Derivatives & Exchange

Derivatives

Strategies

1. Covered Call Strategy (long Stock, short Call)

i.e. $S=2000$$S=2000$

• Yield Enhancement 为长期持有, $K=3000$$K = 3000$

  write a OTM call option.

• Reduce a Position 为了卖, $K=1800$$K=1800$

• Target Price Realisation 近期到期 combine of above two $K=2500$$K=2500$

As the covered call strategy is (long S, short Call)

$Profit=\underset{longS}{\underbrace{(S_T-S_0)}}+\underset{LongCall}{\underbrace{C-max(S_T-K,0)}}$$Profit = \underbrace{(S_T - S_0)}_{long \ S} + \underbrace{ C - \max(S_T-K,0)}_{Long\ Call}$

• if $S_T>K$$S_T > K$, then, $Profit=K-S_0+C$$Profit = K - S_0 + C$ (行权)

• if $S_T\le K$$S_T \leq K$, then, $Profit=S_T-S_0+C$$Profit = S_T -S_0 + C$ (不行权)

• if Break-even, $S_T-S_0+C=0$$S_T - S_0 + C =0$

• if max loss, $S_T=0$$S_T = 0$, $Loss(Profit)=-S_0+C$$Loss (Profit) = -S_0 + C$

• if max gain, $S_T>X$$S_T > X$, $Gain(Profit)=K-S_0+C$$Gain (Profit) = K - S_0 + C$

2. Protected Put (long Stock, long Put)

$Profit=\underset{longStock}{\underbrace{(S_T-S_0)}}+\underset{LongPut}{\underbrace{max(K-S_T,0)-P}}$$Profit = \underbrace{(S_T - S_0)}_{longStock} + \underbrace{\max(K - S_T,0) - P}_{LongPut}$

Protected Put is a hedge, but is not a perfect hedge, because there is still the maximum loss part.

3. Collar (long Stock, long Put, short Call, options are all out-the-money OTM)

$Profit=\underset{LongStock}{\underbrace{(S_T-S_0)}}+\underset{ShortCall}{\underbrace{C-max(S_T-K)}}+\underset{LongPut}{\underbrace{max(K-S_T,0)-P}}$$Profit = \underbrace{(S_T - S_0)}_{LongStock} + \underbrace{C - \max(S_T-K)}_{Short Call} + \underbrace{\max(K-S_T,0)-P}_{LongPut}$

P.S. Put-Call Parity

• Case 1:

  $C+\frac{K}{1+r}=P+S$$C+ \frac{K}{1+r} = P + S$

  Fiduciary Call = Protective Put

• Case 2:

  $S-C=\frac{K}{1+r}-P$$S-C = \frac{K}{1+r} -P$

  Covered Call = Cash Secured Put

4. Vertical Arbitrage (different exercise price X/K)

• Bull Spread:

  • long $Cal{l}_L$$Call_L$ short $Cal{l}_H$$Call_H$

  • Short $Pu{t}_H$$Put_H$ long $Pu{t}_L$$Put_L$

• Bear Spread:

  • long $Pu{t}_H$$Put_H$ short $Pu{t}_L$$Put_L$

  • Short $Cal{l}_L$$Call_L$ long $Cal{l}_H$$Call_H$

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。

5. Horizontal Arbitrage (different Expire Date)

• Long Calendar: 买长期 long $Cal{l}_{longer}$$Call_{longer}$ short $Cal{l}_{shorter}$$Call_{shorter}$

• Short Calendar: 买短期 long $Cal{l}_{shorter}$$Call_{shorter}$ short $Cal{l}_{longer}$$Call_{longer}$

