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OptionsOptions

Presented By..

PremSundar

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Options

• Special type of financial instrument

• Buyer has the right to buy or sell; no obligations

• Acquires an advantage for which he pays premium

• Option writer is induced by the premium to offer this advantage to the buyer

• Options can be American or European

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Terminologies• Buyer- induces the writer by offering a premium to give him

the right to buy or sell the underlying asset • Call buyer- for calls • Put buyer – buys from the writer of the put by paying the

premium• Writer – receives the premium and is obliged to buy from or

sell to the buyer the underlying assets, should the buyer exercise the option to do so.

• Expiration date – is the date specified by which the contract comes to an end

• Strike price – this is the exercise price • In the money options- positive cash flows if options were

exercised immediately• At the money options- zero cash flows• Out of money options – negative cash flows if exercised

immediately.

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Value of call optionValue of call option• Is Zero when S<= X on expiration date or

earlier or

• Is positive when S>X

• Call options C = max ( S-X,0)

• The Gain G = max ( S-X ,0) – P ( where P is the premium paid)

• The Loss is = premium paid when S<X

• The break-even point is where S- (X+P)=0

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125 130 135 140 145 150

450

S>XS<X

X = 125

value of call option

Share price

Value of call option to Buyer

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130 135 140 145 150

450

S>X +PS<= X + P

-5

Gain / Loss

Share price

0

Value of Call option to Buyer

X= 125

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Gain/Loss to call option writer

• Gain so long as S<X+P subject to S-X < P if RLN call is written @130 with Rs. 5 premium , so long as the stock trades <= 135, the writer will incur no loss

• If stock price is between exercise price and exercise price plus premium , there will be a erosion in “gains on writing”

• Max gains for the call writer – premium earned • Max loss for the call writer - UNBOUNDED

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130 135 140 145 150

S>X +P

S<= X + P

5

(Loss area)

0

Value of Call option to call writer

X= 125

P= 5Premium

share price

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Put Options

• Put holder exercises this right when X>S • When X<S , he does not exercise option and

would prefer to lose the premium paid• Value of put option (like call option) can't be

negative • Value of put option = Max ( X-S,0)• The put option curve as we will soon see has a

negative slope ( greater a fall in prices , higher the put options value)

• The max profit to a put holder is when S=0 , Here Gain= Exercise price – premium price

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135

135 140 145 150

Upper boundary ( when S=0 , Gains= X – P)

Gains to Put Holder

In Money

Out of money

Exercise price = 135

130

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Put writer

• Bullish on the prospects of the share• Gain = premium received• BEP = X+P • When S<X , the writer would be obligated to buy the

underlying asset from the put holder Example: If reliance put was written at Rs 125/- on a

premium of Rs 5/- , the BEP is 130/- . If reliance closes above Rs. 125/- , say Rs. 128/- the writer will have to buy Rs 128/- and deliver it at Rs 125/-. The net premium retained is Rs 5- Rs 3 = Rs 2/-

• The max loss the writer incurs is when the share falls to zero , loss in that case = X- premium received

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Gain/ loss to put writer

Loss area

Gain

Loss

Gain Premium

X+P

X-PMax loss

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Options Payoffs• Call option holder- Loss – limited to premium paid - Profits unbounded not exceeding share price.• Call writer - Loss: unbounded profit : limited to premium received

• Put holder – Loss: limited to premium Paid Profit: Max upto X- P ( when share =0)• Put writer - Loss: Max loss = X- Premium received Profit: premium received

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Option boundaries

• Value of an option can never be negative Ex- If reliance is selling Rs 133 with strike price Rs. 125 and call

premium of Rs 5 , arbitrage crops up. The arbitrageurs buys a call for Rs 5/- demands immediately delivery and sells it 133/- immediately, making a risk free profit of Rs 3.

• Thus C>= S-E .• For all call option with time for expiry , C=max(S-E,0) . This is the lower bound.• The upper bound of the call option though unbounded can never be

more than the price of the share itself . This happens when the call option has a very very long expiration period or is unlikely to be exercised . Here the present value of the strike price to be paid ( in the distant future ) approaches zero.

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Factors influencing options valuations

• Current stock price

• Exercise price

• Risk free rate of interest

• Time to expiration

• Price volatility of the share

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Factors influencing option values

• Generalization – the value of a call option is the market price of the share less the present value of the exercise price ( discounted at risk free rate of return). Thus the value of call option is a function of

1) current stock price = +ve correlation 2) Exercise price = -ve correlation 3) Risk Free Rate of Return= +ve correlation 4) time to expiration = +ve correlation 5) price volatility = +ve correlation

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Regarding Risk free interest rate

• C= S- X / (1+r)n ( recollect the generalization)

• If ‘r’ increases , X / (1+r)n becomes a smaller figure , so C goes up.

