Valuation Of Options

Valuation of Options Prof Mahesh Kumar Amity Business School

Introduction  There are number of models for valuation of options.  The most important ones are: 1. The Binomial Option Pricing Model 2. Black-Scholes Option Pricing Model  The difference in the two models of option valuation stems basically from the assumptions made about how prices change over time.

Introduction 

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Binomial model assumes that percentage change in share prices follow binomial distribution, the Black Scholes model is based on the assumption that it follows a log normal distribution. Both the above models are in respect of the European call options. Since it never pays to exercise an American call option before its expiration if the stock involved would not pay dividend before the expiration date or if the call is dividend protected. Thus, an American call, which satisfies this condition, will be just like an European call and evaluated in a similar manner

The Binomial Option Pricing Model  This model is based on the assumption that if a share price is observed at the start and end of period of time, it will take one of the two values at the end of that period i.e., the model assumes that the share price will either increase to a given higher price or decrease to a given lower price.  Although this may seem an extreme simplification, it allows us to come closer to understanding more complicated and realistic models.

The Binomial Option Pricing Model Suppose the stock sell at S0= Rs.100 and the price will either increase by a factor of u=2 to Rs.200 (u stands for ‘up’) or fall by a factor of d=0.5 to Rs.50 (d stands for ‘down’) by year end. A call option on the stock might specify the exercise price of Rs.125 and a time to expiration of 1 year. The interest rate is 8%. At the year end, the payoff to the holder of the call option will be either zero, if the stock price falls or Rs.75, if the stock price goes to Rs.200.  These possibilities are illustrated by the following value ‘trees’ 200 75 100 C 50 0 

The Binomial Option Pricing Model Now if we compare the pay off of the call to that of a portfolio consisting of one share of the stock and borrowing of Rs.46.30 at the interest rate of 8%. The payoff of this portfolio depends on the price at the year end. Value of the stock at year end Rs.50 Rs.200 - Repayment of loan with int. - Rs.50 -Rs.50 Total 0 Rs.150  We know the cash outlay to establish the portfolio is Rs.53.70: Rs.100 for the stock less Rs.46.30 proceeds from borrowing. Therefore the portfolio’s value tree is: 150 53.70 0 

The Binomial Option Pricing Model 

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The payoff from this portfolio is exactly twice that of call option for either value of the stock price. In other words, two call options will exactly replicate the payoff to the portfolio; it follows that two call options should have the same price as the cost of establishing the portfolio. Hence the two calls should sell for the same price as the ‘replicating portfolio’. Therefore 2C=53.70 or each call should sell at C=Rs.26.85 Thus given the stock price, exercise price, interest rate and volatility of the stock prices (as represented by the magnitude of the up or down movements), we can derive the fair value of the call option.

The Binomial Option Pricing Model 

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This valuation approach relies heavily on the notion of replication. With only two possible end-of-year values of the stock, the payoff of the leverages stock portfolio replicate the payoffs to two call options therefore command the same market price. This notion of replication is behind most option pricing formulas. For complex price distribution for stocks, the replication technique is corresponding more complex, but the principle remains the same. One way to view the role of replication is to note that, using the numbers assumed for this example, a portfolio made up of one share of stock and two call options written is perfectly hedged. Its year end value is independent of the ultimate stock price. Stock Value Rs.50 Rs.200 -Obligation from 2 calls written -0 -Rs.150 Total Rs.50 Rs.50

The Binomial Option Pricing Model    

The investor has formed a risk less portfolio, with a payout of Rs.50. Its value must be the recent value of Rs.50 or Rs.50/1.08=Rs.46.30 or C=26.85 The hedge locks in the end-of-year payout, which can be discounted using the risk free interest rate. To find the value of the option in terms of the value of the stock, we do not know either the option’s or stock’s beta or expected rate of return. With a hedged position, the final stock price does not affect the investor’s payoff, so the stock’s risk and return parameters have no bearing.

The Binomial Option Pricing Model  The hedge ratio of this example is one share of stock to two calls i.e. for every call option written, one half share of stock must be held in the portfolio to hedge away the risk.  Hedge ratio is thus the ratio of the range of the values of the option to those of the stock across the two possible outcomes.

The Binomial Option Pricing Model    

The stock, which originally sells for S0= Rs.100, will be worth either d*Rs.100=Rs.50 or u*Rs.100=Rs.200 for a range of Rs.150 If the stock price increase, the call will be worth Cu=Rs.75 whereas if the stock price decreases, the call will be worth Cd=0, for a range of Rs.75 The ratio of ranges, 75/150, is one half, which is the hedge ratio we have established. Thus we can generalize the hedge ratio for other two-state option problem as H= Cu - Cd uS0-dS0 Where Cu and Cd refers to the call option’s value when the stock prices goes up or down respectively and uS0 and dS0 are the stock prices in two states.

