第11讲：风险参数
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Lecture 11: The Greeks
Delta
• • Delta () is the sensitivity (rate of change) of the option price with respect to the underlying asset price. is the slope of the option value curve. Delta用于衡量期权价格相对于基础资产价格变动程度的敏感性 。是期权价值曲线的斜率。
Lecture 11: The Greeks
5
Naked & Covered Positions（未保护&保护的头 寸）
Naked position Take NO action If S = $60, it costs the bank $1,000,000
Covered position Buy 100,000 shares today If S = $40, the loss is $900,000. Both strategies leave the bank exposed to significant risk
1
Consider the following two scenarios（考虑以下情 景）: • A fund manager purchases put or call options to protect market fall or speculate on bullish market trends. •A bank writes a substantial amount of OTC call or put options to meet investors’ demands. In both cases, a well understanding of the sensitivity of option value to risk factors is important.
。
• Delta of call 1 Delta of put 0
0
Stock Price
-1
Lecture 11: The Greeks
13
Properties of Delta (Cont.)
• Moneyness and Delta
– Deep in-the-money call: delta 1.0（深度实值看涨期权） Reason: it is extremely likely that the option will be in the money at maturity – In-the-money call: delta > 0.5（实值看涨期权） Reason: It has more than 50% chance to end in in the money at maturity – At-the-money call: delta 0.5（两平期权） Reason: There is 50% chance that the option will be in the money – Out-the-money call: delta < 0.5 （虚值看涨期权） Reason: It has less than 50% chance to end up in the money – Deep out-of-the-money call: delta 0 （虚值看涨期权） – Reason: It is extremely likely that the option will never be in the money
Lecture 11: The Greeks
Both are large compared to the risk premiums
6
A Stop-Loss Strategy（止损策略）
x
This involves: • Buying 100,000 shares as soon as price reaches $50 • Selling 100,000 shares as soon as price falls below $50 This deceptively simple hedging strategy does not work well (transactions costs, discontinuity of prices, and the bid-ask bounce kills it)
Lecture 11: The Greeks
12
Properties of Delta（ Delta的性质）
• Signs of delta – delta for call > 0; delta for put < 0.(看涨期权的Delta>0,看跌期权的
Delta<0）
– The negative delta means that a long put position should be hedged with a long position in the underlying stock, and a short put position should be hedged with a short position in the underlying stocks. – 负的Delta意味着多头的看跌期权头寸要用多头的股票来对冲，反之亦然
7
Option Price Slope =
B
A
Lecture 11: The Greeks
Stock price
8
Delta Meaning of Delta
• Delta measures the likely change in the option price given $1 change in the underlying asset. A delta of 0.6 means that when the stock price changes by $1, the option price changes by $0.6. Delta衡量当标的资产价格变化1单位时期权价格的变动 Consider a portfolio: long one call, price c, and short shares: value VP = c - S.
No.ofstocks( 400) C ( $0.8) hedgeratio delta NumberofCalls ( 500) S ($1.0)
• If IBM price goes up by $1 (a gain of $400 on the shares purchased), the option price goes up by $0.8 (a loss of $400), and vice versa.
Lecture 11: The Greeks
3
Introduction
• We have learned that 6 factors affecting option prices. • 4 factors are the main sources of uncertainty in option markets. • 有4个因素是期权市场不确定性的主要来源。 • The measured impact of these 4 factors on option prices are called risk parameters (Greeks). • This lecture focuses on – How these risk parameters are measured（风险参数怎 么测量） – How the values of risk parameters are affected（风险 参数对价值的影响） – How to hedge a derivative portfolio（怎么对冲衍生品 组合的风险）
Lecture 11: The Greeks
10
Delta Hedging（ Delta对冲） • In the above example, the delta of the option position is 0.8 * (-500) = -400; the delta of the stock is 1, and the long position in 400 shares has a delta of 400. • The portfolio is delta-hedged. The total position is termed as delta neutral (with a delta of zero). • Delta hedging allows banks etc to write options without taking risk (wrt asset price changes) • Delta hedging involves maintaining a delta neutral portfolio, and the hedge position must be frequently rebalanced (the slope is changing)
•
V P c 0 S S
Delta is also a hedge ratio (see the example).
