Financial Mathematics Essay

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Financial Mathematics culminating in an Introduction to the Black and Scholes Model.

1.Background to Financial Mathematics

i. Definitions of financial objects

Within the financial services industry there are a multitude of different assets. An asset is a financial entity whose current value is known however in the future it is liable to change. Such assets include shares, commodities and currencies (a).

There are many services available to investors who aim to make money from the financial markets. The one I shall examine in this report is call options. A call option is a financial contract between the buyer and seller of that option. There are two types of call options: a European call option and a European Put Option.

Definition 1:
A European Call Option allows its buyer the right (but no the obligation) to purchase from the seller, of set option, a predetermined asset for a pre-agreed price (strike price) at a particular time in the future (the expiration date) (b). A European Put Option allows its buyer the right (but not the obligation) to sell to the seller, of set option, a predetermined asset for a pre-agreed price (strike price) at a particular time in the future (the expiration date) (b). The price of an option is called a premium (a). We can denote the value of the European call option & put option at their expiration date, by C & P respectively.

It should be clear from their definitions that:
C = max (ST – K, 0)
P = max (K - ST, 0)
Where K is the strike price (the pre-agreed price at which the holder of the option can buy/sell) and ST is the price of the asset at the expiry date, T.

Definition 2.
A portfolio is a term used to describe a combination of: assets, options and cash invested (a). I shall denote the value of any given portfolio as V(t).

Example 1
Given I plan to invest $500 000. It has been suggested that ICAP shares are likely to increase in value, and they have a current value of $14.50. The cost of a 6 month call option on ICAP shares with strike price $15 is $1.76.

I shall consider two investment strategies:
1.Invest all my money in ICAP shares
2.Invest the money in call options on ICAP shares

Calculate the profits of the strategies if the market value of ICAP shares in six months is (a) $18.75 (b) $14.68

Solution
Current position:
Portfolio 1: Buy 34 482 ICAP shares (500000/14.50 = 34482.76) Portfolio 2: Buy 284 090 call options on ICAP shares; with strike price $15 and Maturity 0.5 years (500000/1.76 = 284090.91)

In 6 months time:
(a)ICAP shares are selling at $18.75
Portfolio 1: Each share has increased in value by $4.25
Profit = 34 482 x 4.25 = $146 548.50
Amount invested = 34 482 x 14.50 = $499 989
Percentage profit = (146548.50/499989) *100 = 29.31%
Portfolio 2: The call options are exercised since they are ‘in the money’. By buying shares at $15 and selling at $18.75, the payoff from each option is $3.75 Payoff = 284 090 x 3.75 = 1 065 337.50

Amount invested = 284 090 x 1.76 = 499 998.40
Percentage profit = (1065337.5/499998.40)*100 = 213.07%

(b)ICAP shares are selling at $14.68
Portfolio 1: Each share has increased in value by $0.18
Profit = 34 482 x 0.18 = $6206.76
Percentage profit = (6206.76/499989)*100 = 1.24%
Portfolio 2: None of the calls are exercises, since they all have: C = max (0, -32) = 0
This exercise highlights the tremendous potential for call options, with in this case a percentage profit seven times that of simply investing in the equity market and simply buying shares. However it shows the associated additional risk of investing in options.

Definition 3
The idea is that the buyer of either such option will exercise the option to buy or sell their shares respectively at the expiry date if there is the opportunity to make a profit. It is clear that options themselves therefore have some value, since otherwise there would be an arbitrage opportunity. An arbitrage opportunity is one, which can lead to...
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