6. Straddle

同买同卖 at the same strike price. Long straddle => long volatility

• Long Straddle = + Call + Put

• Short Straddle = -Call - Put

In Sum

Sell option <=> sell volatility, long option <=> expect vol

Long call <=> long bull

Greek

Delta of Option

Call: $\mathrm{\Delta }\in [0,1]$$\Delta \in [0,1]$

• Out-of-Money, $\mathrm{\Delta }\to 0$$\Delta \to 0$

• In-the-Money, $\mathrm{\Delta }\to 1$$\Delta \to 1$

• At-the-Money, $\mathrm{\Delta }=0.5$$\Delta = 0.5$

Put: $\mathrm{\Delta }\in [-1,0]$$\Delta \in[-1,0]$

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

  $C+\frac{K}{1+r}=p+S$$C+ \frac{K}{1+r} = p +S$

  Take derivative w.r.t. S

  ${\mathrm{\Delta }}_C+0={\mathrm{\Delta }}_P+1$$\Delta_C + 0 = \Delta_P + 1$

• By Black-Scholes Model

  $Call=S\cdot N(d_1)+K\cdot e^{-rt}\cdot N(d_2)$$Call = S\cdot N(d_1) + K\cdot e^{-rt}\cdot N(d_2)$

  ${\mathrm{\Delta }}_C=N(d_1)$$\Delta_C = N(d_1)$

Delta Hedging

Delta neutral hedging is a dynamic process

$+S-\frac{1}{\mathrm{\Delta }}C=0$$+S - \frac{1}{\Delta} C = 0$

$+\mathrm{\Delta }S-C=0$$+\Delta S - C = 0$

Gamma <- Dynamic Hedging

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.

Vega <- w.r.t. volatility, Vega > 0

Theta <- w.r.t. time, Theta < 0

Rho < w.r.t. risk-free rate, >0 for call, <0 for put

Volatility Smile

Volatility Smile and Volatility Skew Smirk

1. The Black-Sholes model assumes constant Volatility

2. Empirically for foreign currency options, when at-the-money, implied volatility is lowest

   • 结论是：OTM 的 Option 的 implied vol 更大

   

3. Equity Option, Skew (Smirk)

   • Reasons for the Smile in Equity Options

   1. Crashophbia 崩盘 market crash 可能，因为option就是用来应对危机的

   2. Leverage. As equity declines in value, company's leverage increases.

   3. Volatility Feedback Effect. 反身性，相当于 负向 accelerator

   

4. 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)

5. Implied Volatility is compared with $\frac{K}{S_0}$$\frac{K}{S_0}$ or $\frac{K}{F_0}$$\frac{K}{F_0}$ (相当于去除量纲)

Risk Reversal

• 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

Term Structure of Volatility

• 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.

Swaps, Forward, Futures

Manage Interest Rate Risks

Interest Rate Swap

Payer and Receiver 都指的是 对Float 的 pay / receive

Duration of the Swap

$D_{payerFloat}=D_{Fix}-D_{Float}$$D_{payerFloat} = D_{Fix} - D_{Float}$, vice veresa

• 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

Market Value Risks and Cash Flow Risks

• 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

Formula

$M{V}_p\times M{D}_r=M{D}_p+N_s\times M{D}_s$$MV_p \times MD_r = MD_p + N_s \times MD_s$

$D_p$$D_p$ duration, 利率变动 价格变多百分比

$D{D}_p=D_p\times V$$DD_p = D_p \times V$ dollar duration, 利率变动，价格变动多少元

$D_p=w_1{D}_1+w_2{D}_2$$D_p = w_1 D_1 + w_2 D_2$

$D{D}_p=D{D}_1+D{D}_2$$DD_p = DD_1 + DD_2$, coz $D_p=w_1{D}_1+w_2{D}_2$$D_p = w_1 D_1 + w_2 D_2$ multiply $V_p$$V_p$ from the both sides,

$D_p{V}_p=w_1{D}_1{V}_p+w_2{D}_2{V}_p$$D_pV_p = w_1 D_1 V_p + w_2 D_2 V_p$

$D{D}_p=V_1{D}_1+V_2{D}_2$$DD_p = V_1D_1 + V_2 D_2$, coz $V_1=w_1{V}_p$$V_1 = w_1 V_p$

$D{D}_p=D{D}_1+D{D}_2$$DD_p = DD_1 + DD_2$

如何通过 swap 达到目标 Durations.

