Question

Q1. Uniform Distribution You believe stock price will follow uniform distribution with mean of 100 and MAD 20. You are pricing a CALL option with strike at 110. a. what is the mean and range of the distribution? b. what is the probability that the call will be ITM at expiration (ie stock price ends above strike at 110)? c. what is the conditional mean of stock price when CALL is ITM (aka stock price is above strike 110)? d. what is the conditional CALL option average payment when the CALL is ITM? e. what is the fair value of the CALL today, ie the unconditional average payment today? (additional practice: redo for 90 strike CALL, 90 strike PUT, 110 strike PUT) Q2. Implied MAD Today the stock is trading at 100, which is also the expected value of future stock price. If a 110 strike CALL is priced at $2, what is the implied MAD? Q3. Normal distribution Stock price today at 100. Stdev is 20. a. How much should a 110 BINARY CALL be priced today? b. How much should a $110 binary PUT be priced today? Q4. Stock price today at $100. A binary PUT with strike of 85 is priced at $0.15. What is the implied standard deviation? Q5. Value at Risk (VaR) A VaR is the loss a portfolio is expected to lose at a certain frequency. For example a 1% VaR is expected to happen at 1% frequency, or once every 100 time periods. A strategy you developed is expected to generate 10 bps of mean return with 100 bps of stdev for return at daily frequency. a. If you manage $100M, What is $ value for 5%, 1%, and 0.1% VaR for the strategy on a daily basis? b. If your trading desk's 1% VaR is set at $10M by the chief risk officer of the company, how much money can you deploy for your strategy?

Answer 1

a) Mean and range of distribution:

Mean of the distribution = 100

Let's say the range is (a,b)

Since mean is the mid-point of range in a uniform distribution,

We can write a= 100-k, b=100+k (k > 0)

Variance of the uniform distribution with range (a,b) is given by:

Also, = 20² = 400

Thus, k²/3= 400 or k²= 400 X 3 =1200 or k= = 34.64

a= 100-34.64= 65.36 ; b= 100+ 34.64 = 134.64

Hence, the range of distribution is given by ( 65.36, 134.64)

b) The call option is in the money if the underlying price is greater than 110. In a uniform distribution, the probability of each point within range is the same. Hence, the probability of call option being ITM is :

or there is 35.5% chance that call option will be ITM.

c) Conditional mean of stock price when call is ITM=

= (134.64+ 110) / 2 = 244.64/2= 122.32

d)  Let X be the stock price, then the call option payment (payoff) is X-110 when the call is ITM.

As the call option starts getting ITM at 110, thus lower bound of call option payment = X-110 = 110-110 = 0

The upper bound of call option average payment will be when the underlying reaches its highest value. This value is given by:

= 134.64-110 = 24.64

This being uniform distribution, conditional call option average payment is mean of lower and upper bound.

Or conditional call option average payment = (0 + 24.64) / 2 = 12.32

e) When the call option is Out-of-money, there is no payment.

Thus, unconditional average payment today = Probability of call option getting ITM X Call option average payment when the call is ITM

Thus, unconditional average payment today = 0.355 x 12.32 = 4.3736