Solution: In this case F=19,X=20, r=0.12, σ=0.20, T-t=0.42,
d1
ln(F
/
X)
( 2 / T t
2)(T
t)
0.33
d2 d1 0.2 0.4167 0.46
N(0.33) 0.6293, N(0.46) 0.6772
N (d1) N (0.33) 0.6293, N (d2 ) N (0.46) 0.6772
In this case, S=50, X = 52,σ = 0.3, Δt =1, r=0.05 , the parameters necessary to construct the tree are
u e t 1.35, d 1 0.74 , e0.05*1=1.10 p e0.05*1 d 0.51, 1 p 0.49
(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, F Ser(Tt) 40e0.1 44.21，
The initial value of the forward contract is zero. f 0
f er(T t) E[ fT ] er(T t) K[N(d21 ) * N(d22 )]
d21
ln(S1
/
X1) (r 12 1 T t
/
2)(T
t)
, d22
ln(S 2
/
X
2
)
(r
2 2
2 T t
/
2)(T
t)
,
九、Use two-step tree to value an American 2-year put option on a
(1) 400 of the first traded option (2) 6,000 of the second traded option. And short 3240 underlying asset.
八 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 N (d2 ) ,
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：
答案： buy a put with the strike prices $65 and buy a call with the strike prices
$70, this portfolio would need initial cost $10.
The pattern of profits from the strangle is the following:
The price of the European put is p Xer(T t) N (d2 ) Fer(T t) N (d1) 20e0.120.42 0.6772 19 0.6293e0.120.42 1.51
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.
Stock Price
Payoff from
Payoff from
Total Payoff
Total Profits
Range
Long Put
Long Call
ST ≤65
65- ST
0
65- ST
55 - ST
65 ＜ ST ＜70
0
0
0
-10
ST >70
0
ST-70
ST-70
ST-80
当 50<ST<80时，组合会带来损失
N (
ln
X
ln
S
(r 2 T t
/
2)(T
t))
N(d
2)
Since T :
fT
K
0
S
1 T
X1, ST 2
X2
else
and
p(ST X ) N(d2 )
Where
E[ fT ] K
S
1 T
X1, ST 2
X22 )
K[P(ST1 X1) *P(ST 2 X 2 ) ] K[N(d21) * N(d22 )]
(a) The delivery price K in the contract is $44.21. The value of the forward contract after six months is given:
f S Ker(T t) 45 44.21e0.10.5 2.95
The forward price, F Ser(T t) 45e0.10.5 47..31
七 Consider a portfolio that is delta neutral, with a gamma of -5,000 and a vega of -8,000. Suppose that a traded option has a gamma of 0.5, a vega of 2.0, and a delta of 0.6.
T:
fT
K
0
S
1 T
X1, ST 2
X2
else
Solution: Since ln ST ~ N ln S (r 2 / 2（) T t), 2 (T t)
p(ST X ) p(lnST ln X ) 1 p(lnST ln X )
1 N (ln X ln S (r 2 / 2)(T t)) T t
ST n ? 3) The varaince of ST is D(ST ) S e (e 2 2 (Tt) 2 (Tt) 1) . What is the variance of ST n ? 4) Using risk-neutral valuation to value the derivative, whose payoff at maturity is
p Xer(T t) N (d2 ) Seq(T t) N (d1 )
d1
ln(S
/
X
)
(r
q T
t
2
/
2)(T
t)
d2 d1 T t
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?
四、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？
五、 基于同一股票的有相同的到期日敲定价为 $70 的期权市场价格为 $4. 敲 定价$65 的看跌期权的市场价格为 $6。解释如何构造底部宽跨式期权.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，宽跨式期权将导致损 失？
Another traded option with a gamma of 0.8, a vega of 1.2, and a delta of 0.5. What position in the traded two call options and in the underlying asset would make the portfolio gamma ,vega and delta neutral？ Solution: If , w1 ,w2 , ,w3 are the amounts of the two traded options and underlying asset included in the portfolio, we require that
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）