$D_{original}+\underset{InterestRateSwap}{\underbrace{D_{Swap}}}=D_{target}$$D_{original} + \underbrace{D_{Swap}}_{InterestRateSwap} = D_{target}$

Solution: by Dollar Duration 能相加

$D{D}_{target}=D{D}_{original}+D{D}_{swap}$$DD_{target} = DD_{original} + DD_{swap}$ expand it

$D_{target}{V}_{target}=D_{original}{V}_{target}+D_{swap}\times N_{swap}$$D_{target} V_{target} = D_{original}V_{target} + D_{swap} \times N_{swap}$ (Notional value of the Swap)

因为在签订 swap 时， swap value = 0, 所以 $V_{target}=V_{original}+V_{swap}=V_{target}+0$$V_{target} = V_{original} + V_{swap} = V_{target} + 0$

Finally, 所以

$D_{target}{V}_{target}=D_{original}{V}_{target}+D_{Swap}{N}_{swap}$$D_{target} V_{target} = D_{original} V_{target} + D_{Swap} N_{swap}$

$N_s=\frac{D_t-D_p}{D_{swap}}\times V_p$$N_s = \frac{D_t - D_p}{D_{swap}} \times V_p$

or

$M{V}_p\times MDu{r}_p+N_s\times MDu{r}_s=M{V}_p\times MDu{r}_t$$MV_p \times MDur_p + N_s \times MDur_s = MV_p \times MDur_t$ ($M{V}_t=M{V}_p$$MV_t = MV_p$ ， 因为市场投资组合的价值 和加入swap的一样，swap在建立时value = 0)

$N_s=\frac{MDu{r}_t-MDu{r}_p}{MDu{r}_s}M{V}_p$$N_s = \frac{MDur_t - MDur_p}{MDur_s} MV_p$

• Implication:

  $D_t-D_p$$D_t - D_p$ 与 $D_{swap}$$D_{swap}$ 同向变动

Interest Rate Forwards (FRA)

Interest Rate Future (Euro-Dollar Future)

• 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

$\mathrm{\$}1,000,000\times \frac{90}{360}\times 1bp=\mathrm{\$}25$$\$1,000,000 \times \frac{90}{360}\times 1bp = \$25$

Fixed-Income Futures

Treasury-bond is the underlying asset, we do not use corporate bond as there is less liquidity and more credit risks.

Delivery

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

Clean + Accured Int = Dirty

$Clean=\text{Quoted Futures Settlement Priec}\times CF$$Clean = \text{Quoted Futures Settlement Priec} \times CF$

$Dirty=QuotedPrice\times CF+AI$$Dirty = QuotedPrice \times CF + AI$

CTD

Futures contract Sell has the right to choose the CTD (cheapest-to-delivered)

$CTD=$$CTD =$$QuotedPrice\times CF$$Quoted Price \times CF$

Market Yield v.s. Notional Yield

• 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 Trading

Basis = S - F , in alternatives 为空头方的交割成本，basis converge to zero with time

$Basis=S-F\times CF$$Basis = S - F \times CF$

• 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)

Fixed-Income Future Hedging

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

Basis Point Value for Bonds ( BPV

• $BP{V}_{CTD}=MDu{r}_{CTD}\times M{V}_{CTD}\times 0.01\mathrm{\%}$$BPV_{CTD} = MDur_{CTD} \times MV_{CTD} \times 0.01\%$

  • $M{V}_{CTD}=(CTDQuotedPrice)\times Future$$MV_{CTD} = (CTD\ QuotedPrice) \times Future$

• $BP{V}_F=MDu{r}_F\times Pric{e}_F\times 0.01\mathrm{\%}$$BPV_F = MDur_F \times Price_F \times 0.01\%$

P.S. recall, irrelevent with the other content

• $MaculyDuration$$Maculy Duration$ 为时间（年）

• $MDu{r}_F$$MDur_F$ 修正久期: 1% 利率变动 => 投资期货合约 收益变动% ($MDur$$MDur$)

• $D{D}_F$$DD_F$ 美元久期: 1% 利率变动 => 收益变动 amount $，BPV 类比 DD，为value

Hedge Ratio <- Number of Future Contract

$\mathrm{\Delta }P=HR\times \mathrm{\Delta }F$$\Delta P = HR \times \Delta F$ => $HR=\frac{\mathrm{\Delta }P}{\mathrm{\Delta }F}$$HR = \frac{\Delta P}{\Delta F }$