• Similar is the reason for period to maturity to be positively correlated

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Price volatility

• Significantly influences option value

• The greater the possibility of extreme outcomes ,greater is the call option value to the holder, other factors remaining constant.

• The greater the standard deviation / variance , the greater the option value

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Assigning probabilities

• An investor is considering call options on two Shares X and Y . Particulars Probability Share Price Expected Price

Share X 0.10 90 9

0.25 108 27

0.30 120 36

0.25 132 33

0.10 150 15

120

Share Y 0.10 60 6

0.25 90 22.5

0.30 120 36

0.25 150 37.5

0.10 180 18

120

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Determination of Call Options values

Particulars Expected share price

Exercise price

Call value probability Expected call value

Share X 90 115 0 0.10 0

108 115 0 0.25 0

120 115 5 0.30 1.50

132 115 17 0.25 4.25

150 115 35 0.10 3.50

9.25

Share Y 60 115 0 0.10 0

90 115 0 0.25 0

120 115 5 0.30 1.50

150 115 35 0.25 8.75

180 115 65 0.10 6.50

16.75

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ConclusionConclusion

• The greater the dispersion in the possible outcomes, the greater the option value

• The range of expected share price for share X is 150- 90 = Rs 60 .

• The range of expected share price for Share Y is 180- 60 = Rs. 120

• Greater the dispersion , greater the variability

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Sensitivity of Options

• Sensitivity analysis deals with measurement of changes in options price due to changes in the underlying parameters that determine the option price.

• These parameters include stock price , time period , interest rate , volatility .

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Contd.

Factor Change Effect on call price

Effect on put price

Price of the underlying

asset

Increase

Decrease

Increase

Decrease

Decrease

Increase

Volatility Increase

Decrease

Increase

Decrease

Increase

Decrease

Time to Expiration

Decrease Decrease Decrease

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Effect of Changing interest rate on option values

Interest If Interest rates rise If Interest rate fall

Calls on stocks Increase in value Decrease in value

Puts on stocks Decrease in value Increase in value

This is because, in a rising interest rate regime , the cash outlay for a stock acquisition increases and people would like to buy calls instead, So calls increase in value due to higher demand. In a rising interest rate regime it makes sense to sell stocks and lend out the money to earn higher rate of interest . Since selling pressure comes on stocks the same comes on to puts and the pressure of selling decreases puts value.

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Sensitivity measures- ( Greeks)• Delta – it is mathematical derivative of the value

function with respect to the price of the underlying asset.

= dV/ dS v= value function and s = stock price

• It is the ratio of the change in the price of the option to change in the price of the underlying asset. Other factors , remaining constant , an increase in the price of the underlying asset causes the premium on calls to increase and the premium on puts to decrease. The bullishness in the underlying stock gets passed on to calls and bearishness on the stock gets transferred to puts

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Contd.

• The Delta of any option gives an idea of the no. of units of a stock that an investor should add or hold to create a risk less hedge.

• Since the value of calls increase when stock price increases, the delta of a call is always positive. But the value of put increases with decrease in stock price, so the delta of a put is negative.

• An option can never gain or lose faster than an underlying asset , So the upper bound for delta is 1.

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Contd.

• An option can’t move in a direction opposite to that of the underlying asset. So the lower bound of the call is 0.

• Since the put has opposite characters of a call , the boundary of delta ranges from -1 to 0. When the price of the underlying asset increases , the price of the put falls and when the ratio of the increase in the price of the put and the fall in the price of the underlying stock are the same , it is -1.

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Delta of a Call on Non-Dividend Paying Stock

1.0

Stock Price

Delta

X

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Delta of a Put on Non-Dividend Paying Stock

-1.0

0X

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Contd. ( Delta)

• It is seen that when changes in call prices happen over small changes in stock prices, change in C/ change in S approximates N(d1) of Black and Scholes

• The Delta of a PUT = change in value of put / change in stock prices

=N (d) -1• A delta neutral portfolio is one where the portfolio value

does not change for small changes in stock price• The concept of delta neutral portfolio earning a risk free

interest rate was the key finding of Black and Scholes

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Delta Interpretation

• If we plot delta on X- axis and stock prices on Y- axis, we find that

1) Delta tends to approach 1 when the call option is DEEP IN THE MONEY

2) Delta tends to approach 0 when the call options DEEP OUT OF THE MONEY

Also delta is MOST SENSITIVE to change in stock prices when the underlying stock price approaches the EXERCISE PRICE

3) Delta of a PUT option is ALWAYS NEGATIVE4) Delta of a PUT option DEEP IN THE MONEY approaches

-15) Delta of a PUT option DEEP OUT THE MONEY

approaches 0

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Delta int. ( Contd.)

