## Summary of call option

Summary of call option

- Ticker Symbol – AAPL (Apple Inc)
- Strike price - $170
- Bid and ask price of the option – $11.50 - $12.80
- Price of the underlying asset – $160
- Expiration date – 20
^{th}May 2022 - Date accessed – 3
^{rd}March 2022

Option valuation technique Black Scholes Merton is one of the most essential ideas in the financial sector. The Black-Scholes Merton model is a differential equation that is often used to price option contracts. While the Black-Scholes Merton model is typically right, it has extra redundancy that might lead to pricing that differs from actual outcomes (Anwar and Andallah 2018).

The Black Scholes model is used to determine the likelihood of an option contract expiring in the money and to value the option based on specific assumptions. The concept is based on the notion that the price of a call option is determined by the underlying stock prices, with the value increasing as the stock price rises and decreasing as the stock price falls (Kreps 2019). The model is focused towards assigning a value to the call option keeping in mind certain assumptions like volatility of the stock, the time left for the option until expiration and the interest rates (Dar and Anuradha 2018).

The following are the assumptions of the Black Scholes Merton model:

- The model assumes that the stock does not pay any dividend throughout the life of the option.
- The movement of the markets are random and cannot be predicted. The market follows the geometric Brownian motion.
- While buying the option, no transaction costs are present.
- One of the major assumptions of the model is that the risk-free rate of return and the volatility of the under lying asset are known and constant.
- The returns of the asset are distributed log normally.
- The primary assumption of the model is that the option is European and can be exercised only at the expiration of the contract.

The model also predicts that the price of assets follows a specific kind of motion known as the geometric Brownian motion having a constant drift and exhibiting constant volatility (Weatherall 2018).

According to the Black Scholes Merton model the following are the inputs required to value an option:

Price of a Share (S), Price of a Strike (K), Expiration Date (t), Rate of Interest (r) and Volatility (sigma).

The price of a call option is derived using the Black-Scholes model as follows:

S * N(d1 ) – Ke(–rt) * N(d2 ) C = S * N(d1 ) – Ke(–rt) * N(d2 )

where:

d1 = [ln (S/K) + (r + 2/2) * t]

d2 = d1 -(sigma* t^0.5) – d1

Once the values for d1 and d2 have been determined, we use the formula =NORMSDIST (d1 or d2) in Excel:

N (d1), N (d2)

This is how the value of a Call Option is calculated using the Black Scholes and Merton Model.

The drawbacks of the model regarding the assumptions of the model are discussed below:

- The assumptions of the model include that the risk-free rate of return and the volatility of the underlying is known and constant, which is an unrealistic assumption. In reality volatility of the stock is not known and might change over time.
- The assumption regarding the no transaction costs involved in trading option is unrealistic as there are liquidity risk present in the market and transaction costs are also present (Janková, 2018).
- The model assumes that the stock price follows a lognormal pattern which is also known as Brownian motion disregarding the volatile nature of the stock prices. There is significant deviation in the stock price of a company in the real world.
- Dividend payments can influence the stock price movement significantly but the model ignores it which is an incorrect assumption.
- The model also assumes that there are no taxes, no penalties on short sale, absence of arbitrage opportunities all of which does not hold true in the real world (Valverde 2015).

Apple Inc is an US based technology company and has operations all around the globe. The company is involved in the business of consumer electronics, online services and software services. Apple Inc has earned the title of the biggest information technology company in the world in terms of revenue and is the fourth largest manufacturer of personal computer. The company is also the second largest mobile phone manufacturing company in the world and is considered as the world’s most valuable company. The company finds itself amongst the five biggest tech companies in the United States and is listed in the NASDAQ stock exchange (Britannica 2022).

## Overview of Black Scholes Merton model

We have chosen to value the call option of Apple Inc which has an expiry date of 25^{th} May and an exercise price of $170. The current price of the stock is around $160 and the date of valuation is 11^{th} March 2022. To carve out the price of the call option according to the Black Scholes Merton model, Microsoft excel was employed. The following are the inputs and assumptions used in the process of valuation:

To calculate the volatility of the stock of Apple, daily stock price date of the company’s stock for the previous five years was ascertained. The daily price return percentage of the company was determined using excel functions. The STDEV.S function of excel was used to find out the daily standard deviation of the company in the past five years which was later converted into annual standard deviation using the formula [(1+r) √ 252] with the number 252 representing the total number of trading days in a year. The resulting annual standard deviation which was equal to 27.0041 percent was used as the proxy for implied volatility of the stock used in the valuation of the call option.