The Binomial Option Pricing Model  Hedge ratio, H, is the ratio of the swings in the possible end-of-period values of the option and the stock.  If the investor writes one option and holds H shares of stock, the value of the portfolio will be unaffected by the stock price.  In this case option pricing is simply setting the value of hedged portfolio equal to the present value of the known payoff.

The Binomial Option Pricing Model  1.

2. 3. 4. 5.

Using our example, the option pricing technique would proceed as follows: Given the possible end-of-year stock prices uS0=Rs.200 and dS0= Rs.50 and the exercise price is Rs.125, calculate Cu=Rs.75 and Cd=0 The stock price range is Rs.150 while option price range is Rs.75 Find that the hedge ratio of 75/150=0.5 Find that the portfolio made up of 0.5 share with one written option would have an year end value of Rs.50 with certainty. Show that the present value of Rs.25 with a 1 year interest rate of 8% is Rs.23.15 Set the value of the hedged position to the present value of certain payoff. 0.5S0-C =Rs.23.15; 50-C=Rs.23.15; C=Rs.26.85

The Binomial Option Pricing Model 

1. 2. 3.

If the option was overpriced, say selling at Rs.30 then investor can make arbitrage profits by following the strategy as given below: Cash flow in 1 year for for each possible stock price Int Cash Flow S1=50 S1=200 Write 2 options Rs.60 0 -Rs.150 Purchase 1 sh. –Rs.100 Rs.50 Rs.200 Borrow Rs.40 Rs.40 -Rs.43.20 -Rs.43.20 At 8% int. Repay in 1 year Total 0 Rs.6.80 Rs.6.80 Although the initial investment is zero, the payoff in one year is positive and risk less.

The Binomial Option Pricing Model 

If the option was under priced, say selling at Rs.24 then investor can make arbitrage profits by following the strategy as given below: Cash flow in 1 year for for each possible stock price Int Cash Flow

1. 2. 3.

Buy 2 options -Rs.48 Short 1 sh. Rs.100 Invest Rs.52 -Rs.52 At 8% int. Repay in 1 year Total 0

S1=50

S1=200

0 -Rs.50 Rs.56.16

Rs.150 - Rs.200 Rs.56.16

Rs.6.16

Rs.6.16

The Binomial Option Pricing Model  It is worth noting that the present value of the profit to

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the arbitrage strategy as discussed equals twice the amount by which the option is overpriced. The present value of risk free profit of Rs.6.80 at an 8% interest rate is Rs.6.30. With two options written in the strategy above, this translates to a profit of Rs.3.15 per option, exactly the amount by which option is over priced: Rs.30 versus the fair value of Rs.26.85

The Binomial Option Pricing Model 

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Let us examine a portfolio consisting of a long position in H shares and a short position in one option. S0u cu S0 c S0d cd If there is an upward movement in the stock price, the value of the portfolio at the end of the life of the option is: S0H-cu If there is an downward movement in the stock price, the value of the portfolio at the end of the life of the option is: S0H-cd

The Binomial Option Pricing Model  

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The portfolio is risk less if the value of H is close so that the final value of the portfolio is same for both the alternatives. This means S0uH-cu=S0dH-cd H= cu-cd S0u-S0d If we denote risk free interest by r, the present value of portfolio is (S0uH-cu)e-rT The cost of setting up the portfolio is S0H-c It follows that S0H-c=(S0uH-cu)e-rT c= S0H-(S0uH-cu)e-rT c = e-rT [pcu+(1-p)cd] where p=erT -d u-d

The Binomial Option Pricing Model  In the example we had taken u=2, d=0.5, r=8% T=1 years, cu=75,cd=0 p=e0.08*1-0.5 = 1.08329-0.5 =0.38886 2-0.5 1.5 c= e-0.08*1[0.38886*75+(1-0.38886)*0] = 0.92312(0.38886*75)= Rs.26.85

Generalizing the Two-State Approach 

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To start, suppose we were to break up the year into two 6-mth segments. Here we suppose stock price S0=100 and it can increase 10% (i.e. u=1.10) or decrease 5% (i.e. d=0.95). The possible paths over the course of the year 121 110 100 104.50 95 90.25 The mid range value of 104.50 can be attained by two paths: an increase of 10% followed by decrease of 5%, or a decrease of 5% followed by a 10% increase.