• A call delta of 0.5 implies that for every 2 call options one underlying asset has to be sold to establish a neutral hedge

• A purchase of a PUT will imply the purchase of the underlying asset to establish a new portfolio. If it is a 0.5 put delta , you need to purchase one underlying asset for every 2 puts purchased.

• A portfolio is said to be delta neutral when the deltas of all the components of that portfolio add upto zero . The portfolio may consist of underlying contracts , calls and puts with different exercise prices and expiration dates.

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Theta

• Is the sensitivity of the derivative to expiration time. If the stock price and other factors of the option pricing models are constant , the price of the option will change with the increase and decrease in OPTION LIFE . Both call and put options go through TIME DECAY – i.e. lose value as EXPIRATION approaches . Theta is the rate at which option loses value as time passes.

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Theta( contd.)

• Theta of a call ( as also that of a PUT) can be greater or less than zero. But in normal circumstances it is always less than zero because as the time to expiration decreases , the value of the option also decrease.

• Being TIME DECAY of an option , theta is expressed in units in time. Though time cannot be negative , theta assumes negative values because options lose value as time passes. Theta is negative for the first derivatives of option price with respect to ‘time remaining till expiration’

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Theta ( contd.)

• A long option position will always have a negative theta and a short option always has a positive theta.

ex- If Reliance stock is Rs 1370 , and a Rs. 1370 exercise call has a premium of Rs. 50/- for a expiration date 1/7/2007 . If reliance continues to trade 1370 by about the first/ second week of June, the call premiums could fall to Rs 40/ less. That is why the Theta of a call option/ put option is negative . But for news driven cases (like mergers/ takeover etc.) as also technical position ( Short) Theta could go positive, as expiration date approaches.

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Vega

• Measures the sensitivity of the option premium with respect to the volatility of the asset, provided other factors determining the option premium are constant.

• Vega is the first derivative of the option price with respect to the volatility of the underlying stock. Since volatility changes over a time period ,the option premium both call and puts are liable to behave accordingly.

• Vega of a call and a put will always be identical and positive because all the options gain value with rising volatility

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Interpretation of Vega ( Kappa , Lamba / Sigma)

• If an option has a Vega of 0.3 , for each percentage increase or decrease in volatility, option calls gains ( loses) theoretical value of 0.3. If the option has a theoretical value of Rs. 3.5 at a volatility of 20% , then at a volatility of 21% , the option will have a value of Rs.3.8. If the Vega of any option is very high, the value of an option will be very sensitive to even a small changes in the volatility of the underlying assets

• Vega of all options decline as expiration approaches• Therefore a long term option will always be more

sensitive to changes in volatility than a short term option with similar characteristics

• Vega will be the highest for a near- the money option• For deep in the money and deep out of the money option

Vega will be low and tends to approach zero.

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Vega

X

Vega

Price of a Stock

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RHO• Rho can be defined as a measure of the sensitivity of option

value to change in interest rates. Rho is the first derivative of option premium with respect to interest rate. A call option is positively related to interest rate while a put option is negatively related to interest rates. Thus, the Rho of any call is always positive and the Rho of any put is always negative

• Rho will be lower for deep out of money call• Rho will be higher for deep in the money call ( because profit

booking emerges in deep in the money and even small changes in interest rates can make a big change in a Rho.

• Rho is a sensitive to changes in stock price when the call is in the money

• Rho of a put is sensitive to changes in stock price when the put is ‘in the money’ ( because people would like to take profits)

• Rho of a call/ put changes with expiration time and tend to approach 0 on expiration date or near by.

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Rho

Rho of a call option

Rho of a put option

Stock Price

Rho of an option

-10

-20

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Rho ( contd.)• It is interesting to note that Rho is high for a Deep in the

money call but low for deep out of the money call. On the other hand Rho will be lower for deep in the money PUT and generally higher for Deep out of the money PUT ( because a short seller in profit covers up his open position fast if bad news like lower interest rate crops up). Suppose A brought reliance put at Rs. 1370 and the stock is now trading 1450 .If bad news regarding interest rate comes , the person who has bought calls which are in the money will book profits leading to a change in the price and a higher RHO in that process.

• Rho of a call and put will change , with the expiration of time and it tends to approach ‘0’ at expiration time.

