期权期货复习

复习题

Explanation 名词解释

speculators wish to take a position in the market.Either they are betting that a price will go up or they are betting that it will go down. They use derivatives to get extra leverage

Hedgers are interested in reducing a risk that they already face. Arbitrage involves locking in a risk-less profit by entering simultaneously into transactions in two or more markets.

A call option gives the holder the right to buy an asset by a certain date for a certain price.

Put option: A put option gives the holder the right to sell an asset by a certain date for a certain price.

Futures （forward）contract: It is an agreement to buy or sell an asset for a certain price at a certain time in the future.

short selling: T he investor’s broker borrows the shares from another client’s account and sells them in the usual way. To close out the position the investor must purchase the shares. The broker then replaces them in the account of the client from whom they were borrowed.

In-the-money option/At the money/ Out out of the money

Time value, intrinsic value, option value,

risk-neutral valuation: Firstly, assume that the expected return from the stock price is the risk-free rate r, then calculate the expected payoff from the option, at last, discounting the expected payoff at the risk-free rate

Factors affecting stock option pricing: stock price, strike price, risk-free interest rate, volatility, time to maturity, and dividends.

Long position of forward:

A callable bond（可提前赎回债券）: It contains provisions(条款) that allow the issuing firm(发行公司) to buy back the bond at a predetermined price at certain times in the future.

risk-neutral valuation:Firstly, assume that the expected return from the stock price is the risk-free rate r, then calculate the expected payoff from the option, at last, discounting the expected payoff at the risk-free rate

Swaps：Swaps are private agreements between two companies to exchange cash flows in the future according to a prearranged formula.

新型期权的常见产品

Asian option, barrier option, lookback option, compound option, forward

start option, as you like option(choose option), convitable bond(可转换债券

Lookback options : the payoffs from lookback options depend on the maximum or minimum stock price reached during the life of the option

‘Out’ options(敲出期权)：Option dies if stock price hits barrier before option maturity

‘In’ options(敲入期权)：A knock-in option ( 敲入期权) is an option that comes into existence only if the underlying asset price reaches a certain barrier before option maturity .

Barrier options are options where the payoff depends on whether the underlying asset's price reaches a certain level before option maturity.

Forward start options ：are options that are paid for now but will start at some time in the future. The strike price is usually equal to the price of the asset at the time the option starts ,ie, the option is at the money.

Asian options：the payoffs from asian options depend on the average price of the underlying asset during at least some part of the life of the option.

价差组合期权

bottom vertical strangle: a bottom vertical strangle can be created by buy a put with lower strike prices and buy a call with higher strike prices.

Bull spreads: A bull spread can be created using two call options with the same maturity and different strike prices. The investor buys the call option with the lower strike price and shorts the call option with the higher strike price. Bull spreads can also be created by buying a put with a low strike price and selling a put with a high strike price.

Bear spreads: A bear spread can be created by selling a call with one lower strike price and buying a call with another higher strike price Butterfly spreads: A butterfly spread involves positions in options with three different strike prices: buying two call options with strike prices X1 and X3, and selling two call options with a strike price X2, X1＜X2 ＜X3

2 .Explain the differences between forward contract and futures contract?

3. 签订一份期货合约对open interest(开放权益)和volum of trade （交易量）的影响

如果交易双方签订一个新合约，那么未平仓合约数增加1个。如果交易双方就同一个合约进行平仓，那么未平仓合约数减少一个。如果一方订立一个新合约，而另一方同时将已有合约平仓，那么未平仓合约数不变。交易量都是增加。

4

解：公司A 的比较优势在英镑而需要美元借款，公司B 相反，因此存在互换的基础。英镑上的利差为0.4％，美元上的利差为0.8％，因此互换的总获利为0.4％。已知银行获利0.1％，则两个公司各获利0.15％。因此A 实际上以6.85％的利率借美元，而B 实际上以10.45％的利率借英镑。在银行承担所有市场风险的情况下，互换安排如下图：