Generalizing the Two-State Approach 

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The three possible end-of-year values for the stock and therefore for the option is: Cuu Cu C Cud Cd Cdd From the above we could value Cu from the knowledge of Cuu and Cud, then value Cd from the knowledge of Cdu and Cdd and finally the value of C from the knowledge of Cu and Cd The above interval of 1 year can be broken into four 3 month units or 12 1-month units or 365 1-day each units

The Binomial Option Pricing Model 

Example: Suppose the risk free interest rate is 5% per 6 month period and wish to value a call option with exercise price Rs.110 on the stock with value S0=100 u=1.10 d=0.95

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The call can rise to the expiration date with value of Cuu =Rs.11 (since the stock price is u*u*S0= Rs.121)

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The call can fall to the expiration date value of Cud =0 (since at this point stock price is u*d*S0= Rs.104.50 which is less than the exercise price i.e. Rs.110). Thus the hedge ratio at this point is H=Cuu -Cud = Rs.11-0 = 2

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uuS0-udS0

121-104.50

3

Thus the following portfolio will be worth Rs.209 at the option expiration regardless of the ultimate stock price.

The Binomial Option Pricing Model t=0

udS0=104.50

uuS0=Rs.121

Buy 2 shares at price uS0=Rs.110

Rs.209

Rs.242

Write 3 calls at price Cu

0

-33

Total Payoff

Rs.209

Rs.209

The portfolio must have a current market value equal to the present value of Rs.209 2*110-3Cu= Rs.209/1.05= Rs.199.047, therefore Cu=6.984 Next we find the value of Cd. This value is zero because if we reach this point (corresponding to stock price of Rs.95), the stock price will be either Rs.104.50 or Rs.90.25; in either case, the option will expire out of the money.

The Binomial Option Pricing Model Now we calculate the value of C.  Cu-Cd=6.984-0=6.984 uS0-dS0= Rs.110-Rs.95= Rs.15 H=6.984/15=0.4656

Action today (time=0) Buy 0.4656 shares at price S0= Rs.100

Value in Next Pd as function of Stock Price dS0=Rs.95

uS0= Rs.110

44.232

51.216

Write one call option at 0 price C0

-6.984

Total Payoff 44.232 44.232 The portfolio must have a market value of Rs.44.232 PV of Rs.44.232 at 5% rate of return = Rs.44.232/1.05=Rs.42.126 0.4656*100-C0= Rs.42.126;C0= Rs.46.56-42.126=4.434

The Binomial Option Pricing Model 

S0

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As we break the year into progressively finer sub intervals, the range of possible year end stock prices expands and, in fact, will ultimately take a familiar bell-shaped distribution. The event tree for the stock for a period with three sub intervals can be shown as: u3S0 u2S0 uS0 u2dS0 udS0 ds0 ud2S0 d2S0 d3S0 From the above we notice that As the number of sub interval increases, the number of possible stock prices also increases. Extreme events such as u3S0 and d3S0 are relatively rare as they require either three consecutive increases or decreases in three sub intervals. More moderate or mid range value like u2dS0 are more likely.

The Binomial Option Pricing Model 

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The probability of each outcome is described by the binomial distribution, and this multi period approach to option pricing is called the binomial model. Let us take the example of stock price movements over three period and let S0= Rs.100, u=1.05 and d=0.97 Cuuu Cuu Cu Cuud

C

Cud Cd Cudd Cdd Cddd

The Binomial Option Pricing Model 

3 2 1 3 

There are eight possible combinations for the stock price movements in three period: uuu, uud, udu, duu, udd, dud, ddu and ddd. Each has probability 1/8. Therefore the probability distribution of stock prices at the end of the last interval would be: Event Prob. Final Stock Prices up movements 1/8 100*(1.05)3=115.76 up and 1 down 3/8 100*1.052*0.97=106.94 up and 2 down 3/8 100*1.05*0.972=98.79 down movements 1/8 100*(0.97)3=91.27 From the above it can be seen that the mid range values are three times as likely to occur as the extreme values.

The Binomial Option Pricing Model 

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If we keep on sub dividing the interval in which stock prices are posited to move up and down the possible stock price movement within that time interval would be correspondingly small and would resemble lognormal distribution. At any node, one still could set up a portfolio that would be perfectly hedged over the next tiny time interval. By continuously revising the hedge position, the portfolio would remain hedged and would earn a riskless rate of return over each interval. This is called dynamic hedging, the continued updating of the hedge ratios as the time passes.