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Gamma

• The gamma of any option is the rate of change of the options delta with respect to the price of the underlying stock. Gamma DOES NOT measure sensitivity of option premium with respect to UNDERLYING factors / parameters that decide the option value. It only indicates the sensitivity of delta with respect to stock prices.

• Gamma is the Second order derivative of the option premium with respect to stock prices ; the first order derivative being delta.

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Gamma ( contd.)

• Gamma is the rate of change of delta to changes in stock prices

• Higher gamma reflects greater sensitivity of option delta with respect to stock prices

• Gamma of call & puts are the same• Gamma also varies with the time remaining till expiration

, for an option that is near the money , gamma INCREASES as expiration approaches for ‘in the money options’.

• For an option that is deep out of the money, the gamma tends to fall as expiration comes nearer ( till the expiration comes nearer there will be some activity built on hopes and upsets, which stop as the expiration gets nearer)

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Interpretation of Gamma

• Is a measure of how fast an option changes its directional characteristics acting more or less like the underlying stock

• A high gamma indicates high risk / rewards

• A low Gamma shows low risks.

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The Greeks

• Reflects the sensitivity of the option value to factors affecting stock prices

• With a proper use of Delta , Gamma , Theta , Rho and Vega one can strive to bring in an element of risk neutrality into the portfolio. This is especially helpful for funds which manage assets benchmarked to a particular index/ indicators. Constant review of the relevant Greeks helps the fund manager to stay course.

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Option pricing models.

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Option pricing• Boundary space for call option :• The value of the call option is directly proportional to the price of the

underlying asset and the time to expiration and inversely proportional to the exercise price.

• Therefore ,the value of the call option will be the highest for an option with zero exercise price and infinite time to expiration.

• Similarly the value of the call option will be the lowest for an option of a higher exercise price and the shortest time to expiration.A call that is about to expire with t =T will be the call with the lowest price, for a given price of the underlying asset and a given expiration date where t= the remaining time to expire and T = expiration date.

• Hence the lower bound for the value of any call option will be its value at expiration .

• The upper bound for the value of any call option will be its stock price. Though technically, the upside is unbounded, it can never exceed the stock price.

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Option pricing(contd.)

• Boundary space for put option

• The value of put option is max(0,X – S) where X is the exercise price of the option and S, the price of the underlying asset .This is the lower boundary.

The upper boundary of a put option (short) is the exercise price .This happens when the stock price is Zero.

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Put call parity

• Studies have shown that the prices of the call and put option on the same stock and quite interrelated.

• Consider two portfolios :

Portfolio A

European call option + cash equivalent of strike price of the option .

Portfolio B

European put option + Stock

Portfolio A & B have both been created out of the same sums of money .

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Value at expiration dateportfolio S>X S<X

A. Call + cash (S-X) + X = SValue of call + strike price .

0 + X = XCall value + cash/Strike is 0 price

B. Put + stock 0 + S = S (X –S)+S = XValue of put

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Contd..

• From the table it is seen that the portfolios have equal future values.So their present values must be equal too.

• That is Call + Exercise price = Put + Stock

c + x = p + s……………………………(1)

This effectively denotes a parity between calls and puts.

• (1) can also be written as p= c+x – s . This is for an European option.For an American option , P (premium) is always > p(premium in European option).

• Thus , P > c+x – s .

• Here c= European call price, p = European put price ,x = exercise price and s= Stock price ,P = American put price .

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Binomial Option pricing model.

• This model was built by Cox, Ross and Rubenstein in 1979.

• Model can be used to estimate a fair value of a call or put option.

• Can be understood easily .

• The assumptions are :

1) The underlying stock does not pay dividend during the life of the option.

2) The options are European Options.

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Single period Binomial model

• Let us assume that the current price of a stock is Rs.100 and that the price may increase to 110 or fall to 90 by the end of one year.Assume a risk free interest rate of 8% and exercise price of Rs.90.

• To value this Call option the following portfolio is constructed which ensures that the owner receives the same return (ie. 0 Rs after one year) , whether the stock sells at Rs90 or Rs.110.Let us denote the value (premium) of a call as C and the stock price after one year as S.

Portfolio Cash flow at t=0

Flow at 1 year.

S=90 Rs.

Flow after 1 year. S=110 Rs.

Write 2 calls + 2c 0 -20

Buy 1 stock(delivery)

-100 +90 +110

Borrow 83.34@8%

+83.34 -90 -90

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Contd..

• The flows at t = 1 year is the same .ie.0 Rs.,both at s=90 and s=110.

• 2c is the premium received for writing calls. Rs 90 & Rs 110 are the realizations by selling the stock bought.

