3. Fomula-Financial Market and Product

1. 对冲比率

1.1 The optimal hedge ratio

$$HR={\rho }_{S,F}\frac{{\sigma }_S}{{\sigma }_F}$$

$$\beta =\frac{Cov(S,F)}{{\sigma }_F^2}=\frac{\rho {\sigma }_F{\sigma }_S}{{\sigma }_F^2}=\frac{\rho {\sigma }_S}{{\sigma }_F}=h^{\ast }$$

一份现货用多少个期货来对冲

1.2 Optimal Number of Futures Contracts

$$N^{\ast }=\frac{h^{\ast }{N}_A}{Q_F}$$

一份现货要用多少份期货合约来对冲

2. 股指期货对冲

Hedging an existing equity portfolio with index futures

$$\text{number of contracts}=({\beta }^{\ast }-\beta )\frac{V_A}{V_F}$$

target beta 可以是任意的值

3. 利率

$$FV=A{\left(1+\frac{R}{m}\right)}^{mn}⟺FV=A{e}^{Rn}$$

注意离散复利和连续复利之间的转换

4. 远期利率

$$R_{\text{Forward}}=\frac{R_2{T}_2-R_1{T}_1}{T_2-T_1}$$

通过画图法来进行记忆

5. 利率报价

$$\text{Full Price}=\text{Clean Price}+\text{Accrued Interest}$$

$$\text{Accrued Interest}=\frac{\text{Number of days between dates}}{\text{Number of days in reference period}}\times \text{Interest earned in redference period}$$

注意使用计算器计算时间跨度

6. 利率互换 Interest Rate Swaps

$$C=\frac{1-B_n}{B_1+B_2+\cdots +B_n}$$

求swap rate, 会算discount factor, 常规考法

7. 股票期权的性质

Put-Call Parity (European)

$$c+X{e}^{-rT}=S+p$$

记忆方法 $CK=PS$

8. 股票组合的期权策略

8.1 Covered Call

Short Call + Long Stock

Profit = ( S T − S 0 ) − [ m a x { 0 , ( S T − X ) } − C ] \text{Profit}=(S_T-S_0)-[max \{0,(S_T-X)\}-C] Profit=(ST​−S0​)−[max{0,(ST​−X)}−C]

8.2 Protective Put

Long Put + Long Stock

Profit = ( S T − S 0 ) + [ m a x { 0 , ( X − S T ) } − P ] \text{Profit}=(S_T-S_0)+[max \{0,(X-S_T)\}-P] Profit=(ST​−S0​)+[max{0,(X−ST​)}−P]

8.3 Bull Spread

Bull Call Spread = Long Call at $X_L$ + Short Call at $X_H$

Profit = [ m a x { 0 , ( S T − X L ) } − C L ] − [ m a x { 0 , ( S T − X H ) } − C H ] \text{Profit}=[max \{0,(S_T-X_L)\}-C_L]-[max \{0,(S_T-X_H)\}-C_H] Profit=[max{0,(ST​−XL​)}−CL​]−[max{0,(ST​−XH​)}−CH​]

Bull Put Spread = Long Put at $X_L$ + Short Put at $X_H$

Profit = [ m a x { 0 , ( X L − S T ) } − P L ] − [ m a x { 0 , ( X H − S T ) } − P H ] \text{Profit}=[max \{0,(X_L-S_T)\}-P_L]-[max \{0,(X_H-S_T)\}-P_H] Profit=[max{0,(XL​−ST​)}−PL​]−[max{0,(XH​−ST​)}−PH​]

8.4 Bear Spread

Bear Call Spread = Short Call at $X_L$ + Long Call at $X_H$

Profit = − [ m a x { 0 , ( S T − X L ) } − C L ] + [ m a x { 0 , ( S T − X H ) } − C H ] \text{Profit}=-[max \{0,(S_T-X_L)\}-C_L]+[max \{0,(S_T-X_H)\}-C_H] Profit=−[max{0,(ST​−XL​)}−CL​]+[max{0,(ST​−XH​)}−CH​]

