Note:

· Two hours

· This exam is open books and open notes

· Searching information related to the exam questions online is not allowed

· Students are prohibited to discuss with each other during the exam by any means

· A scientific/financial calculator is allowed

· This exam consists 5 problem-solving questions Q1 (16 points)

Q2 (20 points)

Q3 (14 points)

Q4 (21 points)

Q5 (29 points)

· Please write down your name and student number on each page of your submitted solution report. Please scan your solution report using a scanning app on your cell phone as instructed before the exam. You must submit your solution report via your Moodle account by 12:15pm. Late submission will not be graded.

·  In case you experience a technical problem when submitting via Moodle, you can email me your solution report at [email protected] and cc Emma at  [email protected] by 12:15pm and then submit via Moodle later. In that case,  only the version at the earliest submission is graded.

· In any case, submission later than 12:15 pm on Dec 14, 2019 will not be graded and you will receive zero on this exam.

1. (16 points)  Given the following inputs, please roughly draw the price curve of both a European call and a European put as a function of the underlying stock price on the same graph assuming all Black and Scholes assumptions hold. Please also draw the lower bound and higher bound of the price of the options.

Both the call and the put will mature in one month, the risk-free rate r=1.8%, the return volatility of the stock s = 20% , and the strike price K = 40 .

(The value of pi =p =3.141592654.)

2. (20 points) Suppose you hold a delta-hedged portfolio comprised of a stock and its derivatives. The portfolio’s delta will drop by 3000 when the underlying stock price goes up by $1 and the portfolio value will drop by $60 when the stock’s return volatility goes up by 100 basis points. There are two traded options A and B written on the same underlying stock as the portfolio. When the underlying stock price increases by $1, the value of options A and B will increase by $0.50 and $0.10, respectively, and the delta of options A and B will increase by 0.60  and 0.50, respectively. In addition, when the stock’s return volatility goes up by 100 basis points, the value of options A and B will go up by $0.02 and $0.015, respectively. What is your hedging strategy if you want to make the portfolio delta-neutral, gamma-neutral and vega-neutral simultaneously? Suppose you decide to use the options A and B and the underlying stock as hedging instruments. You need to show clear derivation steps.

3. (14 points) A variable, x, follows the process:

dx = mdt + s dBt

where m and s are constants and Bt is a standard Brownian motion under

measure P. Find the process followed by

y = ea x-b t

under measure P, where a

and b are constants. You need to show clear derivation steps.

4. (21 points) The payoff function of a contract at maturity T is given below,

where K is the strike price and St

is the current price of a risky stock. The stock

does not pay dividends. K is a constant parameter.

In the Black-Scholes framework, price this contract by martingale approach. You need to show clear derivation steps.

5. (29 points) The payoff function of a d -option at maturity T is given below,

ì0,

C (S ,T ) = ï 1

ï

ïî0,

where K is the strike price and  St

ST < K - e

, K - e < ST < K + e

ST > K + e

is the current price of a risky stock. The stock