The present value of Rs.90 at 8% discounting is 83.34.This is the same as the exercise price of Rs.90 for the call .

• The portfolio is constructed in such a way that the investor receives nothing at the end of the year whether the stock move up or down.

• Therefore,since returns are zero it is an acceptable argument that investment required will also be zero.

• This means 2c- 100 +83.34 = 0 or c=8.33.Therefore the value of the call is 8.33.

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Contd..• Now if 8.33 is not sacrosanct , process of arbitrage sets in .

• To confirm this, take 2 different values of C, one more than Rs.8.33 and the other less than Rs.8.33.Let us say they are C=Rs.5 and C=Rs.15.

• If c=Rs.5 , the call is underpriced and leads to an arbitrage where traders (A)buy 2 calls

(B) short the stock.

(C) lend an amount of Rs.83.34 which is equivalent to the present value of the exercise price.

• Based on this arbitrage we will work out the cash flows for the portfolio when the call price is Rs.5.

Portfolio Flow at t=0 Flows at t =1

S=Rs90

Flow at T=1

S=Rs110

Buy 2 calls -10 (2calls of rs 5 each outflow)

0 +20

Short 1 stock +100 -90.

Buy it back @90

-110.buy it back at 110.

Lend 83.34 for 1 year@8%

-83.34 +90. +90

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Contd.

• Please note that the total of column no.2 representing cash flow at t = 0 is Rs.6.66 .The cash flow totals in column 3 & 4 are both 0.

• This indicates that there has been a positive inflow of Rs.6.66 when the call price is Rs.5.This is unlike the case where cash flow was zero for a call price of Rs.8.33.

• We shall now examine the effect of arbitrage when the call price is Rs.15.If call at Rs.8.33 is the correct call ,then a price of Rs.15 represents an overpriced call leading to an arbitrage .Hence the arbitrager :

• 1) writes 2 calls .

• 2) buys 1 stock

• 3) borrows Rs.83.34 for 1 year.

See the next table for results.

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Contd.

Effect of arbitrage when call price is Rs.15

portfolio Cash flow at

T=0

Flow at t=1

S=Rs.90

Flow at t=1

S=Rs.110

Write 2 calls +30

(Rs.15 per call written )

0 -20

Short 1 stock -100 +90 +110

Borrow 83.34

For 1 year.

83.34 -90 -90

Total 13.34 0 0

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Contd.

• CONCLUSION

• The call price is 8.33 Rs. .This is because ,for call price in excess or less than Rs.8.34,an arbitrage profit results at the beginning (t=0) itself.This is assured .

• In both the cases .ie.for call price=Rs.5 and call price=Rs.15,the net cash flow after 1 year is 0 , but there is a net cash inflow at t=0 which is Rs.6.66 and Rs.13.34 respectively , reflecting arbitrage profits .

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Hedge Ratio

• Making use of the concept of no payment portfolio as discussed earlier,we arrive at the hedge ratio ; a ratio used in option pricing models like the Black and Scholes model.

• To replicate a zero portfolio pay-off concept , we buy n stocks and sell calls in such a way that the pay-off from the calls and the stocks will be the same irrespective of the stock price one year down the line.For this we will have to find the number of stocks to be bought per call written to make the pay-off from the portfolio

• 1) Equal to Zero• 2) Independent of stock price.• The number of stocks required to achieve this is called the hedge ratio.• Symbols used :• S --------- stock price at time t• X --------- exercise price of the option.• u ---------- 1 + % increase in stock price from time 0 to time 1.• d ---------- 1 + % decrease in stock price from time 0 to time 1.• C ---------- call price• Cu --------- value of call if stock price increases• Cd---------- value of call if stock price decreases.• r ----------- risk free rate of return.

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Hedge ratio..contd

Cash flow @ t=0 Cash flow at t=1

S0 = uS0

Cash flow at t=1

S=dS0

Sell calls c -Cu -Cd

Buy ‘n’ stocks - nS0 nuS0 ndS0

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Hedge ratio …..contd.• The inflow of premium to the writer is C .Hence both Cu and Cd are negative.• The outflow of cash for ‘n’ stocks is –nS0 .On sale after year 1 it is nuS0 and ndS0

are both +ve .• Now if the value of the portfolio is independent of stock prices , - Cu + nuS0 = - Cd + ndS0 ie. n( uS0 – dS0) = Cu – Cd.• n = (Cu – Cd) / So(u – d).• Now if Cd = o , Cu = 10 ,u = 1.10 , d =.90, So = Rs.100 • n = 10 – 0 / 100(1.10 – 0.90) = ½.• So, to have the call and stock of the same payoff, no matter what happens to the

stock price at the end of year one , we have to write 2 calls for each stock bought .• If we use ‘n’ as the hedge ratio , the cash flow after year 1 will be the same for the

above mentioned portfolio ie.- Cu + nuSo = -Cd + ndSo(see the previous table).• Now to make a portfolio cash flow = zero ,we have to borrow an amount at t =o so

that we owe Cd – ndSo { -(-Cd + ndSo)} or (Cu – nuSo) at time t=1 year(see the previous table).

