Risk Management and Financial Institution Chapter 8 —— How Traders Manage Their Risk

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Risk Management and Financial Institution Chapter 8 —— How Traders Manage Their Risk

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• 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
  • S_T = 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 Γ_T，则portfolio 总Gamma 等于：

• 使上述函数等于0 ，求W_T= −Γ / Γ_T，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 / V_T

• 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