typora-copy-images-to: Risk and institurion
文章目录
- Risk Management and Financial Institution Chapter 8 —— How Traders Manage Their Risk
Risk Management and Financial Institution Chapter 8 —— How Traders Manage Their Risk
-
交易功能一般是指金融机构的前台，front office
-
金融机构关注资本合规以及监管合规等部分风险的，称之为middle office
-
记录的保存功能称之为后台，back office
-
本章主要是说明前台的独立对冲
-
Each of the Greeks measures a different aspect of the risk in a trading position
8.1 Delta
- In general, the delta of a portfolio with respect to a market variable is
-
δS is a small increase in the value of the variable,δP is the result change in the value of the portfolio
-
delta 是组合价值关于变量的偏导数
-
When the hedging trade is combined with the existing portfolio the resultant portfolio has a delta of zero. Such a portfolio is referred to as delta neutral，对冲交易加上已存组合delta 为 0 ，则称为delta 中性
8.1.1 Linear products
-
底层资产与组合价值在任何时候都是线性相关的，linearly dependent
-
forward contract and gold is a linear product,option is not
-
Once the hedge has been set up, never needs to be changed
8.1.2 Nonlinear Products
-
两个理由非线性的产品更难对冲：
- delta neutral 只有在底层资产又小变动的时候有效
- 我们不能 hedge and forget situation，只能dynamic hedging
-
例子：
- S0 = 49
- ST = 50
- R = 0.05
- σ = 0.2
- T = 20weeks
- 从图中可以得出，delta随着股价的变化不断变化，从0 变化到1 是切线的关系
-
the trader can hedge by buying 52,200 shares
-
rebalancing,to preserve delta neutrality,the hedge has to be adjusted periodically
-
对于short call 的持有者来说，δ下降，对冲所需要的股票下降，borrowing cost下降，delta上升，则对冲所需要的股票上升，borrowing cost上升
-
As rebalancing takes place more frequently, the variation in the cost of hedging is reduced
8.1.3 Where the Cost Comes From 成本从哪里产生
-
在上述例子中，对冲所使用的的多头持仓，rebalance过程包括：
- 当股价下挫的时候，卖出股票
- 当股价上升时候，买入股票
-
240000元的期权价值包括了股价支付与股价实现的平均差异
8.1.4 Economies of Scale 规模经济
- 基于单一资产的规模极大的衍生品组合更容易达到delta 中性，因为仅有的单一的底层资产交易能够使整个组合的delta 中性，对冲成本被不同的交易利润覆盖，因此可以说，衍生品交易是有规模效应的，大交易商主宰着衍生品市场
8.2 Gamma
- γ gamma ， Γ，measures the extent to which large changes cause problems
- Gamma is the rate of change of the portfolio’s delta with respect to the price of the underlying asset，是关于资产价格的二阶导数
- if gamma is small delta changes slowly
- if gamma is large in absolute large ，delta is highly sensitive to the price of the underlying asset
- The difference between C′ and C″ leads to a hedging error，This error depends on the curvature of the relationship between the option price and the stock price. Gamma measures this curvature，曲率造成对冲误差，gamma 衡量这种曲率
- Gamma is positive for a long position in an option ,gamma 随着股价的变化：
- Gamma is greatest for options where the stock price is close to the strike price K
8.2.1 Making a Portfolio Gamma Neutral
- 例：suppose delta - neutral portfolio has a gamma Γ，a traded option has a gamma equal to ΓT，则portfolio 总Gamma 等于：
- 使上述函数等于0 ，求WT= −Γ / ΓT，gamma对冲完，带来的的delta，需要rebalance
- 结论：Delta neutrality provides protection against relatively small asset price moves between rebalancing. Gamma neutrality provides protection against larger movements in the asset price between hedge rebalancing
8.3 Vega
-
The volatility of a market variable measures our uncertainty about the future value of the variable
-
options value are liable to change because of movements in volatility as well as the asset price change and the passage of time
-
Vega 定义，is the rate of change of the value of the portfolio with respect to the volatility, σ, of the underlying asset price
-
if vega is high in absolute terms, the portfolio’s value is very sensitive to small changes in volatility − V / VT
-
If a hedger requires a portfolio to be both gamma and vega neutral, at least two traded derivatives dependent on the underlying asset must usually be used，如果想要gamma neutral and vega neutral 两种交易衍生品必须被使用
- The volatilities of short‐dated options tend to be more variable than the volatilities of long‐dated options
8.4 Theta
-
定义，Θ，is the rate of change of the value of the portfolio with respect to the passage of time，又被称作为 time decay
-
特性，Theta is usually negative for an option
- There is uncertainty about the future price of the underlying asset, but there is no uncertainty about the passage of time
8.5 Rho
-
==定义：==Rho is the rate of change of a portfolio with respect to the level of interest rates
-
外汇有两个Rho，本币与外币的汇率，当债券以及利率衍生品是组合一部分时，交易者通常必须仔细考虑利率期限结构的变化
8.6 希腊字母的计算
- Use RMFI software，可以用来计算美式欧式期权以及一些奇异期权
- Example： for long position
- 解释：
- 当股价上升0.1时，其余不变，option price 上升0.522 * 0.1，short position value decreases 5200
- 当股价上升0.1时，其余不变，delta 上升0.066，option delta 下降660
- volatility 上升0.5%时，option price 上升0.5 * 0.121，value of short position 下降6050
- 每过一天，option price 下挫0.012，value of short position 上升1200
- 利率上升100基点，也就是1%，option price 上升0.089，value of a short potion 下降8900
8.7 Taylor Series Expansion 泰勒展开式
- Taylor series expansions can be used to show how the change in the portfolio value in a short period of time depends on the Greek letters，展示了某一短时间内，各个希腊值在交易组合变化中所起的不同左右
- ΔP and ΔS are the change in P and S in a small time interval Δt
- 以下等式忽略了误差等级高于delta T的项
8.8 The Realities of Hedging
- When managing a large portfolio dependent on a single underlying asset, traders usually make delta zero, or close to zero at least once a day
- zero gamma and zero vega are less easy to achieve,没有对应的有价格优势且适量的option 可供交易
8.9 Hedge Exotic Options
- 奇异期权可以使用简单产品的对冲方式进行对冲
- 静态期权复制，我们建立一个简单期权的组合P，使其在给定的标的资产价格和时间的边界点上，具有和奇异期权相同的价值Static options replication
- 亚市看涨期权对冲相对容易，因为观察到越来越多的资产价格，作为平均值作为计算
- 障碍期权的对冲难度相对较大，障碍期权的Delta在障碍值附近不连续
8.10 Scenario Analysis 情景分析
- 银行持有的汇率期权组合，考虑两个主要变量：汇率及汇率波动
- 情景列表：
- 最大损失为表的右下角，但这一特性并不永久正确，当Gamma 为正的时候，，最大的损失对应于标的资产市场价值不变的时候