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Spreads. A spread is a combination of a put and a call with different exercise prices.

Spreads

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- A spread is a combination of a put and a call with different exercise prices.
- Suppose that an investor buys simultaneously a 3-month put option at an exercise price of Rs 95 and a call option at an exercise price of Rs 105 on a company’s share. What will be the investor’s positions if the share price is Rs 120? How much will be the investor’s pay-off if the share price is Rs 90?

Pay-off for a spread buyer

Pay-off for a spread seller

- The price spread or the vertical spread involves buying and selling options for the same share and expiration date but different strike (exercise) prices. For example, you may buy a BPCL December option at a strike price of Rs 215 and sell a BPCL December option at a strike price of Rs 210.
- The calendar spread or the horizontal spread involves buying and selling options for the same share and strike price but different expiration dates. For example, you may buy a Tata Power December 2002 option at a strike price of Rs 95 and sell a Tata Power January option at a strike price of Rs 90.

- An investor maybe expecting the price of an underlying share to rise. But she may not like to take higher risk. Therefore, she would buy the higher-priced (premium) option on the share and sell the lower-priced option on the share.

Pay-off for a spread combining long position and short position on a call

- An investor, who is expecting a share or index to fall, may sell the higher-priced (premium) option and buy the lower-priced option. For example, you may sell a BPCL December option at Rs 10 (premium) with a strike price of Rs 210 and buy a BPCL December option at Rs 5 (premium) with a strike price of Rs 220.

- A long butterfly spread involves buying a call with a low exercise price, buying a call with a high exercise price and selling two calls with an exercise price in between the two. Thus, there are three call contracts with different strike prices.
- A short butterfly spread involves the opposite position; that is, selling a call with a low exercise price, selling a call with a high exercise price and buying two calls with an exercise price in between the two.

Pay-off to a butterfly spread

- A collar involves a strategy of limiting a portfolio’s value between two bounds.
- It is a strategy that would let pay-off to range within a band, irrespective of the price fluctuations

- Exercise price and the share (underlying asset) price
- Volatility of returns on share
- Time to expiration
- Interest rates

The value of the options will lie between Max and Min lines

- Important determinants of options are the value of the underlying asset and the exercise price.
- If the underlying asset were a share, the value of a call option would increase as the share price increases.
- The excess of the share price over the exercise price is the value of the option at the expiration of the option.

- The option will be worthless if the share price remains at strike price at maturity.
- It will be valuable if there are chances that the share price may rise above the strike price.
- The probability of a higher price of the share causes the option to be worth more.

- The figure below shows graphically the effect of the volatility of the underlying asset on the value of a call option. The underlying assets in the example are share of two companies—Brightways and Jyotipath. Both shares have same exercise price and same expected value at expiration. However, Jyotipath’s share has more risk since its prices have large variation. It also has higher chances of having higher prices over a large area as compared to Brightways’ share. The greater is the risk of the underlying asset, the greater is the value of an option.

Volatility of the share and the value of a call option

- The present value of the exercise price will depend on the interest rate and the time until the expiration of the option.
- The value of a call option will increase with the rising interest rate since the present value of the exercise price will fall.
- The effect is reversed in the case of a put option. The buyer of a put option receives the exercise price and therefore, as the interest rate increases, the value of the put option will decline.

- The present value of the exercise price will be less if time to expiration is longer and consequently, the value of the option will be higher.
- Further, the possibility of share price increasing with volatility increases if the time to expiration is longer.
- Longer is the time to expiration, higher is the possibility of the option to be more in-the-money.

- The DCF approach does not work for options because of the difficulty in determining the required rate of return of an option. Options are derivative securities. Their risk is derived from the risk of the underlying security. The market value of a share continuously changes. Consequently, the required rate of return to a stock option is also continuously changing. Therefore, it is not feasible to value options using the DCF technique.

- Simple binomial tree approach to option valuation.
- Black-Scholes option valuation model.