• If ‘r’ is the borrowing rate , we should borrow Cd – ndSo / 1+ r (present value of Cd – ndSo at r% discounting .

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Table after borrowing

Cash flow at t=0 Cash flow at t=1

S1=uSo

Cash flow at t=1

S1=dSo

Write call C -Cu -Cd

Buy n stocks -nSo nuSo ndSo

borrow - { Cd – ndSo/ 1+r } Cu – nuSo Cd – ndSo

C- nSo – ( Cd + ndSo / 1+r)

0 0

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Contd.• Now since the cash flow from the portfolio after year 1 is zero , the intial

investment required to set up the portfolio must also be = 0.Therefore C – nSo – (Cd – ndSo / 1+r ) = 0 If we put 1 +r = R and rearrange , we get C = nSo R + Cd – ndSo / R.This equation can be written as C = Cu p+ Cd (1-p)/r where p = (R – d / u-d).• This is called a Single period Binomial Option model . Here , as we will later

find out , p can be taken to be the probability of stock prices moving up or down resulting in Cu or Cd as cash flows.

• Presumably because there are only two probabilities – p and 1-p it is called Binomial model .

• This is a single period Binomial option model. Here , we have not assigned probabilities to the stock price . We can assign probabilities by assuming a Risk neutral economy where expected returns on stocks are equal to risk free rate of interest.

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contd• To reiterate , in a Risk neutral economy , the return on stock = Risk free rate

of return.

• Now if the probability of upward movement of a stock = p where p = ( R – d/ u –d) , then the probability of downward movement is 1- p and the stock price after upward movement at time t is , Su = (S x u) and the stock price after the downward movement Sd = (S x d). The expected price S t is

S t = p ( S x u) +( 1-p )Sd

= p S( u –d) + Sd

= ( R – d ) / u - d X S( u – d) + Sd

= (R - d ) S + S d

= ( RS – Sd + Sd)

= ( 1 + r ) S

• This means that in a risk neutral economy , the return on stock = risk free rate of return and the expected stock price at t is (1 +r) S .

• In C = Cu p + Cd x (1-p) / R where p = ( R – d / u – d), since Cu p + Cd ( 1- p) represents the expected value of pay –off , its division by R gives the present value of future expected pay-off.

(contd.)

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Contd..

• Since the numerator is the expected value of calls pay off and clearly the probability that the stock price will increase is p where p = ( R – d / u – d)

and the probability that the decrease is 1-p ie. ( u – R) / (u – d). Eg: The current price of a stock A is Rs.50 and it is expected that the stock price after

1 year will be either Rs.60 or Rs.40.Calculate the value of call option on this stock if its exercise price is Rs.50.Find the probability of the stock price increasing or decreasing(Risk free lending and borrowing is 10%).

Cu = Rs.10 ( Stock price up from Rs.50 to Rs.60 and exercise price is Rs.50). Cd = Rs.0 ( Since exercise price is Rs.50). Stock price = Rs.50. R = 1.10. Probability of price increase = (R – d / u- d ) ie. u = 1 + % increase in stock. ie. 1 + 60 – 50 / 50 = 1.20. d= 1 – 10/50 = 0.8. So p = (R-d / u – d) = 1.10 – 0.80 / 1.20 - 0.80 = 0.75. So there is a 75% probability of an increase in prices and 25% probability of a

decrease in prices.Value of call option = Cu * p + Cd ( 1 – p) / r = 10 X 0.75 + 0 X ( 1- p) / 1.10 = 7.5 / 1.10 = Rs.6.82

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Two period binomial

• We can extend the one period binomial to two or more periods.

• Let the price of a stock be Rs.100 and in two time periods let us assume that there exists a probability of 10% in the stock going up or coming down in each period .

• So we can represent the situation in a tree diagram.

A

B

D

E

FC

100

110

121

99

8190

21

0

00

decline decline

decline

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Contd.• Let us now calculate the value of the call at different nodes in this binomial

tree. Let us assume that each step is one year and the Risk free rate of interest is 8% with strike price for the call option at Rs.100.