The yield on the 3-month treasury yield which was equal to 0.44% percent was used as a proxy for risk free rate of return for the purpose of carrying out the valuation. Nd1 and Nd2 were determined using the NORMDIST function of the excel as explained in the steps above. The value of the option was calculated as $3.82.

The following section explains how the inputs used in the valuation of the call options satisfies the assumptions made by the model:

Risk-free rate - The risk-free rate utilized as the discounting factor in the computations is assumed to be known and constant by the model. We utilized the yield on 3-month treasury government issued bonds to value the option, which meets the premise of the risk-free rate being known and constant.

Implied volatility - Implied volatility is a probability measure of a security's expected or predicted price volatility. In case of a bearish markets, investors expect the markets to fall and hence the implied volatility is generally on the higher side. In case of a bullish market, the investors have a positive view regarding the market with an expectation of rising prices of assets and as a result the implied volatility is expected to be on the lower side. Hence, a falling or bearish market is considered to be riskier than a rising or bullish market because of the expectations of the market Participants (Mijatovi?, and Tankov 2016).

## Assumptions of Black Scholes and Merton Model

Implied volatility is considered to be one of the primary necessities in valuing an option as the premium of the options is a function of the implied volatility embedded in the underlying security. The price of an option would be higher if the implied volatility is higher and lower if the implied volatility is lower. Implied volatility can only be used to estimate or forecast future prices because of the probabilistic nature of it. Investors evaluate implied volatility while making investment decisions. Regular investors are well aware that the price of an option does not always follow a predictable pattern.

To calculate the implied volatility, we have used the past six-month daily returns of the stock of Apple Inc and used the annual standard deviation of the returns as the proxy for implied volatility. This is consistent with the assumption of the model for the implied volatility of the underlying stock being known and constant.

Dividend- The assumption regarding no dividend payment from the company is fulfilled as the company has not paid any dividend during the period of analysis.

No transaction costs – During the calculations of the theoretical price of the call option, no transaction costs were assumed satisfying the assumption made by the model of no transaction costs being involved in the process of buying and selling an option contract.

The theoretical value of the call option which was determined using the inputs discussed in the above section of the report in the Black Scholes and Merton model, came out to be $3.82 whereas the market observed price of the call option was equal to $11.50 as of 11^{th} March 2022. The implied volatility that resulted in the theoretical value of the call option was around 27.0041 percent using historical data and the risk-free rate was assumed to be 0.44 percent.

Based upon the market price of the call option price which is equal to $11.50, the implied volatility was back calculated using the excel solver function. The implied volatility that resulted in a price of call option which is equal to the market price, was equal to 55.49 percent. The difference in the value of call option can be attributed to the differences in the implied volatility as the market assumes that the stock of the company is more volatile than the historical volatility of the stock in the past six months. The Participants of the market believe the market to be more volatile than it should be based upon the BSM model.

## Derivation of call option value using Black Scholes and Merton Model

The choice of the risk-free rate of return based on government bond yield or the choice of the period of stock returns can also be a reason behind the distinct values of call price observed.

Financial derivatives are securities or contracts whose value is derived from an underlying asset, a collection of assets, or a benchmark. These contracts are drawn up between two or more Parties and can be exchanged over the counter or on exchanges. These contracts are used to speculate on the market or to hedge an individual's or an organization's current financial position (Gupta 2017). Options, Swaps, Futures, and Forwards are some of the numerous forms of derivatives accessible on the market (LiPuma 2017). The merits and downsides of different financial derivatives such as Options, Swaps, Futures, and Forwards are examined in this portion of the paper.

Option contracts are financial contracts whose value is determined by the price of an underlying security. A buyer of an options contract has the right to purchase or sell the underlying asset at a pre-determined price, depending on the kind of contract. The contract's features are defined by the exchange on which it is traded (Tompkins 2016). Market Participants employ options contracts for hedging or speculative purposes. Options contract have several benefits associated with it which are discussed below:

- Low upfront costs – Options allow an investor to pay a lower price in exchange for the advantages of a Particular stock. If an investor wants to buy a stock but doesn't have the funds to do so, he or she can buy a call option on the stock for a tiny premium, which is determined by the option's maturity, moneyness, and implied volatility. When the option's strike price is breached, the investor receives advantages and has the opportunity to purchase the shares at the option's strike price. The investor would simply lose the premium if the stock price did not breach the strike price.
- Offers flexibility to the investors and traders – Investors can use options to execute their trading strategy without restriction, since they can exercise the option and purchase the company's stock, sell the shares acquired through options, or even sell the option contract itself to another investor, among other things.
- Options strategies – There are a variety of option strategies accessible to investors, such as the spread strategy, butterfly strategy, strangle, and straddle strategy, which may be used depending on market conditions and allow an investor to achieve above-average gains. The aforementioned tactics can also be used to inexpensively hedge a portfolio position against market volatility.