Black Scholes Option Valuation  An option pricing formula is far easier to use than the complex algorithm involved in binomial model.  Fischer Black, Myron Scholes & Robert C Merton provided a workable option pricing model which is popularly known as Black Scholes formula.  Scholes and Merton shared the 1997 Nobel Prize in Economics for their accomplishment.

Black Scholes Option Valuation The Black Scholes formula for the prices of European calls and puts on nondividend paying stocks are: c=S0N(d1)-Xe-rT N(d2) p=Xe-rT N(-d2)-S0N(-d1) Where d1=ln(S0/X)+(r+σ 2/2)T σ (T)1/2 d2=ln(S0/X)+(r-σ 2/2)T = d1- σ (T)1/2 σ (T)1/2 

Where c=current call option value p=current put option value N(d)=the probability that a random draw from a standard normal distribution will be less than d e=2.71828, the base of natural log function r=risk free interest rate (annualized continuously compounded rate on a safe asset with the same maturity as the expiration of the option, which is distinguished from rf, the discrete period interest rate. T=time to maturity of options in years. ln=Natural logarithm function σ = Standard deviation of annualized continuously compounded rate of return on the stock.

Black Scholes Option Valuation  From the above formula it can be seen that the option value does not depend on the expected rate of return on the stock. In a sense, this information is already built into the formula with the inclusion of stock price, which itself depends on the stock’s risk and return characteristics.  A full proof of the Black Scholes formula is beyond the scope of this course but we can explain it at a somewhat intuitive level.

Black Scholes Option Valuation  

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N(d) terms can loosely taken as risk-adjusted probabilities that the call option will expire in the money. First if we assume N(d) are close to 1.0, that is, when there is high probability that the option will be exercised. Then the value of call option is S0-Xe-rT, which is what we called earlier the adjusted intrinsic value S0PV(X). This makes sense, if exercise is certain, we have a claim on a stock with a current value S0 and an obligation with present value PV(X), or with continuous compounding, Xe-rT Second, if we assume N(d) terms are close to zero, meaning the option certainly will not be exercised. Thus the equation confirms that the call is worth nothing.

Black Scholes Option Valuation 

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For middle range values of N(d) between 0 and 1, Black Scholes equation tells us that the call value can be viewed as the present value of the call’s potential pay off adjusting for the probability of in-the-money expiration. How do the N(d) terms serve as risk adjusted probabilities? In the Black Scholes formula ln(S0/X) which appears in the numerator of d1 and d2 is approximately the percentage amount by which the option is currently in or out of the money. Example if S0=105 and X=100, the call option is 5% in the money and ln(105/100)=0.049 Similarly if S0=95 then option is 5% out of the money and ln(95/100)=-0.051

Black Scholes Option Valuation  The denominator σ (T)1/2, adjusts the amount by which the option is in or out of the money for the volatility of the stock price over the remaining life of the option.  An option in the money by a given percent is more likely to stay in the money if both stock price volatility and time to maturity are small.  Therefore N(d1) and N(d2) increase with the probability that the option will expire in the money.

Black Scholes Option Valuation  S0=100, X=95, r=0.10(10% p.a.), T=0.25 (3mths.) Standard deviation=0.50% (50% per year) d1=ln(100/95)+ [0.10+0.52/2].025 = 0.43

0.5(0.25)1/2 d2=0.43-0.5(0.25)1/2=0.18 N(0.43)=0.6664 N(0.18)=0.5714 C=100*0.6664-95e-0.10*0.25*0.5714 =66.64-52.94=13.70

Assumptions Black Scholes Option Formula  Following are the assumptions in the formula underlying Black Scholes Option pricing model: 1. The stock will pay no dividend until after the option expiration date. 2. Both the interest rate, r, and the variance rate, σ , of the stock are constant. 3. Stock prices are continuous, meaning that sudden extreme jumps such as those in aftermath of a takeover attempt are ruled out.

Assumptions Black Scholes Option Formula  

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Variants of Black Scholes have been developed to deal some of these limitations. Four of the five variables- S0, X, T, r- are straight forward. The stock price, exercise price and time of maturity are readily determined. The interest rate used is the money market rate for a maturity equal to that of the option and the dividend payout is reasonably predictable, at least over short horizons. Standard deviation, the last input is estimated from historical data. The discrepancies between an option price and its Black Scholes value are simply artifacts of error in the estimation of the stock’s volatility.

