 1.5 % - 2 % = - 3.5 %$ .
Required strike: $K = 1, 000 \times (1 - 0.035) = 965$ ; rounded up to the 25-point exchange interval, $K = 975$ .
A higher $β$ needs not only MORE contracts (11 vs 6) but also a HIGHER strike (975 vs 900)!

外汇期权 – 定价原理（Garman-Kohlhagen 模型）

外国货币可视为提供连续红利率为 $r_{f}$ （外国无风险利率）的金融资产。
- 直接使用连续红利股票的期权定价公式，其中 $S_{0}$ 为即期汇率（每单位外币折合的本币金额），红利率设定为 $q = r_{f}$ 。
买卖权平价定理：
$c + K e^{- r T} = p + S_{0} e^{- r_{f} T}$
二叉树模型：
$f_{0} = e^{- r Δ t} [p f_{u} + (1 - p) f_{d}], 其中 p = \frac{e^{(r - r_{f}) Δ t} - d}{u - d}$
Black-Scholes 公式（Garman-Kohlhagen 1983）：
$c = S_{0} e^{- r_{f} T} N (d_{1}) - K e^{- r T} N (d_{2}), p = K e^{- r T} N (- d_{2}) - S_{0} e^{- r_{f} T} N (- d_{1})$
其中： $d_{1} = \frac{\ln (S_{0} / K) + (r - r_{f} + \frac{1}{2} σ^{2}) T}{σ \sqrt{T}}, d_{2} = d_{1} - σ \sqrt{T}$

Currency Options – Pricing (Garman-Kohlhagen)

A foreign currency is an asset that provides a continuous dividend yield $r_{f}$ (foreign risk-free interest rate).
- Use the formula for an option on a stock paying a continuous dividend yield, where $S_{0}$ is the exchange rate (domestic currency per unit of foreign currency) and $q = r_{f}$ .
Put-Call Parity:
$c + K e^{- r T} = p + S_{0} e^{- r_{f} T}$
Binomial Model:
$f_{0} = e^{- r Δ t} [p f_{u} + (1 - p) f_{d}], with p = \frac{e^{(r - r_{f}) Δ t} - d}{u - d}$
Black-Scholes Formula (Garman-Kohlhagen 1983):
$c = S_{0} e^{- r_{f} T} N (d_{1}) - K e^{- r T} N (d_{2}), p = K e^{- r T} N (- d_{2}) - S_{0} e^{- r_{f} T} N (- d_{1})$
with: $d_{1} = \frac{\ln (S_{0} / K) + (r - r_{f} + \frac{1}{2} σ^{2}) T}{σ \sqrt{T}}, d_{2} = d_{1} - σ \sqrt{T}$

简单例子 – 外汇期权买卖权平价

数字取最简单的：即期汇率 = 执行价 = $1.20，期限 1 年

已知：1 年期欧式欧元看涨期权，执行价格 $K = $ 1.20$ ，市价 $c = $ 0.1107$ （ $σ = 20 %$ ）；即期汇率 $S_{0} = $ 1.20$ /欧元；美元利率 $r = 5 %$ ，欧元利率 $r_{f} = 2 %$ （连续复利）。求同执行价的欧式看跌期权价格 $p$ 。

\begin{aligned} c + K e^{- r T} & = p + S_{0} e^{- r_{f} T} \\ \Rightarrow p & = c + K e^{- r T} - S_{0} e^{- r_{f} T} \end{aligned}

折现执行价格（按美元利率 $r = 5 %$ ）： $K e^{- r T} = 1.20 \times e^{- 0.05} = 1.20 \times 0.95123 = $ 1.14148$
折现即期汇率（按欧元利率 $r_{f} = 2 %$ ）： $S_{0} e^{- r_{f} T} = 1.20 \times e^{- 0.02} = 1.20 \times 0.98020 = $ 1.17624$
看跌期权价格： $p = 0.1107 + 1.14148 - 1.17624 = $ 0.0759$ （约 7.6 美分）

由于

r (5 %) > r_{f} (2 %)