Bear Put Spread = Short Put at $X_L$ + Long Put at $X_H$
Profit = − [ m a x { 0 , ( X L − S T ) } − P L ] + [ m a x { 0 , ( X H − S T ) } − P H ] \text{Profit}=-[max \{0,(X_L-S_T)\}-P_L]+[max \{0,(X_H-S_T)\}-P_H] Profit=−[max{0,(XL​−ST​)}−PL​]+[max{0,(XH​−ST​)}−PH​]

8.5 Butterfly Spread

Butterfly Spread Using Calls = long call at $X_L$ +long call at $X_H$+short two calls at $X_M$

Profit = [ m a x { 0 , ( S T − X L ) } − C L ] + [ m a x { 0 , ( S T − X H ) } − C H ] − 2 [ m a x { 0 , ( S T − X M ) } − C M ] \text{Profit}=[max \{0,(S_T-X_L)\}-C_L]+[max \{0,(S_T-X_H)\}-C_H]-2[max \{0,(S_T-X_M)\}-C_M] Profit=[max{0,(ST​−XL​)}−CL​]+[max{0,(ST​−XH​)}−CH​]−2[max{0,(ST​−XM​)}−CM​]

Butterfly Spread Using Puts = long put at $X_L$ +long put at $X_H$+short two puts at $X_M$

Profit = [ m a x { 0 , ( X L − S T ) } − P L ] + [ m a x { 0 , ( X H − S T ) } − P H ] − 2 [ m a x { 0 , ( X M − S T ) } − P M ] \text{Profit}=[max \{0,(X_L-S_T)\}-P_L]+[max \{0,(X_H-S_T)\}-P_H]-2[max \{0,(X_M-S_T)\}-P_M] Profit=[max{0,(XL​−ST​)}−PL​]+[max{0,(XH​−ST​)}−PH​]−2[max{0,(XM​−ST​)}−PM​]

9. Greeks

$$\Delta =\frac{\partial c}{\partial S}=N(d_1)\sim (0,1)$$

$$\Delta =\frac{\partial p}{\partial S}=N(d_1)-1\sim (-1,0)$$

$$\text{portfolio delta}={\Delta }_p=\sum _{i=1}^n{w}_i{\Delta }_i$$

10. 二叉树

10.1 Synthetic Call Replication

$$\text{call price}=\text{hedge ratio}\times [\text{stock price}-PV(\text{borrowing})]$$

$$HR=\frac{c_U-c_D}{S_U-S_D}$$

括号中内容为 bankruptcy-free-portfolio.

10.2 Risk-Neutral Valuation

$$p=\frac{e^{r\Delta t}-d}{u-d}\to u=e^{\sigma \Delta t},d=1/u=e^{-\sigma \Delta t}$$

$$S{e}^{r\Delta t}=pSu+(1-p)Sd$$

记住$u$和$d$，学会构建二叉树模型

11. BSM模式

$$d_1=\frac{In(S_0/T)+[R_f^c+(0.5\times {\sigma }^2)]\times T}{\sigma \times \sqrt{T}}$$

$$d_2=d_1-\sigma \sqrt{T}$$

$$C_0=[S_0\times N(d_1)]-[X\times e^{-R_f^{c}T}\times N(d_2)]$$

Valuation of Warrants

$$\frac{N}{N+M}\times \text{value of regular call option}$$

$N$ 为之前股票在外发行的数量，$M$ 为权证行权导致的新发现的股票数量

12. 大宗商品投资-期货定价

12.1 Convenience Yield

$$F=(S+U){\left(\frac{1+R}{1+Y}\right)}^T\to Y={\left(\frac{S+U}{F}\right)}^{\frac{1}{T}}(1+R)-1$$

$U$是storage costs的现值

12.2 Lease Rate

$$F=S{\left(\frac{1+R}{1+l}\right)}^T\to l={\left(\frac{S}{F}\right)}^{\frac{1}{T}}(1+R)-1$$

13. 外汇管理

13.1 名义利率和实际利率

$$(1+R_{nom})=(1+R_{real})(1+R_{inf})\to R_{nom}\approx R_{real}+R_{inf}$$

13.2 Covered Interest Parity

$$F=S{\left[\frac{(1+R_{YYY})}{(1+R_{XXX})}\right]}^T$$

$$F=S\times e^{(R_{YYY}-R_{XXX})T}$$