The following are the drawbacks associated with options contract:

- Risk of amplified losses – Option sellers, unlike option purchasers, are exposed to greater losses if the stock price moves against the option seller's expectations. Although the option seller receives a premium for each option sold, it also faces the risk of losing a significant amount of money if the stock price rises or falls depending on the type of option sold. Purchasing options offers the investor the right to purchase or sell at a certain price, whereas option sellers must adhere to the terms of the contract.
- Time is limited for an option contract – While buying or selling an option contract based upon an investment thesis or strategy, an investor has to keep in mind the time after which the strategy would bear result as option contracts have limited time after which they expire. If a strategy takes more time than the maturity of the option contract to bear fruits, the investor can face immense losses after the expiry of the contract.
- Operational hindrances and costs associated with options – Prior authorization and paperwork from the broker are required before an investor may trade in options. The investor or trader must also keep a certain amount of margin money on hand, which will be changed according to the transactions made by the trader.

Swaps are financial derivatives contracts that are agreed upon by two Parties and swapped after a defined period of time. Two of the Parties agree to swap the instrument at a specific time. Many investors regard these sorts of contracts to be a hedging mechanism because they are traded over the counter (Mixon, Onur and Riggs 2018). Since the financial crisis of 2008-09, currency swaps were entered between central bank with the USD beginning to get decentralized and other countries’ gaining increased prominence (Mingpi 2016). Two common types of swaps are discussed below:

- Interest rate swaps – When a person or a company needs money, they hunt for different ways to borrow money and prioritizes sources with lower interest rates. A fixed rate loan is converted into a variable rate debt or vice versa using an interest rate swap, making it simpler for the borrower or lender to profit from the transaction.
- Currency swap – It is an arrangement between two Parties in which the principal and interest rate of one currency are swapped for the principal and interest rate of another currency.

The following are the benefits associated with swaps:

- Lowering of borrowing costs – Swaps has an advantage as it allows for cheaper borrowings for the lender. The comparative advantage of one borrower is exchanged for the comparative advantage of the other borrower. Both Parties will be able to get cash at a lesser cost as a result.
- Provides access to new financial markets – Swaps can be utilized by lenders and borrowers to get access to new financial markets for acquiring funds by assessing the comparative advantage of the other Party in Particular markets. Swaps of many types can be used to exploit one Party's comparative advantage, allowing the needed cash to be obtained at lower rates.
- Hedging risks – Swaps may be used to hedge against risk in general, and they can also be used to hedge against interest rate risk by maintaining a perfect balance of fixed and variable debt. Swaps allow both Parties to benefit from the interest-paying transaction by obtaining lower bond rates than a bank would provide.

The following section highlights the drawbacks of swaps:

- It is a difficult to find a counter Party to get involved into a swap agreement with another Part It is difficult to match the amount of both lenders and borrowers with the exact match of maturity.
- The swap contract cannot be terminated without the approval of all Parties involved, and there is a significant danger of default, which means that the Parties engaged may refuse to make needed payments at Particular times.
- Swap markets, like primary markets for equities and debt, are not fully developed. Due to the restricted nature of swap transactions, swap markets have extremely little liquidity. This has an influence on swap prices since investors want more liquid marketplaces.
- Because swaps are an over-the-counter market that is not regulated by an exchange, the level of risk is quite high. As a result, Parties concerned must exercise extra caution when dealing with counter Parties while doing due diligence on the other Party's creditworthiness. There is no guarantee that the Parties concerned will follow through on their commitments.