，远期汇率

F_{0} = 1.20 \times e^{0.03} = $ 1.2365 > S_{0}

（远期升水），所以看涨期权明显比看跌期权贵（$0.1107 对比 $0.0759）。课本原题完整解答见《Hull 习题与解答》第 9 讲。

Simple Example – Put-Call Parity for Currency Options

Simplest numbers: spot = strike = $1.20, 1-year term

Given: a 1-year European call on the euro, strike $K = $ 1.20$ , price $c = $ 0.1107$ ( $σ = 20 %$ ); spot $S_{0} = $ 1.20$ /euro; USD rate $r = 5 %$ , euro rate $r_{f} = 2 %$ (continuously compounded). Find the European put price $p$ with the same strike.

\begin{aligned} c + K e^{- r T} & = p + S_{0} e^{- r_{f} T} \\ \Rightarrow p & = c + K e^{- r T} - S_{0} e^{- r_{f} T} \end{aligned}

Discounted strike (USD rate $r = 5 %$ ): $K e^{- r T} = 1.20 \times e^{- 0.05} = 1.20 \times 0.95123 = $ 1.14148$
Discounted spot (euro rate $r_{f} = 2 %$ ): $S_{0} e^{- r_{f} T} = 1.20 \times e^{- 0.02} = 1.20 \times 0.98020 = $ 1.17624$
Put price: $p = 0.1107 + 1.14148 - 1.17624 = $ 0.0759$ (about 7.6 cents)

Since

r (5 %) > r_{f} (2 %)

, the forward

F_{0} = 1.20 \times e^{0.03} = $ 1.2365 > S_{0}

(forward premium), so the call is much pricier than the put ($0.1107 vs $0.0759). Original Hull problem with full solution: see Hull Exercises & Solutions, Lecture 9.

外汇期权 – 合约规格

费城股票交易所 (PHLX) 澳大利亚元期权产品规格（代码：XDA）

项目	PHLX 规格详情
说明	PHLX 美元结算的澳大利亚元期权，以每单位基础货币（澳元）对应的美元数报价；权利金以美元支付和收取。
合约规模	10,000 澳大利亚元
交易代码	主代码：XDA • 替代代码：AZD • 半点执行价代码：AND • 半点替代代码：DOA • 结算价代码：ADW。若在主代码下找不到某个执行价，请改用替代代码查找。
CUSIP® 编号	69391M 11 1
行权方式	欧式
到期日	到期月份第三个周五之后的周六。
到期循环	3 月季度循环月加上两个近月（任何时候共六个到期月份）。
结算货币	美元。
到期合约的结算价	到期前最后一个交易日美国东部时间中午 12:00:00 的即期汇率；以代码 ADW 发布（PHLX 规则 1057）。
最后交易日	到期月份的第三个周五。
合约点值	$100（即 0.01 × 10,000）。
执行价格间距	由交易所设定固定点位间距；通常为半美分（PHLX 规则 1012）。
权利金报价	1 点 = $100，因此报价 2.13 即 $213。权利金最小变动 0.01 = $1.00。

Currency Options – Contract Specifications

PHLX Australian Dollar Product Specifications (Symbol: XDA)

Item	PHLX Specification
Description	PHLX U.S. dollar-settled Australian dollar currency options are quoted in U.S. dollars per unit of the underlying currency (Australian dollar); the premium is paid and received in U.S. dollars.
Contract Size	10,000 Australian dollars
Symbols	Trading symbol: XDA • Alternate: AZD • Half-point strike: AND • Half-point alternate: DOA • Settlement value: ADW. If no strike is listed under the trading symbol, try the alternate symbol.
CUSIP® Number	69391M 11 1
Exercise Style	European
Expiration Date	Saturday following the third Friday of the expiration month.
Expiration Cycle	Quarterly on the March cycle plus two additional near-term months (six months at all times).
Settlement	U.S. dollars.
Settlement Value for Expiring Contracts	Spot price at 12:00:00 Eastern Time (noon) on the last trading day prior to expiration; disseminated under the symbol ADW (PHLX Rule 1057).
Last Trading Day	The third Friday of the expiration month.
Contract Point Value	$100 (i.e. 0.01 × 10,000).
Strike Price Intervals	Fixed-point intervals set by the Exchange; generally half-cent intervals (PHLX Rule 1012).
Premium Quotation	One point = $100, so a quote of 2.13 is $213. The minimum change in a premium is 0.01 = $1.00.