- Sell a call option on the share. We can create a portfolio of certain number of shares (let us call it delta, D) and one call option by going long on shares and short on options that there is no uncertainty of the value of portfolio at the end of one year.
- Formula for determining the option delta, represented by symbol D, can be written as follows:
Option Delta = Difference in option Values / Difference in Share Prices.

- The value of portfolio at the end of one year remains same irrespective of the increase or decrease in the share price.
- Since it is a risk-less portfolio, we can use the risk-free rate as the discount rate:
PV of Portfolio = Value of Portfolio at end of year / Discount rate

- Since the current price of share is S, the value of the call option can be found out as follows:
Value of a call option = No. of Shares (D) Spot

Price – PV of Portfolio

- The value of the call option will remain the same irrespective of any probabilities of increase or decrease in the share price. This is so because the option is valued in terms of the price of the underlying share, and the share price already includes the probabilities of its rise or fall.

- Investors are risk-neutral. They would simply expect a risk-free rate of return.

- The rates of return on a share are log normally distributed.
- The value of the share (the underlying asset) and the risk-free rate are constant during the life of the option.
- The market is efficient and there are no transaction costs and taxes.
- There is no dividend to be paid on the share during the life of the option.

- The B–S model is as follows:
where

C0 = the current value of call option

S0 = the current market value of the share

E = the exercise price

e = 2.7183, the exponential constant

rf = the risk-free rate of interest

t = the time to expiration (in years)

N(d1) = the cumulative normal probability density

function

where

ln = the natural logarithm;

σ= the standard deviation;

σ2 = variance of the continuously

compounded annual return on the share.

- Black–Scholes model has two features-
- The parameters of the model, except the share price volatility, are contained in the agreement between the option buyer and seller.
- In spite of its unrealistic assumptions, the model is able to predict the true price of option reasonably well.

- The model is applicable to both European and American options with a few adjustments.

- The hedge ratio is a tool that enables us to summarise the overall exposure of portfolios of options with various exercise prices and maturity periods.
- An option’s hedge ratio is the change in the option price for a Re 1 increase in the share price.
- A call option has a positive hedge ratio and a put option has a negative hedge ratio.
- Under the Black–Scholes option valuation formula, the hedge ratio of a call option is N (d1) and the hedge ratio for a put is N (d1) – 1.

- Rakesh Sharma is interested in writing a six-months call option on L&T’s share. L&T’s share is currently selling for Rs 120. The volatility (standard deviation) of the share returns is estimated as 67 per cent. Rakesh would like the exercise price to be Rs 120. The risk-free rate is assumed to be 10 per cent. How much premium should Rakesh charge for writing the call option?

- First we calculate d1 and d2
- Then, we obtain the values of N(d1) and N(d2) as follows:
- We obtain the call and put values as given below:

- Implied volatility is the volatility that the option price implies. An investor can compare the actual and implied volatility. If the actual volatility is higher than the implied volatility, the investor may conclude that the option’s fair price is more than the observed price.

- We can use slightly modified B–S model for this purpose. The share price will go down by an amount reflecting the payment of dividend. As a consequence, the value of a call option will decrease and the value of a put option will increase.
- We also need to adjust the volatility in case of a dividend-paying share since in the B–S model it is the volatility of the risky part of the share price. This is generally ignored in practice.

- The limited liability feature provides an opportunity to the shareholders to default on a debt.
- The debt-holders are the sellers of call option to the shareholders. The amount of debt to be repaid is the exercise price and the maturity of debt is the time to expiration.
- The shareholders’ option can be interpreted as a put option. The shareholders can sell (hand-over) the firm to the debt-holders at zero exercise price if they do not want to make the payment that is due.

- Excel Corporation is currently valued at Rs 250 crore. It has an outstanding debt of Rs 100 crore with a maturity of 5 years. The volatility (standard deviation) of the Excel share return is 60 per cent. The risk-free rate is 10 per cent. What is the market value of Excel’s equity? What is the current market value of its debt?

The market value of debt is : Market value of debt= Value of firm – value of equity

= 250 – 200 = Rs 50 crore.