A future contract is an agreement between two Parties for the purchase or sale of a commodity, currencies or stocks at a certain date in the future. The price at which the transaction will be completed is pre-determined and agreed upon by both Parties. When an investor or trader buys a futures contract, they are obligated to buy the commodity at the pre-agreed price when the contract expires, and the same is true for the seller of the futures contract (Bekkermen and Tejeda 2017). There are multiple benefits involved with futures contracts, some of them are discussed below:

- Open up new markets - It allows new investors who are not accustomed to financial markets to enjoy the benefits of it. It opens up new opportunities for commodity traders and stock investors or traders to hedge against prices going against them.
- Margin requirements - Futures markets are highly regulated and efficiently managed hence commodities and currencies are well established in the markets. The margin requirements are also well established and favorable for all the Parties involved.
- No issues of time decay - As compared to options, the futures contracts have an advantage that it does not has any time decay. A trader or investor involved in future contract transactions does not have to lose profits due to the problem of time decay as compared to options contract.
- Highly liquid market – The market for futures contract are highly liquid as there are multiple transactions related to currencies, stocks, indexes and commonly traded commodities. High liquidity in a market are highly valued by investors and traders as it allows them to enter and exit their position quite easily.
- Not complex like options – Futures pricing is simple and straightforward when compared to option pricing techniques such as the Black-Scholes option pricing model. Futures are priced using the cost of carry model, which entails adding the cost of carry model to the asset's spot price.
- Diversification benefits – It helps in diversify an investment portfolio of an individual client or an institutional investor. If an investor feels that the stocks of a Particular sector are expected to do well in the coming days, futures contracts are used to get exposure to that Particular sector without adding the stocks in the portfolio.

## Drawbacks of the model

The following section highlights the drawbacks of the futures contracts –

- Problems related to leverage – The future markets and the future prices of different assets experience increased levels of fluctuations due to the high level of leverage involved in trading. The prices of the assets move up and down very frequently damaging capital invested by many of the traders and inflicting losses to them.
- No control over events that are going to occur in the future – Investing in future contracts is also considered to be risky as control over future events is not possible. A potential hazardous event that might take place in the future can damage the prospects of the investment and inflict significant losses to investors and traders.
- The futures contract also comes with an expiration date and investors runs the risk of valuing the contracts as potentially worthless at the time of expiry.

A forward contract can be defined as a financial agreement entered into by two counterParties to buy or sell an asset at some point in future at a price determined today. A forward contract is used for purposes like hedging and speculating. A forward contract is quite similar to the futures contracts with the only difference being that the futures contract is standardized and exchange traded whereas a forward contract is non-standardized which makes it apt for hedging purposes only (Islam and Chakraborti 2015). Forward contracts have advantage over futures contract in the fact that it can be customized based on the requirements of the associated Parties. Forward contracts are not traded on centralized exchanges like the futures contract and over the counter instruments. The following are the benefits of a forward contract:

- Forward contract can fix a price of the asset that would be traded between the Parties and can help in efficient cost management practices by the involved Part The counterParties are better able to manage the supply and control the risk associated with it.
- Currency risks can be hedged using forward contracts and companies can manage their foreign currency transactions in with an increased efficiency. The interest rate risks associated with the company also can be managed using forward contracts.
- Forward contracts are extremely customized and can be tweaked according to the preferences of the Parties involved including matching the maturity of both Part
- A future cash flow can be predicted with a certain confidence if forward contracts are deployed to hedge the cash flow. The uncertainty regarding the fluctuations in the cash flow is eliminated with the help of forward contract.

The following sections defines the drawbacks associated with the forward contracts:

- Forward contract involves the regular risk which is associated with unfavorable movement of prices of the asset. It can happen that an investor or trader may end up paying an unfavorable price for the contract entered.
- As forward contracts are non-standardized, there is a risk of default associated with forward contracts as either of the counterParty may choose to dishonor the agreement at the expiry date.
- The forward markets are illiquid as they are not regulated by any governing authority like in the case of futures contract. Hence, traders are disincentivized from using the forward markets.
- As the forward contract is unorganized, non-standardized and the risk of defaulting on the obligations of a contract is high, traders finds it difficult to find an appropriate and suitable counterParty to enter into a contract with.

Conclusion

After considering all of the advantages of derivatives contracts stated above, it can be concluded that the advantages of derivatives contracts exceed the disadvantages of derivatives contracts. Typically, investors employ derivatives for three reasons to hedge a position, raise leverage, or speculate on the movement of an asset. Hedging a position is frequently done to safeguard or insure against an asset's risk. All of the aforementioned derivatives contracts are primarily utilized by businesses, individual investors and managers to hedge their investment positions against various market risks. Swaps are used in managing interest rate risks and are increasingly used by corporates all around the world. Interest rate swaps are used by various fund managers to hedge against the interest rate risk. Because derivatives contracts, as noted above, are necessary for market efficiency, they have grown in importance in financial markets. It enables firms to gain access to financial markets that they would otherwise be unable to access. Derivatives contracts lower transaction costs, which would otherwise be higher if traded directly. Swaps

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