期货期权 – 收益结构

期货看涨期权： 赋予持有者在约定到期日或之前以执行价格 $K$ 建立期货合约多头的权利。
- 行权时，持有者获得一份对应期货合约的多头头寸，并
- 获得一笔等于最新期货结算价 $F^{*}$ 超过执行价格部分额度的现金： $max (F^{*} - K, 0)$ 。
期货看跌期权： 赋予持有者在约定到期日或之前以执行价格 $K$ 建立期货合约空头的权利。
- 行权时，持有者获得一份对应期货合约的空头头寸，并
- 获得一笔等于执行价格超过最新期货结算价部分额度的现金： $max (K - F^{*}, 0)$ 。
若行权后获得的期货头寸立即平仓：
$看涨期权收益 = max (F_{T} - K, 0), 看跌期权收益 = max (K - F_{T}, 0)$
其中 $F_{T}$ 为行权时刻的期货市场价格。
期权的到期日通常设定在对应期货合约交割日当天或交割日前数个交易日。

Futures Options – Payoffs

Call Option on Futures: Right to enter into a long futures contract by a certain date at strike $K$ .
- When exercised, the holder acquires a long position in the futures contract, plus
- A cash amount equal to the excess of the most recent futures settlement price $F^{*}$ over strike: $max (F^{*} - K, 0)$ .
Put Option on Futures: Right to enter into a short futures contract by a certain date at strike $K$ .
- When exercised, the holder acquires a short position in the futures contract, plus
- A cash amount equal to the excess of strike over the futures settlement price: $max (K - F^{*}, 0)$ .
If the resulting futures position is closed out immediately:
$Call Payoff = max (F_{T} - K, 0), Put Payoff = max (K - F_{T}, 0)$
where $F_{T}$ is the futures price at the time of exercise.
The option's maturity is usually on, or a few days before, the delivery date of the underlying futures contract.

期货期权 – 定价（Black 模型）

在风险中性世界中，期货合约的预期漂移率为零：
- 进入期货合约的初始成本为零；零成本投资的期望利润必须为零。
- 直接使用连续红利收益率定价公式，代入 $S_{0} = F_{0}$ 以及 $q = r$ 。
买卖权平价定理：
$c + K e^{- r T} = p + F_{0} e^{- r T}$
期货二叉树模型： $f_{0} = e^{- r Δ t} [p f_{u} + (1 - p) f_{d}], 其中 p = \frac{e^{(r - r) Δ t} - d}{u - d} = \frac{1 - d}{u - d}$
Black (1976) 期货期权定价模型：
$c = e^{- r T} [F_{0} N (d_{1}) - K N (d_{2})], p = e^{- r T} [K N (- d_{2}) - F_{0} N (- d_{1})]$
其中： $d_{1} = \frac{\ln (F_{0} / K) + \frac{1}{2} σ^{2} T}{σ \sqrt{T}}, d_{2} = d_{1} - σ \sqrt{T}$

Futures Options – Pricing (Black's Model)

In a risk-neutral world, the expected drift on a futures contract is ZERO:
- The initial cost to enter a futures contract is zero; expected profit from a zero-cost investment must be zero.
- Use the continuous dividend yield formula by setting $S_{0} = F_{0}$ and $q = r$ .
Put-Call Parity:
$c + K e^{- r T} = p + F_{0} e^{- r T}$
Binomial Model for Futures: $f_{0} = e^{- r Δ t} [p f_{u} + (1 - p) f_{d}], with p = \frac{e^{(r - r) Δ t} - d}{u - d} = \frac{1 - d}{u - d}$
Black's (1976) Pricing Model:
$c = e^{- r T} [F_{0} N (d_{1}) - K N (d_{2})], p = e^{- r T} [K N (- d_{2}) - F_{0} N (- d_{1})]$
with: $d_{1} = \frac{\ln (F_{0} / K) + \frac{1}{2} σ^{2} T}{σ \sqrt{T}}, d_{2} = d_{1} - σ \sqrt{T}$

美式期货看跌期权：两步二叉树与提前行权检验

数字取最简单的：

F_{0} = K = $ 100

，每期涨 20% 或跌 20%

参数设定： 期货价格 $F_{0} = $ 100$ ，执行价格 $K = $ 100$ ；每期涨 20% 或跌 20%，共两步，每步半年；连续复利无风险利率 $r = 5 %$ 。

