Derivatives & Exchange



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

i.e. S=2000

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

Profit=(STS0)long S+Cmax(STK,0)Long Call

2. Protected Put (long Stock, long Put)


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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)


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P.S. Put-Call Parity

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

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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)

6. Straddle

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

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In Sum

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

Long call <=> long bull

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Delta of Option

Call: Δ[0,1]

Put: Δ[1,0]

Thus, both strategies are not delta neutral.

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Delta Hedging

Delta neutral hedging is a dynamic process



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

    Screenshot 2024-01-10 at 20.47.28

  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

    Screenshot 2024-01-10 at 20.48.20

  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 KS0 or KF0 (相当于去除量纲)

Risk Reversal

Term Structure of Volatility

Swaps, Forward, Futures

Manage Interest Rate Risks

Interest Rate Swap

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

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Duration of the Swap

DpayerFloat=DFixDFloat, vice veresa

Market Value Risks and Cash Flow Risks


Dp duration, 利率变动 价格变多百分比

DDp=Dp×V dollar duration, 利率变动,价格变动多少元


DDp=DD1+DD2, coz Dp=w1D1+w2D2 multiply Vp from the both sides,


DDp=V1D1+V2D2, coz V1=w1Vp


如何通过 swap 达到目标 Durations.


Solution: by Dollar Duration 能相加

DDtarget=DDoriginal+DDswap expand it

DtargetVtarget=DoriginalVtarget+Dswap×Nswap (Notional value of the Swap)

因为在签订 swap 时, swap value = 0, 所以 Vtarget=Voriginal+Vswap=Vtarget+0

Finally, 所以




MVp×MDurp+Ns×MDurs=MVp×MDurt (MVt=MVp , 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)


Interest Rate Forwards (FRA)
Interest Rate Future (Euro-Dollar Future)

Euro Dollar Futures have fixed notional value and fixed term (, as it is standardised).


Fixed-Income Futures


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

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=Quoted Futures Settlement Priec×CF



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


Market Yield v.s. Notional Yield
Basis Trading

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


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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

P.S. recall, irrelevent with the other content

Hedge Ratio <- Number of Future Contract



负号不能丢,表示 hedge 与 underlying 的变动是相反的

Adjust Portfolio Duration


通过加入 N×BPVFuture 项,调将 portfolio 的久期调成 target 的数额

N is named BPVHR,根据上式,解出 BPVHR


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Manage Equity Risk


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

Equity Swap

returns are from

  1. a single stock

  2. a basket equity (ETF)

  3. an equity index


Adjust Beta

Future price is quoted as: F=fQuoteFuturePrice×Multiplier (f is %, multiplier convert the F into $ amount)





Or =βTβpβfMVpF

Ff×Multiplier are $ amount

Cash Equilisation

make βp or βT as "0".

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 β=0,for Stocks, BPV=0,for Bonds

所以此时,并不是直接 Sell bond,而是 make BPVT=0

不是直接 buy equity,而是 make β=0

Manage Currency Risks

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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

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

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=Amount of Currency to be ExchangedFuture Contract Size


Sell EUR/USD 相当于卖USD

Manage Volatility Risks

VIX Futures


VIX and Equity returns are mostly negative correlated

P.S. Cost of Carry model does not work on VIX F=SX(1+rf)TCB+CC, because VIX spot is not able to be invested (VIX is an index, is calculated)

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

Swap 的双方为 σ2 and K2. 如果市场上实际的 variance, σ2>K2 the strike price squared.

那么 每份swap 的payoffσ2K2 相当于用K2σ2.

整个 swap 的 payoff 是,即 settlement amount = Nvariance×(σ2K2)