Similarly

• Basis Point Value Hedge Ratio (BPVHR) 即为 N

  • $BPVHR=-\frac{BP{V}_P}{BP{V}_F}$$BPVHR = - \frac{BPV_P}{BPV_F}$

  • $BP{V}_F\times CF=BP{V}_{CTD}$$BPV_F \times CF= BPV_{CTD}$

  • => $BPVHR=-\frac{BP{V}_P}{BP{V}_{CTD}}\times CF$$BPVHR = - \frac{BPV_P}{BPV_{CTD} } \times CF$

  • 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

Adjust Portfolio Duration

$BP{V}_{Portfolio}+N\times BP{V}_{Future}=BP{V}_{Target}$$BPV_{Portfolio} + N\times BPV_{Future} = BPV_{Target}$

通过加入 $N\times BP{V}_{Future}$$N\times BPV_{Future}$ 项，调将 portfolio 的久期调成 target 的数额

N is named BPVHR，根据上式，解出 BPVHR

$BPVHR=\frac{BP{V}_T-BP{V}_P}{BP{V}_F}$$BPVHR = \frac{BPV_T - BPV_P}{BPV_F}$

• $BP{V}_T=M{V}_P\times D_{Target}\times 1bp$$BPV_T = MV_P \times D_{Target} \times 1bp$

• $BP{V}_P=M{V}_P\times D_{Portfolio}\times 1bp$$BPV_P = MV_P \times D_{Portfolio} \times 1bp$

• $BP{V}_F\times CF=BP{V}_{CTD}$$BPV_F \times CF= BPV_{CTD}$ (Accured Interest is ignored just now)

• Finally, $BPVHR=\frac{BP{V}_T-BP{V}_P}{BP{V}_{CTD}}\times CF$$BPVHR = \frac{BPV_T - BPV_P}{BPV_{CTD}}\times CF$

Manage Equity Risk

管理 Equity Risks 的方法 (1) Swap (2) Future and Forward <- adjust target portfolio beta & cash equitisation

Equity Swap

• fixed rate <-> equity return

• float rate <-> equity return

• equity return <-> another equity return (high risks)

returns are from

1. a single stock

2. a basket equity (ETF)

3. an equity index

Futures

Adjust Beta

Future price is quoted as: $F=\underset{QuoteFuturePrice}{\underbrace{f}}\times Multiplier$$F = \underbrace{f}_{QuoteFuturePrice} \times Multiplier$ ($f$$f$ is %, multiplier convert the $F$$F$ into $ amount)

$N_f=\frac{{\beta }_T-{\beta }_p}{{\beta }_f}\frac{M{V}_p}{F}$$N_f = \frac{\beta_T-\beta_p}{\beta_f} \frac{MV_p}{F}$

$N_f=\frac{{\beta }_T-{\beta }_p}{{\beta }_f}\times \frac{M{V}_p}{f\times Multiplier}$$N_f = \frac{\beta_T-\beta_p}{\beta_f} \times \frac{MV_p}{f\times Multiplier}$

${\beta }_p\times M{V}_p+\mathrm{\#}\times {\beta }_f\times \underset{F(\mathrm{\$}amount)}{\underbrace{(f\times Multiplier)}}={\beta }_T\times M{V}_p$$\beta_p \times MV_p+ \# \times \beta_f \times \underbrace{(f\times Multiplier)}_{F (\$ amount)} = \beta_T \times MV_p$

$\mathrm{\#}=N_f=\frac{{\beta }_T\times M{V}_p-{\beta }_{p}M{V}_p}{{\beta }_f\times f\times Multiplier}=\frac{{\beta }_T-{\beta }_p}{{\beta }_f}\frac{M{V}_p}{f\times Multiplier}$$\# =N_f= \frac{\beta_T \times MV_p - \beta_p MV_p}{\beta_f \times f \times Multiplier} = \frac{\beta_T -\beta_p}{\beta_f}\frac{MV_p}{f\times Multiplier}$

Or $=\frac{{\beta }_T-{\beta }_p}{{\beta }_f}\frac{M{V}_p}{F}$$= \frac{\beta_T -\beta_p}{\beta_f}\frac{MV_p}{F}$

$F\iff f\times Multiplier$$F \Leftrightarrow f \times Multiplier$ are $ amount

Cash Equilisation

make ${\beta }_p$$\beta_p$ or ${\beta }_T$$\beta_T$ as "0".