单步参数：

u = 1.2, d = 0.8

p = \frac{1 - d}{u - d} = \frac{1 - 0.8}{1.2 - 0.8} = 0.5

每步折现因子

e^{- 0.05 \times 0.5} = 0.9753

期货价格树：

F_{0} = 100

第 1 步：

F_{u} = 120, F_{d} = 80

第 2 步：

F_{u u} = 144

，

F_{u d} = 96

，

F_{d d} = 64

到期行权收益（ $K = 100$ ； $P$ 为各节点看跌期权价值，非组合价值）：
$P_{u u} = 0$ ， $P_{u d} = 100 - 96 = $ 4$ ， $P_{d d} = 100 - 64 = $ 36$ 。

第 1 步倒推估值与美式提前行权检验：

上涨节点（ $F_{u} = 120$ ）： 持有价值 $= 0.9753 \times (0.5 \times 0 + 0.5 \times 4)$ $= $ 1.95$ ；行权收益 $= 0$ 。→ 不行权（ $P_{u} = $ 1.95$ ）。
下跌节点（ $F_{d} = 80$ ）： 持有价值 $= 0.9753 \times (0.5 \times 4 + 0.5 \times 36)$ $= $ 19.51$ ；行权收益 $= 100 - 80 = $ 20$ 。
→ 提前行权最优！（ $P_{d} = $ 20$ 高于持有的 $19.51，多拿 $0.49）。
根节点（ $t = 0$ ）： 持有价值 $= 0.9753 \times (0.5 \times 1.95 + 0.5 \times 20)$ $= $ 10.70$ ；立即行权收益 $= 0$ 。→ 期权价格 = $10.70。

对比：欧式版本只值 $0.9753 \times (0.5 \times 1.95 + 0.5 \times 19.51)$ $= $ 10.46$ ，提前行权特权值 $0.24。课本原题完整解答见《Hull 习题与解答》第 9 讲。

American Futures Put: 2-Step Binomial Tree & Early Exercise Check

Simplest round numbers:

F_{0} = K = $ 100

, up or down 20% each step

Setup: futures price $F_{0} = $ 100$ , strike $K = $ 100$ ; each step moves up 20% or down 20%, two steps of half a year each; continuously compounded risk-free rate $r = 5 %$ .

Single-step parameters:

u = 1.2, d = 0.8

p = \frac{1 - d}{u - d} = \frac{1 - 0.8}{1.2 - 0.8} = 0.5

Per-step discount factor

e^{- 0.05 \times 0.5} = 0.9753

Futures price tree:

F_{0} = 100

Step 1:

F_{u} = 120, F_{d} = 80

Step 2:

F_{u u} = 144

F_{u d} = 96

F_{d d} = 64

Payoffs at expiry ( $K = 100$ ; $P$ = put value at each node, not portfolio value):
$P_{u u} = 0$ , $P_{u d} = 100 - 96 = $ 4$ , $P_{d d} = 100 - 64 = $ 36$ .

Step 1 backward induction & American early-exercise check:

Up node ( $F_{u} = 120$ ): value of holding $= 0.9753 \times (0.5 \times 0 + 0.5 \times 4)$ $= $ 1.95$ ; exercise payoff $= 0$ . → Do not exercise ( $P_{u} = $ 1.95$ ).
Down node ( $F_{d} = 80$ ): value of holding $= 0.9753 \times (0.5 \times 4 + 0.5 \times 36)$ $= $ 19.51$ ; exercise payoff $= 100 - 80 = $ 20$ .
→ Early exercise is optimal! ( $P_{d} = $ 20$ beats the $19.51 from holding, an extra $0.49).
Root node ( $t = 0$ ): value of holding $= 0.9753 \times (0.5 \times 1.95 + 0.5 \times 20)$ $= $ 10.70$ ; immediate exercise payoff $= 0$ . → Put price = $10.70.

Comparison: the European version is worth only $0.9753 \times (0.5 \times 1.95 + 0.5 \times 19.51)$ $= $ 10.46$ , so the early-exercise privilege is worth $0.24. Original Hull problem with full solution: see Hull Exercises & Solutions, Lecture 9.

