i.e.
Yield Enhancement 为长期持有,
write a OTM call option.
Reduce a Position 为了卖,
Target Price Realisation 近期到期 combine of above two
As the covered call strategy is (long S, short Call)
if
if
if Break-even,
if max loss,
if max gain,
Protected Put is a hedge, but is not a perfect hedge, because there is still the maximum loss part.
Case 1:
Fiduciary Call = Protective Put
Case 2:
Covered Call = Cash Secured Put
Bull Spread:
long
Short
Bear Spread:
long
Short
P.S. 如果 options 是 out-of-money,那么是 期权费之差。如果 option 是 in-the-money, 那么还要考虑行权价格的差 + 期权费。总之,现推导。
Long Calendar: 买长期 long
Short Calendar: 买短期 long
同买同卖 at the same strike price. Long straddle => long volatility
Long Straddle = + Call + Put
Short Straddle = -Call - Put
Sell option <=> sell volatility, long option <=> expect vol
Long call <=> long bull
Call:
Out-of-Money,
In-the-Money,
At-the-Money,
Put:
Thus, both strategies are not delta neutral.
Delta of Covered Call = Delta of Stock (=1) + Delta of Call ([0,1])
Delta of Protective Put = Delta of Stock + Delta of Put ([-1,0])
By put-call parity
Take derivative w.r.t. S
By Black-Scholes Model
Delta neutral hedging is a dynamic process
How delta change w.r.t. S. Lower the gamma, easily the delta hedging. Because delta is less affected.
Gamma is positive, as the curve convex.
Gamma is the maxima while option is at-the-money.
The Black-Sholes model assumes constant Volatility
Empirically for foreign currency options, when at-the-money, implied volatility is lowest
结论是:OTM 的 Option 的 implied vol 更大
Equity Option, Skew (Smirk)
Reasons for the Smile in Equity Options
Crashophbia 崩盘 market crash 可能,因为option就是用来应对危机的
Leverage. As equity declines in value, company's leverage increases.
Volatility Feedback Effect. 反身性,相当于 负向 accelerator
同样是 OTM ,put option 价格比 call option 贵。因为,put以上三个原因。所以Put Price 大,则 implied vol for Put 大。
OTM Put 的 vol 低, OTM Call 的 vol 高. Buy OTM Call (underpriced) and sell OTM put (overpriced)
Implied Volatility is compared with
Long Risk Reversal: 预期因为上述情况,implied volatility 被高估,那么应该
short put, long call 挣 put 高估和call 低估的钱
Then, short stocks 为了只保留 vega risks, 去除 delta risks, we then need to
Short Risk Reversal: 反之
The term structure of volatility is often in contango, (因为期限越远,风险越高,所以upward sloped)
(the implied volatilities for long-term options are higher)
When market is stress, the term structure inverts.
Payer and Receiver 都指的是 对Float 的 pay / receive
The duration of float bond is nearly zero, thus the duration of payer float is positive, the duration of the receiver Float is negative.
Empirically, float-rate bond duration = time period of reset . E.G. if half-year to reset, then duration = 0.5
Float Bond is s.t. Cash Flow Risks 因为每一期现金流都不确定
Fixed Bond is s.t. Market Value Risks 因为pay fixed 的部分,受 market rate 影响
If change from float to fixed, then CF risks is neutralised and then be s.t. market value risks. Vice Versa. 如果通过swap 把float 换成了 fixed,那么从 expose cf risks 转为 expose to market value risks
如何通过 swap 达到目标 Durations.
Solution: by Dollar Duration 能相加
因为在签订 swap 时, swap value = 0, 所以
Finally, 所以
or
Implication:
As it is future, it's standard (not OTC as FRA).
Guaranteed by a clearinghouse, so counterparty risk is nearly zero. 有保证金,所以交易对手方风险基本上为0
Cash settled
P.S. interest rate futures are short-term, but fixed-income futures are more long time.
Euro Dollar Futures have fixed notional value and fixed term (, as it is standardised).
Notional value: 1 million USD
Term: 3 months
Treasury-bond is the underlying asset, we do not use corporate bond as there is less liquidity and more credit risks.