• The Objective• To calculate the option price at node A.• We start with the right hand end branch of the tree at the top ie. Node ‘D’.• The Stock price at ‘D’ is Rs.121.Therefore the option price will be 121 – 100

=Rs.21.• Now at nodes ‘E’ and ‘F’ the option is ‘OUT OF THE MONEY’ and has a

zero value .• At node ‘C’ also the option is ‘OUT OF THE MONEY’ and hence has zero

value.• To calculate the value of the option at ‘B’ we look at nodes ‘D’ and ‘E’.Here

u = 1 + 0.10 = 1.10. d similarly is 0.90 and ‘R’ is 1.08.• Therefore ‘p’ = R – d / u – d = 1.08 – 0.90 / 1.10 – 0.90 = 0.18 / 0.20 = 90

%.• The value of option at ‘B’ is C = Cu p + (1 – p) Cd / R = (0.90 * 21 + (1 -

0.90) * 0 / 1.08 = Rs.17.5 (contd.)

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Contd.• Now we have to find the value of call option at node ‘A’.We concentrate on

nodes ‘B’ and ‘C’.The value of call option at ‘B’ is Rs.17.50 while at ‘C’ it is 0.

• The value of call option at ‘A’ is 17.50 * 0.90 + (1 – 0.90) * 0 / 1.08. = 15.75 / 1.08 = Rs.14.58• The Binomial model has only two possible outcomes p and 1 –p whereas

the two period model has four possible stock price patterns.We can say that there will be 2 to the power of i if i is the number of periods to expiration.

As the number of periods increase , the number of branches also increases.• We have considered European option so far . American options can also be

valued just the same way except with a small difference.In American options ,the final nodes will carry the same values as that as European options.To value an American option at a node OTHER THAN THE FINAL NODE we need to take the HIGHER of the two values.

1) The value given by one period binomial model. 2) The pay off received by an early exercise of option.

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American Option

• Consider a two year American option with a strike price of Rs.50 on a Stock whose current price is also Rs.50.Assume that there are two time periods of one year each and each year , the stock price can go up or come down by 20%.The Risk free rate of interest is 20%

A

50

B60

D72

E48

F32

C40

22

0

0

0

13.49

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Contd.• The value of American Options at nodes ‘D’ ‘E’ and ‘F’ will be equal to the

value of European options ie. Rs.,22,Rs.0,Rs.0 respectively.Using the single period binomial, the value of the call option at node ‘B’ is C = Cu p + C d (1-p)/ R

= 22 X 0.65( ie. value of p given below) + 0 X 0.35 / 1.08 = Rs.13.49

P = R- d / u –d = 1.08 – 0.80 / 1.20 - 0.8 = 0.65 .

• Now at node ‘B’ the pay-off from an early exercise of call option in American options will result in Rs.10 +ve(Rs.60 – Rs.50).This Rs.10 pay – off is less than that arrived at by binomial model.Hence at node ‘B’ an early exercise is not preferable.

• If the value of an early exercise had been higher , it would have been taken as the value of the call option.

• Similarly at node ‘A’ the value of call option includes Cu p + Cd (1-p) / R

• Looking at nodes ‘B’ and ‘C’ it is 13.49 * 0.65 + 0.35 * 0 / 1.06 = Rs.8.27

• This is the value of a call at node ’A’ at time t = 0.

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Contd.• Consider a two year American PUT Option with a strike price of Rs.105 on a

stock whose current market price is Rs.100 .We assume that there are two time steps of one year and in each year the stock price moves up or down by 15 % .The risk free rate of interest is 6 %.

A 100

B 115

D132.25

E 97.75

F 72.25 C 85

0

7.25

32.7520

2.05

7.014

X = Rs.105

115 -15%(115)

105 – 97.75

85 – 15%(85)

105 – 72.25

X = 105

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Contd.• Using a single period model the probability of price increase is p = R - d/u –

d = 1.06 – 0.85 / 1.15 – 0.85 = 0.70 .ie. There is a 70% chance of upward movement.

• The value of European PUT options at nodes ‘D’,’E’ and ‘F’ are 0, 7.25 and 32.75. The value of PUT option at node ‘C’ using single period model focusing attention on nodes ‘E’ and ‘F’

• = P = Pu P + P d( 1 – p) / 1.06. • This is equal to 7.25 X 0.70 + 32.75 X 0.30 / 1.06 = Rs.14.06• (Note : Pu is Rs.725 and Pd is Rs.32.75)• The value of the put option is Rs.14.06.However an early exercise at node

’C’ will give 105 - 85 = Rs.20.Since this is higher ,we take the put option value at ’C’ as Rs.20 and not Rs.14.06.In contrast to node ’C’ an early exercise at node ‘B’ will give nothing ie.0 , and hence an early exercise is not profitable. At node ‘B’ the value of American put option is P = Pu * p + Pd(1-p / 1.06) ie. Pu = 0 and Pd = 7.25 .

