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### Option as Derivatives

The figure above shows a call option payoff graphs. Options are derivatives instruments. A derivative is a financial instrument whose value is ‘derived’ from the underlying asset. In this case, the option is on an underlying equity (stock). A call option gives the buyer (holder) of the option the right to buy the asset (stock) at a pre-determined strike price (also called exercise price). The option also has a maturity or time-frame within which it is valid.

To summarize: S0 = is current stock price, St= stock price at expiry, X = Exercise price at which holder of call option can buy the stock T = Maturity of the option, rf= risk free interest rate

### Call Option

A Call option gives the owner the right to buy a security. In the graph above, the strike price is $20. As long as the stock price is below $20 at maturity/expiry of the option the option expires worthless. That is, if the stock is trading at $18 in the market, the holder of the option will not pay $20 to buy it by exercising the option. Therefore, the option expires without being exercised and is worthless to the holder of the option.

Consider, if the stock is trading at $22 on maturity date the owner of the option will benefit from exercising it. In this case the owner will buy the stock at $20 which is worth $22 in the market, giving him a payoff of $2. The payoff from a call option is thus given by Max{ St – X, 0}. This formula implies that the payoff will be the larger value of 0 and a positive number. Option payoff can thus never be negative. Buyer of a call option profits when the stock price goes up. Put differently, a buyer of a call option bets of the stock price to goes up. Work with our finance tutor in London for an in-depth understanding of pricing and payoffs of call options.

### Put Option

The figure above shows a put option payoff in red and a call option payoff in blue. Put option is a right to sell an asset at a pre-determined strike price. Buyer of the put options profits when the stock price declines. In other words, a put option buyer is betting stock price to go down. Finance tutor in London will cover Put-Call parity in detail.

**This question is about option pricing**

The current price of stock XYZ is £100, and each month it either goes up by 15% or goes down by 10%. The monthly riskless interest rate is 5%. Assume that the stock does not pay dividends. Our finance tutor in London can practice similar questions for courses like corporate finance, financial management, and financial markets.

- [5 marks] Construct a binomial tree with three dates 0, 1, and 2 for stock price movements and calculate risk neutral probabilities of the up and down moves.
- [5 marks] Calculate the time-0 price of a European call with strike price of £100 and maturity at date 2.
- [5 marks] Determine the delta of the call option in part b) at time zero. Using put-call parity determine the time-0 price of a put option with strike price £100, which matures at date 2.
- [5 marks] What is the price of a bull spread with strikes and maturing at date 2? Assume that the price movements of the stock are described by the binomial tree in part a). (Hint: a bull spread is constructed as a long position in one European call with strike Kl and a short position in one European call with strike ).
- [5 marks] Consider a European security, which pays at the terminal date . That is, the payoff is the minimum of two numbers: stock price and . Assume that Plot the payoff diagram. Construct a portfolio of stocks and call options, which replicates the payoff of this security at date .

### Bear spread using call options

We can also combine multiple options to create exposures suitable for our views. For example a bear spread is a strategy used to gain from downside. We can sell a call at a lower strike price (K1) and buy a call at a higher strike price (K2). The diagram below shows the payoff of a bear spread. Can you write an expression for the payoff with a short call at K1 and a long call at K2? Our finance tutor in London will go over commonly used option strategies such as bear spread, box spread, collar, butterfly, straddle, and strangle.

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