1）互换收益的分配在公司A 、公司B 和银行间的比例分别为 35%、35%、30%。如何设计 ？

2）若分配比例为50%、25%、25%，如何设计 ？

Private contract between 2 parties

Exchange traded Non-standard contract

Standard contract Usually 1 specified delivery date

Range of delivery dates Settled at maturity Settled daily Delivery or final cash

settlement usually occurs Contract usually closed out prior to maturity FORWARDS

FUTURES 美元6.85％ 美元6.2％ 英镑11％ 英镑11％ 美元6.2％

英镑10.45％ 公司A 银 行 公司B

美元6.86％

美元6.2％

英镑11％ 英镑11％ 美元6.2％ 英镑10.46％ 公司A

银 行 公司B

5.

6 基于同一股票的看跌期权有相同的到期日.执行价格为$70、$65和$60，市场价格分为$5、$3和$2. 如何构造蝶式差价期权.请用一个表格说明这种策略带来的盈利性.股票价格在什么范围时，蝶式差价期权将导致损失？

7 基于同一股票的有相同的到期日敲定价为 $70的看涨期权价格为 $4. 敲定价$65 的看涨期权的价格为 $6。解释如何构造bull/bear spread option.请用一个表格说明这种策略带来的盈利性.股票价格在什么范围时，价差期权将导致损失？

8. Oil: An Arbitrage Opportunity?

Suppose that:

- The spot price of gold is US$95

- The quoted 1-year futures price of gold is US$125

- The 1-year US$ interest rate is 5% per annum

Is there an arbitrage opportunity ？

Suppose that:

- The spot price of gold is US$95

- The quoted 1-year futures price of gold is US$80

- The 1-year US$ interest rate is 5% per annum

Is there an arbitrage opportunity?

8 远期/期货价格公式及其价值公式

()()()()()r T t r T t r q T t F Se S I e Se ----==-=

B-S 公式的使用()()12()()q T t r T t c Se N d Xe N d ----=-

()()21()()r T t q T t p Xe N d Se N d ----=---

美元6.8％ 美元6.2％

英镑11％ 英镑11％ 美元6.2％ 英镑10.5％ 公司A

银 行 公司B

21d =

21d d =-1).What is the price of a European call option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility is 35% per annum, and the time to maturity is six months?

2). Suppose the current value of the index is 500, continuous dividend yields of index is 4% per annum, the risk-free interest rate is 6% per annum . if the price of three-month European index call option with exercise price 490is $20, What is the price of a three-month European index put option with exercise price 490?

by put-call parity

3) What is the price of a European futures put option ：current futures price is $19, the strike price is $20, the risk-free interest rate is 12% per annum, the volatility is 20% per annum, and the time to maturity is five months? （保留2位小数）

Solution: In this case F=19,X=20, r=0.12, σ=0.20, T -t=0.42,

210.33d ==-

210.46d d =-=-

(0.33)0.6293,(0.46)0.6772N N ==

12()(0.33)0.6293,()(0.46)0.6772N d N N d N -==-== The price of the European put is

()()210.120.420.120.42()()

200.6772190.6293 1.51r T t r T t p Xe N d Fe N d e e -----?-?=---=?-?=

4) A one-year-long forward contract on a non-dividend-paying stock is entered into when the stock price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding.

(a) What are the forward price and the initial value of the forward contract?

(b) Six months later, the price of the stock is $45 and the risk-free interest rate is

still 10%. What are the forward price and the value of the forward contract? The forward price, 21.44401.0)(===-e Se F t T r ，

The initial value of the forward contract is zero.0=f

(a) The delivery price K in the contract is $44.21. The value of the forward

contract after six months is given:

95.221.44455.01.0)(=-=-=?---e Ke S f t T r

The forward price, 31..47455.01.0)(===?-e Se F t T r

5) 计算基于无红利支付股票的欧式看跌期权价格，其中执行价格为＄50，现价为＄50，有效期3个月期，无风险年收益率为10%，波动率为每年30%。

若在两个月后预期支付的红利为＄1.50，则计算会有何变化

解 050,50,0.1,0.3,0.25S X r T σ=====

10.2417d ==

210.0917d d =-=

欧式看跌期权价格是

0.10.250.10.2550(0.0.0917)50(0.2417)