• Synthetic Risk-free Asset ${\beta }_T=0$$\beta_T=0$

  $N_f=\frac{0-{\beta }_p}{{\beta }_f}\frac{M{V}_p}{f\times Multiplier}$$N_f = \frac{0 - \beta_p}{\beta_f}\frac{MV_p}{f\times Multiplier}$

• Synthetic Equity ${\beta }_p=0$$\beta_p = 0$

  $N_f=\frac{{\beta }_T-0}{{\beta }_f}\frac{M{V}_p}{f\times Multiplier}$$N_f = \frac{\beta_T -0}{\beta_f}\frac{MV_p}{f\times Multiplier}$

Using Derivatives to Alter Asset Allocation

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 $\beta =0,forStocks$$\beta = 0, for\ Stocks$, $BPV=0,forBonds$$BPV = 0, for\ Bonds$

所以此时，并不是直接 Sell bond，而是 make $BP{V}_T=0$$BPV_T=0$

不是直接 buy equity，而是 make $\beta =0$$\beta=0$

Manage Currency Risks

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）

1. 在inception期初 换本金 付出 USD principal 收到 EUR principal

2. 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

3. 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

$\frac{F}{S}=\frac{1+r_f}{1+r_d\pm (basis)}$$\frac{F}{S} = \frac{1+r_f}{1+r_d \pm(basis)}$

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.

Cross-Currency Basis Swap 期初还两个币种的本，期末换回来

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

Synthetic Borrowing 不换本金

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

Currency Forward and Futures

$HR=\frac{\text{Amount of Currency to be Exchanged}}{\text{Future Contract Size}}$$HR = \frac{\text{Amount of Currency to be Exchanged}}{\text{Future Contract Size}}$

Buy EUR/USD 相当于买USD

Sell EUR/USD 相当于卖USD

Manage Volatility Risks

• Volatility Derivatives

  1. VIX Futures

  2. VIX Options

  3. Volatility index

• Variance Swap

VIX Futures

VIX and Equity returns are mostly negative correlated

P.S. Cost of Carry model does not work on VIX $F=SX(1+r_f{)}^T-CB+CC$$F = S X(1+r_f)^T - CB + CC$, because VIX spot is not able to be invested (VIX is an index, is calculated)

• 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

  • $RollYield=\frac{F-S}{S}$$RollYield = \frac{F-S}{S}$

• 如果是 backwards，在 roll的时候，price会提升，可以挣 price 提升的钱。

Variance Swap

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), $K^2$$K^2$: the implied vol at the beginning

  • $Strike=K,\text{and }variancestrike=K^2$$Strike = K , \text{and } variance \ strike = K^2$

• Realised Variance, ${\sigma }^2$$\sigma^2$: Actual Vol over the life of the swap

Swap 的双方为 ${\sigma }^2$$\sigma^2$ and $K^2$$K^2$. 如果市场上实际的 variance, ${\sigma }^2>K^2$$\sigma^2>K^2$ the strike price squared.

那么 每份swap 的payoff 是 ${\sigma }^2-K^2$$\sigma^2 - K^2$ 相当于用$K^2$$K^2$买${\sigma }^2$$\sigma^2$.

整个 swap 的 payoff 是，即 settlement amount = $N_{variance}\times ({\sigma }^2-K^2)$$N_{variance}\times (\sigma^2 - K^2)$

• Vega Notional 的定义: $N_{vega}=\frac{Gain-Loss}{2}$$N_{vega} = \frac{Gain - Loss}{2}$ 因为loss是负数，我们要absolute amount所以用 -loss 调整为正的

  • Gain: $(K+1{)}^2\times N_{variance}$$(K+1)^2\times N_{variance}$

  • Loss: $(K-1{)}^2\times N_{variance}$$(K-1)^2\times N_{variance}$

• So, $N_{vega}=\frac{4K}{2}=2K\times N_{variance}$$N_{vega} = \frac{4K}{2}=2K\times N_{variance}$