Assumptions Black Scholes Option Formula 

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Market participants often give the option valuation problem a different twist. Rather than calculating the Black Scholes option value for a given stock’s standard deviation, they ask instead: What standard deviation would be necessary for the option price that I observe to be consistent with the Black Scholes formula? This is called implied volatility of the option, the volatility level for the stock that the option price implies. Through implied volatility investors can judge whether they think the actual stock standard deviation exceeds the implied volatility. If it does, the option is considered a good buy; if actual volatility seems greater than the implied volatility, its fair price would exceed observed price.

Assumptions Black Scholes Option Formula 

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The option with the higher implied volatility would be considered relatively expensive, because a higher standard deviation is required to justify its price. The analyst might consider buying an option with the lower implied volatility and writing the option with the higher implied volatility.

Dividends and Call Option Valuations 

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In case of dividend paying stocks, the Black Scholes formula is adjusted by replacing S0 with S0PV(dividends). Suppose the underlying asset pays a continuous flow of income. This might be a reasonable assumption for options on stock indexes, where different stocks in the index pay dividends on different days, so that the dividend income arrives in more or less continuous flow. If the dividend yield, denoted ‘d’ is constant then value of S0 is replaced by S0e-dt in the original formula. The above approximation is good for European call option.

Dividends and Call Option Valuations 

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Example: Suppose that a stock selling at Rs.20 will pay a Re.1 dividend in four months whereas the call option on the stock expires in 6 months. The effective annual interest rate is 10% so that the present value of the dividend is Rs.1/(1.10)1/3= Rs.0.97. Black suggests that we can compute the option value in one of the two ways: Apply the Black Scholes formula assuming early exercise, thus using the actual stock price of Rs.20 and a time to expiration of 4 months. (the time until dividend payment). Apply the Black Scholes formula assuming no early exercise, using the dividend adjusted stock price of Rs.20Re.0.97 = Rs.19.03 and a time to expiration of 6 months.

Dividends and Call Option Valuations 

The greater of the two values is the estimate of the option value. In other words, the so called pseudo-American call option value is maximum of the value derived by assuming that the option will be held until expiration and the value derived by assuming that the option will be exercised just before an ex-dividend date.

Hedge Ratios and Black Scholes Formula      

The call option position is more sensitive to swings in stock’s price that is the all-stock position. To quantify these sensitivities hedge ratios are used. An option’s hedge ratio is the change in the price of an option for a Re.1 increase in the stock price. A call option, therefore has a positive hedge ratio and put option a negative hedge ratio. The hedge ratio is commonly called the option’s delta. If we plot a graph of the option value as a function of the stock value, hedge ratio is simply the slope of the value curve evaluated at the current stock price.

Hedge Ratios and Black Scholes Formula 

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If the hedge ratio of a stock at S0=120 is say 0.60, it implies that as the stock increase in value by Re.1, the option value increase by approximately Re.0.60. For every call option written, 0.60 shares of the stock would be needed to hedge the investor’s portfolio. Example: If one writes 10 options and holds 6 shares of stock, according to hedge ratio of 0.60, a Re.1 increase in stock price will result in a gain of Rs.6 on the stock holding whereas the loss on the 10 options written will be 10*0.60= Rs.6 The stock price movement leaves total wealth unaltered, which is what hedged position is intended to do.

Hedge Ratios and Black Scholes Formula 

Black Scholes hedge ratios are easy to compute. The hedge ratio for a call is N(d1) , whereas the hedge ratio for put is N(d1)-1

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Although rupee movements in option prices are less than the rupee movements in the stock price, the rate of return volatility of options remain greater that the stock return volatility because options sell at lower price.

Hedge Ratios and Black Scholes Formula 

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In our example, with the stock selling at Rs.120 and the hedge ratio of 0.6, an option with exercise price Rs.120 may sell for Rs.5 If the stock price increases to Rs.121, the call price would be expected to increase by Re.0.60 to Rs.5.60. The percentage increase in the option value is Rs.0.60/5=12% however, the stock price increase is only Re1/120=0.83%. The ratio of percentage change is 12/0.83=14.4%. That is for every 1% increase in the stock price, the option price increases by 14.4%. This ratio, the percentage change in option price per percentage change in stock price, is called the option elasticity.

Hedge Ratios and Black Scholes Formula 

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When you establish a position in stocks and options that is hedged with respect to fluctuations in the price of the underlying asset, your portfolio is said to be delta neutral, meaning that the portfolio has no tendency to either increase or decrease in value when the stock price fluctuates. Portfolio insurance can be obtained by purchasing a protective put option on an equity position. When the appropriate put is not traded, portfolio insurance entails a dynamic hedging strategy when a fraction of the equity portfolio equal to the desired put option’s delta is sold and put in riskless securities.