500.4634500.4045 2.37

N e N e -?-?---=?-?= 若在两个月后预期支付的红利为＄1.50，在本题中我们在使用BS 公式前必须从股票价格中减去红利的贴现值，因此0S 应该是

0.16670.1050 1.5048.52S e

-?=-= 其他变量不变50,0.1,0.3,0.25X r T σ==== 在本题中

10.0414d ==

210.1086d d =-=-

欧式看跌期权价格是

0.10.250.10.2550(0.1086)48.52(0.0414)

500.543248.520.4045 3.03

N e N e -?-?---=?-?=

6) 求无红利支付股票的欧式看涨期权的价格。其中股票价格为＄52，执行价格为＄50，无风险年收益率为12%，年波动率为30%，到期日为3个月。

在本题中052,50,0.12,0.30,0.25S X r T σ=====

1210.53650.3865

d d d ===-= 欧式看涨期权的价格是

0.0352(0.5365)500.6504 5.06N e --?=

7) 求无红利支付股票的欧式看跌期权的价格。其中股票价格为＄69，执行价格为＄70，无风险年收益率为5%，年波动率为35%，到期日为6个月。

在本题中069,70,0.05,0.35,0.5S X r T σ=====

21210.16660.0809

d d d ===-=- 欧式看跌期权价格为

0.050.50.02570(0.0809)69(0.1666)700.5323690.4338 6.40e N N e -?---=?-?=

9. 试证明标的资产无红利发放时的欧式看涨期权和看跌期权满足的平价公式

()r T t c Ke p S --+=+

10. 试证明标的资产有红利发放时的欧式看涨期权和看跌期权满足的平价公式

()r T t c Ke D p S --++=+

11. 1）证明在风险中性环境下，到期的欧式看涨期权被执行的概率为 )(2d N ,

2） 使用风险中性定价原理，假设股票1的价格和股票2的价格分别服从几何布朗运动，且独立，给到期损益为如下形式的欧式衍生品定价：

121,2 : 0 else

T T T K S X S X T f ?>>?=??? Solution: Since ())(),)2

/(ln ~ln 22t T t T r S N S T ---+σσ（ )N(d )))(2/(ln ln ( )))(2/(ln ln (1)

ln p(ln 1)ln p(ln )p(222=------=------=<-=>=>t

T t T r S X N t

T t T r S X N X S X S X S T T T σσσσ Since 121,22 : and p() N(d )0 else

T T T T K S X S X T f S X ?>>?=>=???

1

21,2212122122()()2122[] P()

K[P() *P() ]K[N(d )*N(d )][][N(d )*N(d )]

T T T T T r T t r T t T E f K S X S X S X S X f e E f e K ----=>>=>>===

Where

1221d =

,2222d =,

12. Use two-step tree to value an American 2-year put option on a

non-dividend-paying stock, current stock price is 50, the strike price is $52, and the volatility of stock price is 30% per annum, the risk-free interest rate is 5% per annum. （保留2位小数）

In this case, S =50, X = 52,σ = 0.3, Δt =1, r=0.05 , the parameters necessary to construct the tree are

11.35,0.74u e d u

====, 0.05*1=1.10e 0.05*10.51,10.49e d p p u d -==-=- {}])1([,max .11.1,j i j i t r j i j j i f p pf e d Su X f +++?---+-=

13 If a stock price, S, follows geometric Brownian motion t SdW Sdt dS σμ+=

1) What is the process followed by the variable n S ? Show that n S also follows geometric Brownian motion.

2)The expected value of ST is =)(T S E )(t T Se -μ. What is the expected value of n T S ?

3) The varaince of ST is =)(T S D )1()()(222---t T t T e e

S σμ.