股指期货期权 – 合约规格

澳大利亚证券交易所 (ASX) SPI 200® 指数期货期权（商品代码：AP）

项目	ASX 规格详情
合约乘数	A$25 / 指数点（例如在 6,000 点时名义价值为 A$150,000）。
合约月份	ASX SPI 200® 指数期权与对应的 ASX SPI 200® 指数期货合约在同一历月到期。看跌和看涨期权均可基于现有期货合约交易。期权可提前四个季月挂牌。
商品代码	AP
上市日期	02/05/2000
最小价格变动	0.5 个指数点（A$12.50）。
执行价格	按 25 个指数点的间距设定。随对应期货价格波动自动生成新的行权价。
最后交易日	对应期货合约的最后交易日。到期合约在最后交易日中午 12:00 停止交易；未到期合约仍按规定的交易时间继续交易（脚注 2）。
现金结算价	对应期货合约的现金结算价。
交易时间	下午 5:10 – 上午 7:00 及上午 9:50 – 下午 4:30（3 月第二个周日至 11 月第一个周日）。下午 5:10 – 上午 8:00 及上午 9:50 – 下午 4:30（11 月第一个周日至 3 月第二个周日）。
结算日	期权可在截至最后交易日（含）的任何营业日行权。只有实值期权在到期时自动行权，除非持有人放弃。行权后，持有人获得对应的 ASX SPI 200® 指数期货头寸，价格为期权执行价。

Options on Index Futures – Contract Specifications

Options on ASX SPI 200® Index Futures (Commodity Code: AP)

Item	ASX Specification
Contract Unit	Valued at A$25 per index point (e.g. A$150,000 at 6,000 index points).
Contract Months	ASX SPI 200® index options expire in the same calendar month as the underlying ASX SPI 200® index futures contract. Put and Call options are available on existing futures contracts. Options are available four quarter months ahead.
Commodity Code	AP
Listing Date	02/05/2000
Minimum Price Movement	0.5 index points (A$12.50).
Exercise Prices	Set at intervals of 25 index points. New exercise prices are created automatically as the underlying futures price fluctuates.
Last Trading Day	The last day of trading of the underlying futures contract. All trading in expiring contracts ceases at 12.00pm on the Last Trading Day. Non-expiring contracts continue to trade as per the stated trading hours (footnote 2).
Cash Settlement Price	The Cash Settlement Price of the underlying futures contract.
Trading Hours	5.10pm – 7.00am and 9.50am – 4.30pm (second Sunday in March to first Sunday in November). 5.10pm – 8.00am and 9.50am – 4.30pm (first Sunday in November to second Sunday in March).
Settlement Day	Options may be exercised on any business day up to and including the Last Trading Day. Only in-the-money options are automatically exercised at expiry, unless abandoned. Upon exercise, the holder receives an underlying ASX SPI 200® index futures position at the option strike price.

期货期权与现货期权的比较分析

相比现货期权，期货期权的潜在竞争优势：
- 期货合约通常比现货资产具有更高的流动性，交易更为便捷（例如国债现货存在最便宜可交割券与应计利息的复杂性）。
- 行权无需立即进行实物现货交割；进入期货合约推迟了现金流出，保留了货币时间价值。
- 现货资产可能难以直接交易或做空（例如大宗农产品、缺乏现货流动性的指数）。
- 期货期权通常享有更低的交易手续费、更窄的买卖价差与交易所统一清算担保。
美式期权估值对比分析：
- 正常市场（期货升水 / Contango， $F_{0} > S_{0}$ ）： 期货美式看涨期权价值高于现货美式看涨期权；期货美式看跌期权价值低于现货美式看跌期权。
- 反向市场（期货贴水 / Backwardation， $F_{0} < S_{0}$ ）： 结论完全相反 – 期货看跌期权价值更高，期货看涨期权价值更低。

Futures Options and Spot Options

Potential advantages of futures options over spot options:
- Futures contracts are frequently more liquid and easier to trade than the underlying physical asset (e.g. government bonds with cheapest-to-deliver and accrued interest complications).
- Exercising does not require immediate physical delivery; entering a futures contract postpones cash outlay, preserving the time value of money.
- The underlying asset may not be readily available for trading or shorting (e.g. agricultural commodities, illiquid spot indexes).
- Futures options often entail lower transaction costs, tighter bid-ask spreads, and uniform clearing.
Valuation Comparison for American Options:
- Normal Market (Contango, $F_{0} > S_{0}$ ): An American call on futures is worth more than an American call on spot. An American put on futures is worth less than an American put on spot.
- Inverted Market (Backwardation, $F_{0} < S_{0}$ ): The reverse is true – futures put is worth more, futures call is worth less.