• 0 X 0.70 + 7.25 X 0.30 / 1.06 = Rs.2.05. Since this is greater than 0 the value of put option at ’B’ is Rs.2.05.

• Put value at node ‘A’ = 2.05 x 0.70 + 20 x 0.30 / 1.06 = Rs.7.014.• The value of put option at node ‘A’ is Rs.7.014.

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Black & Scholes ModelProposed by Fisher Black and Myron Scholes. A Risk free portfolio consisting of a position in the

option and the position in the underlying stock is constructed . The returns from this portfolio will be equal to the risk free rate of interest in the absence of any arbitrage opportunities. For both Put and Call a suitable portfolio of the stock and the option can be constructed, so that the gain from the stock option always offsets the gain/loss from the option position and provides a certain total value of the portfolio over a short period of time.

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B&S ( contd.)

• For ex- consider that any particular time the relationship between the small change in the stock price ds and corresponding change in the price of the European call options dc follows

dc = 0.5 * ds The slope of the line is 0.5. The risk less

portfolio is consist of the following-1) One Long position in 0.5 shares 2) A short position in one call option

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B& S ( Contd.)• In B&S the risk less portfolio only for short

period of time and would require rebalancing. ( for ex- 0.5 * ds might become 0.6 *ds when a long position in 0.6 stock may have to be taken against the short position in one call option. It is however true that the returns from a risk less portfolio over a short period of time should reflect risk free interest rates. This is the essence of B&S model.

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B& S ( Contd.)

• The assumptions:• Short selling in securities is permitted• Absence of transaction cost and taxes• No dividend payments during the life of the

options• Security trading is continuously• Risk free rate of interest is r and is constant for

all maturities• The probability distribution of financial return all

the share is normal distribution. • European terms have been used

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B&S ( Contd.)• Based on the above assumptions B&S formula

is

C = S * N ( d1) – Xe –r *T-t * N (d2)

P = Xe –r *T-t * N (-d2) – S* N(-d2)

Where d1 = In ( S/X)-(r+ 2/2) * ( T- t) (T – t )

Where d2 = In ( S/X)-(r- 2/2) * ( T- t) (T – t )or

d2 = d1- ( T-t)

A

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B&S( Contd.)• C is call option price• P is put option price• S is spot price of the underlying asset• X is strike price of the option• r is a risk free rate of • T-t is the time to expiration expressed in years• is the annualized standard deviations of the returns

on the underlying stock. Measure of volatility • N ( d1) is the cumulative normal distribution. It is call the

delta of the option which is the change in the option price for a unit change in the underlying asset.

• e is exponential function, 2.7183 • In is the natural logarithm. For example- In ( 1.02 ) =

log10 (1.02 * 2.3026). Standard multiplication factor , to convert log10 to natural log is 2.3026.

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Contd.

• Also Standard deviation ( annual) = Std deviation daily * Square root of no of trading days per year ( usually 250 days per year)

• Nd1 represents delta of the options or hedge ratio , it reflects the no of shares to be bought for each option to maintain fully hedged position. Further more , the options trader is expected to be a levered trader and is hence expected to borrow an amount equal to PV of the exercise price at risk free rate of interest.

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Contd.• This aspect of borrowing is represented on the

right hand side of equation A ( slide No 77) , it indicates the present value of the “exercise price times and adjustment factor of N (d2) . In simple terms equation A represents C= (option delta * share price ) – Loan adjusted

• The values of d1 and d2 are in units of volatility because of its division by std root of (T-t) and these are the points on the horizontal axis of the standardized normal distribution for a variable normally distributed with a mean of 0 and the standard deviations of 1 . N(d) = 1-N(-d) can be interpreted as the probability of call options being in the money at expiry.

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Contd• The first part of B&S i.e. S * N ( d1) , represents the expected

benefit from owning a stock outright . This is found by multiply the stock price by the change in the call premium, with respect to a change in the price of the underlying asset . N ( d1)

• The second part Xe –r *T-t * N (-d2) gives the present value of paying the exercise price on the expiration date .

• The fair market value of the call option is then calculated by working out the difference between first and second point above.

• Equation A is in the general form of the equation for call price , i.e. C >S - Xe –r *T-t

• The equation A can be rewritten as follows— C= S * risk factor 1 - Xe –r *T-t *risk factor 2

risk factor 1 and 2 are N ( d1) and N(d2)