There is a portfolio of T-bonds/bills for settlement. As the Fixed-Income Futures are standarised, they are traded in the exchange, and are settled daily. 标准化的future,有一个basket,交易所交割,可以 cash 可以实物交割
the Short side has a dilivery option to choose which bond to deliver 意味着short方可以决定deliver哪个,所以有CF转换 basket里的,所以有CTD
Basket 中的 bonds 有 CF (conversion factor) 用来调节 Basket 中不同 bonds 不同期限的rates等带来的差额, 和 Accured Interest 的差别
Futures contract Sell has the right to choose the CTD (cheapest-to-delivered)
if market yield > notional yield, then long duration bond is likely to be CTD (最合适用来交割的)
因为 market rate 大的 ,duration 长 的折价多
如图,久期长的离 expired date 远,折价越多。但是这只是clean price 方面,还未考虑 Accured Interest
if market yield < notional yield 的, 短久期的 可能为 CTD (更便宜,适合交割)原因见上图,把图沿 y-axis 对称
Basis = S - F , in alternatives 为空头方的交割成本,basis converge to zero with time
if basis is positive, sell the basis , ( long the future and short the bond) 因为 Basis converges to zero, 所以如果 basis 正,那么S会变小,F会变大 使basis趋近于0
if the basis is negative, buy the basis, ( long the bond and short the future)
担心 interest rate 上涨, 带来 T-bond Price 下跌,同时 Fixed-Income Future 价格下跌,所以 Short Futures
P.S. recall, irrelevent with the other content
Similarly
Basis Point Value Hedge Ratio (BPVHR) 即为 N
=>
The number of future to sell to fully hedge the portfolio is 345 future contracts
负号不能丢,表示 hedge 与 underlying 的变动是相反的
通过加入
N is named BPVHR,根据上式,解出 BPVHR
Finally,
管理 Equity Risks 的方法 (1) Swap (2) Future and Forward <- adjust target portfolio beta & cash equitisation
fixed rate <-> equity return
float rate <-> equity return
equity return <-> another equity return (high risks)
returns are from
a single stock
a basket equity (ETF)
an equity index
Future price is quoted as:
Or
make
Synthetic Risk-free Asset
Synthetic Equity
如 Fund has 33.33% Bonds + 66.67% Equity, 想调整为 30% Bonds + 70% Equity。因为若fund很大,直接抛售等买卖会带来资产价格大幅变动。
为了避免价格波动,我们不直接买卖资产,而是买卖对应的 derivatives (futures and forwards),因为这样可以最小化对市场价格的影响
通过 Cash Position make
所以此时,并不是直接 Sell bond,而是 make
不是直接 buy equity,而是 make
本国US investor 要买 EU bond。进入Cross-currency Swap. (现金流结构相当于在期初 花USD 买了 USD bond,收EUR,发行EUR bond)
在inception期初 换本金 付出 USD principal 收到 EUR principal
在 periodic 期中, pay EUR interest, 收到 USD interest (有 basis)一般dealer在哪国就需要哪国的basis。如本国US,那么dealer也在US,那么investor收到的 USD interim interest 就会有 basis扣除,即为 dealer 挣的钱
在期末 换回本金。收到 USD,付出 EUR
P.S. 如果美元涨价,那么外国人要付basis给本国
In this case, US investor pays USD and receives EUR, if the USD appreciates the US investor needs to pay more interest of USD, but why the answer is "the US investor will most likely increase the periodic net interest she receives in US dollars." the US investor pays USD!!!
Cross-currency basis: additional cost of borrowing dollar (most currencies show a negative basis against dollar) 即美元有加点
if USD (domestic) is strong, then receive basis
Sample Text:
Cross currency basis is a measure of dollar shortage in the market. Whenever there’s a higher demand for the dollar, the counterparty lending the dollar will ask for a price premium. It is this amount which is referred to as the “cross currency basis”. The more negative the basis becomes, the more severe the shortage.
For a European investor looking to take a loan (borrow) from a domestic bank (EUR) and enter a EUR/USD currency swap, if the basis is (-) then there is a Dollar shortage, and European investor will have to pay additional basis.
For dollar-funded investors, negative basis can work in their favor when they hedge currency exposures. In order to hedge foreign currency exposure, the dollar-funded investors lend out dollar today and receive it back in the future, earning additional cross currency basis spread on top of the yield of their foreign investments.
Swap the notional principals, 本金可以在beginning换,在ending换回来 but periodic interest payment could not be netted 期间的利息不换
No exchange the principals. Instead,
exchange CF in the future at fixed exchange rate
The amount of exchanged are based on the Exchange Rate and Interest Rate
Buy EUR/USD 相当于买USD
Sell EUR/USD 相当于卖USD
Volatility Derivatives
VIX Futures
VIX Options
Volatility index
Variance Swap
VIX and Equity returns are mostly negative correlated
P.S. Cost of Carry model does not work on VIX
如果是 Contango, 横坐标 term,纵坐标 price,向上倾斜。这意味着:随着期限减少(向 term = 0),价格会降低。
这意味着,在 roll VIX future 时,如以100买 t=10,卖90 在t=9。再roll,卖80在t=8,etc。每次roll的价格都会降低。所以contango的情况会亏 price
如果是 backwards,在 roll的时候,price会提升,可以挣 price 提升的钱。
Payoff of Variance Swaps is based on Variance rather than Volatility (s.d.).
No Exchange of Notional Principal and No interim Settlement periods
Variance Stike (Implied Volatiltiy),
Realised Variance,
Swap 的双方为
那么 每份swap 的payoff 是
整个 swap 的 payoff 是,即 settlement amount =
Vega Notional 的定义:
Gain:
Loss:
So,