. Option pricing

Options Pricing Presented by Rajesh Kumar Sr. Lecturer (Fin.)- Satya College of Engg. & Tech., Palwal. Visiting Faculty (Project finance)- STC, Oriental Bank of Commerce, Sec. 18, Noida. Visiting Faculty (Derivatives)- IIBS, Noida Branch. {Scholar Student of YMCA University, MBA-Fin. (CMS, Jamia Millia Islamia), BA (Eco. Hons.), AMFI, NCFM (A Grade), A Certified Trainer from NSDL (NSE).}

Derivatives Derivatives: A derivative is a standardized financial contract whose value is determined from the underlying assets like security, share price Index, exchange rate, commodity, oil price, etc. Major components : Futures Options Forex Swaps Commodity, etc.

Options Options: An option is a legal contract which gives the holder the right to buy (or not) or sell (or not) a specified amount of underlying asset at a fixed price within a specified period of time. Types of options: Call options: A call option is a legal contract which gives the holder the right to buy or not to buy an asset for a predetermined price on or before a specified date. Put options: A put option is a legal contract which gives the holder the right to sell or not to sell an asset for a predetermined price on or before a specified date.

Options Situations for Call options and Put options: (a)A Call option may face three situations: 1.In-The-Money (ITM), when S 0 ˃ E, or 2.Out-of-The-Money (OTM), when S 0 ˂ E, or 3.At-The-Money (ATM), when S o = E (b) A put option may face three situations: 1.In-The-Money (ITM), when S 0 ˂ E, or 2. Out-of-The-Money (OTM), when S 0 ˃ E, or 3. At-The-Money (ATM), when S o = E

Option Pricing The price (value) or the premium of an option is made up of two components: (IV & TV) 1.Intrinsic value (or parity value) : The IV refers to the amount by which it is in money if it is In-The- Money. An option which is OTM or ATM has zero Intrinsic Value. Intrinsic value of a call option: Max (0, S 0 - E) {Excess of Stock price (S 0 ) over the Exercise price(E)} Intrinsic value of a put option: Max (0, E - S 0 ) {Excess of Exercise price (E) over Stock price (S 0 )}

Option Pricing 2. Time Value (TV): The TV of an option is the difference between the premium of the option and the intrinsic value of the option. Time value of a call option: C – {Max (0, S 0 – E)} Time value of a put option: P – {Max (0, E – S 0 )}

Option Pricing Hence, P = IV + T V Where, P = price or premium of an option, IV = Intrinsic value and TV = Time value.

Option Pricing Factors affecting the price of an option: Exercise price (E) Time to expiry (t) Volatility (σ) Interest rate (r) Dividend

Black & Scholes (B&S) Model: Propounded by Fisher Black and Myron Scholes in Uses of the B&S Model: Valuation of Call & Put options Valuation of Index Hedging etc.

B&S Model: (1) C = S 0 × N(d 1 ) – Ee -ert × N(d 2 ) (2) P = C + Ee -rt – S 0 Or, P = E -rt × N(-d 2 ) – S 0 × N(-d 1 ) Where, d 1 = [ln(S 0 /E) + t(r + 0.5σ 2 )] ÷ σ √ t d 2 = [ln(S 0 /E) + t(r – 0.5σ 2 )] ÷ σ √ t Or, d 2 = d 1 - σ √ t

B&S Model: (1) C = S 0 × N(d 1 ) – Ee -ert × N(d 2 ) (2) P = C + Ee -rt – S 0 Where, C = Current value of Call option, P = Current value of Put option, r = Continuous compounded risk-free rate of interest, S 0 = Current price of the stock, E = Exercise price of the option

B&S Model: (1) C = S 0 × N(d 1 ) – Ee -ert × N(d 2 ) (2) P = C + Ee -rt – S 0 Where, t = Time remaining before the expiration date (expressed as a fraction of a year), σ = Standard deviation of the continuously compounded annual rate of return, ln(S 0 /E) = Natural logarithm of (S 0 /E), N(d) = Value of the cumulative normal distribution.

B&S Model: Assumptions: (1)The option being valued is a European style option, with no possibility of an early exercise. (2)There are no transaction costs and there are no taxes. (3)The risk-free interest rate is known and constant over the life of the option. (4)The distribution of the possible share prices (or index levels) at the end of a period of time is log normal or, in other words, a share’s continuously compounded rate of return follows a log normal distribution.

B&S Model: (5) The underlying security pays no dividends during the life of the option. (6) The market is an efficient one. (7) The volatility of the underlying instrument (Standard Deviation, σ) is known and is constant over the life of the option.

B&S Model: Required inputs: (1)Current price of the stock (S 0 ) (2)Exercise price of the option (E) (3)Time remaining before expiration of the option (t) (4)Risk free rate of interest (r) (5) Standard deviation of the continuously compounded annual rate of return (σ)

B&S Model: Steps involve for the calculation of standard deviation (σ): Step1. calculate the price relative for each week. Step2. Find natural logarithms of each of the PR. Step3. calculate the standard deviation (σ). Step4. convert the weekly standard deviation to a yearly standard deviation by multiplying it by the √52.

. Q.No.1. consider the following information with regard to a call and put option on the stock of ABC Company: Current price of the share, S 0 = Rs 120 Exercise price of the option, E = Rs 115 Time period to expiration = 3 months. Standard deviation of the distribution of continuously compounded rates of return, σ = 0.6 Continuously compounded risk free interest rate, r = 10% pa. With these inputs, calculate the value of the call and put option using the Black and Scholes formula.

Adjustment for Dividend Step 1. calculate the present value of Dividend D 0 = De -rt Step 2. adjust the present value of dividend (D 0 ) with Current market price of the stock (S 0 ) i.e., Price of stock (After dividend) = S 0 – D 0 Step 3. take this value in B&S Model in the place of S 0 Q.No. 2. Reconsider the Q.NO.1. suppose a dividend of Rs will de received from the underlying share 40 days from today. Take all other inputs from Q.NO.1. and calculate the value of call and put options.

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