What is the variance of n T S ? 4) Using risk-neutral valuation to value the derivative, whose payoff at maturity is

50 7.43 50 2 27.44

24.56

91.11 0 37.04

14.96

67.4 0.93

:n T T

T f S = 1)We now use Ito's lemma to derive the process followed by n S ,

Define n S G =,

0,)1(,222

1=??

-=??=??--t G

S n n S G nS S G n n

t SdW Sdt dS σμ+=

2

22)(21dS S G dS S G

dt t G

dG ??+??+??=

2

21)()1(21

)(t n t n SdW Sdt S n n SdW Sdt nS σμσμ+-++=--dt S n n dW S n dt S n n t n n 2)1(21

σσμ-++=

t n n n dW S n dt S n n S n σσμ+-+=])1(21

[2

t n n dW S n dt S n n n dG σσμ+-+=])1(21

[2

t GdW n Gdt n n n σσμ+-+=])1(21

[2

n S G =Θ

So that n S also follows geometric Brownian motion.

2) t SdW Sdt dS ce σμ+=sin

=)(T S E )(t T Se -μ. dG t GdW n Gdt n n n σσμ+-+=])1(21

[2

=?)(T G E )

]()1(21

[2t T n n n Ge --+σμ n S G =Θ

,

=∴)(n T S E )]()1(21

[2t T n n n n e S --+σμ

3) Since t SdW Sdt dS σμ+= and varaince of ST is

=)(T S D )1()

()(222---t T t T e e S σμ.

Similarly, by dG t GdW n Gdt n n n σσμ+-+=])1(21

[2

We get the varaince of n T S is

=)(n T S D =)(T G D ]1[)()]()1(2[2222----+t T n t T n n n e e G σσμ

]1[)()()]()1(2[2222-=---+t T n t T n n n n e e S σσμ

14 In a risk-neutral world, suppose stock prices follow geometric

Brownian motion

,dS rSdt SdW σ=+

1) What is the process followed by the variable n S by Ito’s lemma? Show that n S also follows geometric Brownian motion.

2) The expected value of T S is =)(T S E ()r T t Se -. What is the expected value of n T S ?

4) Using risk-neutral valuation to value the derivative, whose payoff at maturity is

:n T T at T f S =

13. Show that the probability that a European call option will be exercised in a risk-neutral world is, )(2d N . Using risk-neutral valuation to value the complicated digtial option whose payoff at maturity is

else

0X S X K f T T 1T ???<<=2: Solution: Since ())(),)2/(ln ~ln 22t T t T r S N S T ---+σσ（

)N(d )))(2/(ln ln ( )))(2/(ln ln (1)

ln p(ln 1)ln p(ln )p(222=------=------=<-=>=>t

T t T r S X N t

T t T r S X N X S X S X S T T T σσσσ 14. The stock price process assumed satisfies

SdW Sdt dS σμ+=

Suppose that f is the price of a call option or other derivative contingent on S. Using no

arbitrage opportunity to derive the Black-Scholes Differential Equation

rf S f S S f rS t f =??+??+??222221σ

Solution: Since f is a function of S and t, let ：f = f (S , t )，it follows from Ito lemma that

t dW S f S dt S f S S f S t f df ??+???? ????+??+??=σσμ222221

The appropriate portfolio is

Short: one derivative

Long : Δ shares

Value of the portfolio at time t : t t t f S -?=∏

The change of in the value of the portflio in time dt is given:

t t t df dS d -?=∏

[]22221212t t t t t t t t t t t t t t t t t t t t t t t t f f f S dt S dW S S dt S dW t S S f f f S dt S dW S S dt S dW t S S μσμσσμσμσσ?????=?+-++-???????

?????=?+?-++-???????

221()2t t t t t t t t t t f f f S S dW S S S dt S t S σσμσμ?????=?--++-????????

Choosing ?to make the portflio in time dt is iskless ，S f ??=?，thus dt r d t t ∏=∏

dt f S S f r dt S f S t f t t t t t t t t )()21(2222-??=??+??-?σ t t t t t t t t rf S f S S f rS t f =??+??+???222221σ