# 2 (i) What is the difference between entering into a long forward contract when the forward price is \$50 and taking a long position in a call option with a strike price of \$50? (ii) Consider two European call options expiring tomorrow on the same underly- ing asset, one with strike price equal to \$40 and one with strike price equal to \$60. If today’s close price of the underlying asset is \$42 which of the two option has higher price and why? (iii) Suppose that you write a put contract with a strike price of \$40 and an ex- piration date in 3 months. The current stock price is \$41 and the contract is on 100 shares. What have you committed yourself to? How much could you gain or lose? (iv) What is the difference between selling a call option and buying a put option? (v) Consider a geometric Brownian motion with drift u and volatility o: dS = uS dt+oSdW. What is the process followed by the variable s?? Calculate the expected return, the variance and the expected value of S.

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i) Entering into a long forward contract is equivalent to committing today to purchase the asset at \$50 at the expiry no matter what the price of the asset is at the expiry.. So, in this contract there is a binding obligation to purchase the asset at \$50

Whereas , purchasing one call option with Strike \$50 is equivalent to purchasing an option , whereby one can purchase the stock at \$50 if one wants to do so, else one may let the option lapse. So, there is no obligation to purchase the asset at \$50. So, if the asset price is less than \$50 at expiry, the buyer of the option may let the option lapse and not purchase the asset at \$50.

ii) On comaparing the Value of two call options on the same underlying asset with the same expiry, the one with a higher strike price will have a lesser value. As it gives the option to purchase the asset at a higher price, so it has a lesser value. Only at expiry , the values of call options are given by max(St-K,0) where St is the asset price at maturity and K is the strike price

So, out of the two call options , the one with strike \$40 will have a higher value (price) than the one with a strike price of \$60. It can also be seen easily as the stock has a price of \$42 , the \$40 strike option is in the money and \$60 strike option is out the money.

iii) On writing (selling) the put option on 100 shares at \$40 , the committment made is to purchase 100 shares from the buyer of the option at expiry at a price of \$40 per share (no matter what the price). Obviously , the buyer of the option would exercise his/her option only when the price at expiry is less than \$40. So, whenever the option is exercised the seller loses, sometimes more than the premium earned by option selling.

The stock can go as low as 0 (theoretically) . So, the writer can lose upto \$40*100 = \$4000

The gain is when the stock goes up beyond \$40 . The buyer simply does not exercise the option and the gain to the writer is only limited to the premium earned by selling the option

iv) Selling a call option and buying a put option are option strategies for a bearish market outlook i.e. when one expects the markets to go down. However, the payoffs of these two strategies are very different

While Selling a call option gives an immediate cash inflow of premium , the seller takes high risk of the stock going up and loses heavily (theoretically infinite loss is possible) when the stock goes up. On the other hand if the stock goes down, the profit is limited to the premium gained.

Buying a put option makes one pay the premium today (immediate cash outflow). However, if stock goes up then one does not lose, as one can simply ignore the option and let it lapse. So, the downside losses are protected and the upside gainns can be made. This strategy protects the portfolio very well.