(Finding Value at Risk (VAR) of a Portfolio) Suppose we own one share of Dell on June 30, 2015. The current price is $94. From above we have estimated that the growth of the price of Dell stock can be modeled as a Lognormal random variable with and . To hedge the risk involved in owning Dell we are considering buying (for $5.25) a European put on Dell with exercise price $80 and expiration date November 22, 2015. Compute

A) the VAR on November 22, 2015 if we own Dell computer and do not buy a put.

B) the VAR on November 22, 2015 if we own Dell computer and buy the put.

HINT-

(Value at Risk (VaR)) Value at risk of a portfolio at a future point in time is usually considered to be the fifth percentile of the loss in the portfolios value at that point in time. In short, there is considered to be only on chance in 20 that the portfolios loss will exceed the VaR.

I. For example, a portfolio today is worth $100. We simulate the portfolios value one year from now and find there is a 5% chance that the portfolios value will be $80 or less. Then the portfolios VaR is $20 or 20%.

II. Please be noted that if we buy a put, the money we invest (i.e. the value of our portfolio) at the beginning increases, which is Beginning Dell Price + Put Price instead of the Beginning Dell Price only for the case without buying a put.

(Put options.) A put option gives the owner the right but not the obligation to sell the underlying asset at a predetermined price during a predetermined time period. The seller of a put option is obligated to buy if asked. The mechanics of the European put option are the following:

at time 0:

1. The contract is agreed upon between the buyer and the writer of the option,

2. The logistics of the contract are worked out (these include the underlying asset, the expiration date T and the exercise (or strike) price K),

3. The buyer of the option pays a premium to the writer.

at time T:

The buyer of the option can choose whether he/she will sell the underlying asset for the strike price K.

(The European put option payoff.)

If the exercise price K exceeds the final asset price S(T), i.e., if S(T)=K, the put-option owner would be better of selling the asset at the market price. So, he/she will simply walk away from the contract incurring the payoff of 0.

In conclusion, the long-put-option payoff = max(K-S(T), 0).

Note that the premium in our case equals to $5.25.