Lecture 11: The Greeks
9
Example
• If you wrote 500 IBM call options with a delta of 0.8, when IMB share rises by $1, you would lose $400. If you want to hedge the option perfectly, you need to buy 400 IBM shares. • S ($1) * No. of stocks (400) = C ($0.8) * No. of calls (500), or
Delta
• • Delta () is the sensitivity (rate of change) of the option price with respect to the underlying asset price. is the slope of the option value curve. Delta用于衡量期权价格相对于基础资产价格变动程度的敏感性 。是期权价值曲线的斜率。
Lecture 11: The Greeks
5
Naked & Covered Positions（未保护&保护的头 寸）
Naked position Take NO action If S = $60, it costs the bank $1,000,000
Covered position Buy 100,000 shares today If S = $40, the loss is $900,000. Both strategies leave the bank exposed to significant risk
1
Consider the following two scenarios（考虑以下情 景）: • A fund manager purchases put or call options to protect market fall or speculate on bullish market trends. •A bank writes a substantial amount of OTC call or put options to meet investors’ demands. In both cases, a well understanding of the sensitivity of option value to risk factors is important.
。
• Delta of call 1 Delta of put 0
0
Stock Price
-1
Lecture 11: The Greeks
13
Properties of Delta (Cont.)
• Moneyness and Delta
– Deep in-the-money call: delta 1.0（深度实值看涨期权） Reason: it is extremely likely that the option will be in the money at maturity – In-the-money call: delta > 0.5（实值看涨期权） Reason: It has more than 50% chance to end in in the money at maturity – At-the-money call: delta 0.5（两平期权） Reason: There is 50% chance that the option will be in the money – Out-the-money call: delta < 0.5 （虚值看涨期权） Reason: It has less than 50% chance to end up in the money – Deep out-of-the-money call: delta 0 （虚值看涨期权） – Reason: It is extremely likely that the option will never be in the money
Lecture 11: The Greeks
Both are large compared to the risk premiums
6
A Stop-Loss Strategy（止损策略）
x
This involves: • Buying 100,000 shares as soon as price reaches $50 • Selling 100,000 shares as soon as price falls below $50 This deceptively simple hedging strategy does not work well (transactions costs, discontinuity of prices, and the bid-ask bounce kills it)
Lecture 11: The Greeks
12
Properties of Delta（ Delta的性质）
• Signs of delta – delta for call > 0; delta for put < 0.(看涨期权的Delta>0,看跌期权的
Delta<0）
– The negative delta means that a long put position should be hedged with a long position in the underlying stock, and a short put position should be hedged with a short position in the underlying stocks. – 负的Delta意味着多头的看跌期权头寸要用多头的股票来对冲，反之亦然
7
Option Price Slope =
B
A
Lecture 11: The Greeks
Stock price
8
Delta Meaning of Delta
• Delta measures the likely change in the option price given $1 change in the underlying asset. A delta of 0.6 means that when the stock price changes by $1, the option price changes by $0.6. Delta衡量当标的资产价格变化1单位时期权价格的变动 Consider a portfolio: long one call, price c, and short shares: value VP = c - S.
No.ofstocks( 400) C ( $0.8) hedgeratio delta NumberofCalls ( 500) S ($1.0)
• If IBM price goes up by $1 (a gain of $400 on the shares purchased), the option price goes up by $0.8 (a loss of $400), and vice versa.
Lecture 11: The Greeks
3
Introduction
• We have learned that 6 factors affecting option prices. • 4 factors are the main sources of uncertainty in option markets. • 有4个因素是期权市场不确定性的主要来源。 • The measured impact of these 4 factors on option prices are called risk parameters (Greeks). • This lecture focuses on – How these risk parameters are measured（风险参数怎 么测量） – How the values of risk parameters are affected（风险 参数对价值的影响） – How to hedge a derivative portfolio（怎么对冲衍生品 组合的风险）
Lecture 11: The Greeks
10
Delta Hedging（ Delta对冲） • In the above example, the delta of the option position is 0.8 * (-500) = -400; the delta of the stock is 1, and the long position in 400 shares has a delta of 400. • The portfolio is delta-hedged. The total position is termed as delta neutral (with a delta of zero). • Delta hedging allows banks etc to write options without taking risk (wrt asset price changes) • Delta hedging involves maintaining a delta neutral portfolio, and the hedge position must be frequently rebalanced (the slope is changing)
•
V P c 0 S S
Delta is also a hedge ratio (see the example).
Lecture 11: The Greeks
9
Example
• If you wrote 500 IBM call options with a delta of 0.8, when IMB share rises by $1, you would lose $400. If you want to hedge the option perfectly, you need to buy 400 IBM shares. • S ($1) * No. of stocks (400) = C ($0.8) * No. of calls (500), or
