CHAPTER 10 OPTION PRICING - II
Options Pricing II Intrinsic Value and Time Value Boundary Conditions for Option Pricing Arbitrage Based Relationship for Option Pricing Put Call Parity 2
Binomial Option Pricing Model Binomial option pricing model makes simplified assumption of only two possible values of the underlying asset, one up (Rs 125, 25% up), with probability p and another down (Rs 80, 20% down with probability (1 p) from the current price (Rs 100). Price of the Asset under Binomial Option Pricing Model At t = 0 at t = 1 p S 1 = 125 S 0 = 100 (1 - p) S 1 = 80 3
Valuing ATM Call We value at-the-money European call with a strike price, X of Rs 100 and time to expiry of one year. The payoff of call with strike price at X = 100 is At price of Rs 125 : 125-100 = Rs 25, At price of Rs 80 : 0 (the call option expires worthless). Risk free return is 6% per annum. 4
Expected Value of ATM Call We find the expected value of the call at t = 1. Then we discount the value of the call at t = 0. the probability of upside movement, p as 0.90. Therefore, the probability of downside movement (1-p) is 0.10. Expected value of stock (at t = 1) = 0.90 125 + 0.10 80 = Rs 120.80 Expected value of the call (at t = 1) = Higher of (expected value of the stock - exercise price, 0) = 120.80-100.00 = Rs 20.80 Value of the call today = 20.80/1.06 = Rs 19.62 5
Expected Value of Call - Problem Estimating the probability and finding the expected value of the option is fallacious because we do not know the appropriate discount rate. The seemingly valid value of the call offers arbitrage opportunity. 6
Arbitrage and Expected Value With the value of the call at Rs 19.62 one arbitrageur can set up a portfolio of long five shares and write (short) nine calls. The initial investment would be: Action Cash flow Buy 5 shares = - 5 100 = - 500.00 Write 9 calls = + 9 19.62 = + 176.58 Cash outflow at t = 0 = Rs 323.42 The portfolio can be set up with borrowing at 6%. The liability would be 1.06 323.42 = Rs 342.83 after a year. 7
End Position of Portfolio With price at Rs 125 With price at Rs 80 Value of stock would be 5 125 = Rs 625. The liability on calls written would be 9 25 = Rs 225. The net cash inflow is Rs 400. Value of stock would be 5 x 80 = Rs 400 The liability for the calls written is zero. Net cash flow is Rs 400 Portfolio Value at T = 1 (All figures in Rupees) T = 1 5 Shares 9 Calls Total 125.00 625.00-225.00 400.00 100 80.00 400.00-400.00 8
Discount Rate Assigning different probabilities to up and down movement with risk perception (attaching higher/lower chances to price moving upwards/downwards) of stock price in future would provide different values to the call option would offer arbitrage. In such estimation of expected value the discount rate used is not appropriate to the risk assumed. 9
Risk Neutral Valuation Under risk neutral method we assume that the expected value of the stock would provide return equal to risk free rate. E(S 1 ) = (1+ r f ) x S 0, or E(S 1 ) = e rft x S 0 with continuous compounding Alternatively stated we can value the derivative by finding the risk neutral probabilities with which the current market price is nothing but expected value of the underlying asset discounted at risk free rate of return. The valuation of the derivative on the asset too can follow the same method. 10
Risk Neutral Valuation Under binomial approach if the risk neutral probability of the upward movement is p with a gain of u%, and probability of the downward movement is (1- p) with a loss of d% then the expected return must equal risk free rate of return p u +(1- p) d = r rf - d p = u - d In our case risk free rate is 6% with u = 25% and d = -20%, then probability of the upside movement implied is p as f p = rf - d = u - d 6 -(-20) = 0. 5778 = 57. 78% 25 -(-20) 11
Risk Neutral Valuation From the implied probabilities we may value the call option as expected value of the payoff. Value of the call at the end of the option period would be: = Probability of upward movement Payoff + Probability of downward movement Payoff = p Max (S 1 - X, 0) + (1 - p) Max (S 1 - X, 0) Hence value of the call at the end of one year would be: = 0.5778 25 + 0.4222 0 = Rs 14.44 The value of the call today is 14.44/1.06 = Rs 13.63. 12
Risk Neutral Valuation - Approach Under the risk neutral approach we found The risk neutral implied probabilities of up and down movement, Then, calculated the expected payoff of the option at maturity with implied probabilities, and Finally, discounted the expected payoff at the risk free rate to arrive at the current value of the option. 13
Equivalent Portfolio Approach An alternate way, the risk neutral valuation is to construct a portfolio of buying some shares by borrowing so as to have the same payoff as that of the call option. The portfolio of share and borrowing can be valued easily with the interest rate and the spot prices known. This can be set equal to the price of the call option. 14
Option Delta & Equivalent Portfolio Option Delta = Portfolio of 5/9 share and a borrowing that matures to Rs 44.44 after a year would have following payoff: Share Price Rs 125.00 Rs 80.00 Value of 5/9 share (Rs) 69.44 44.44 Maturity value of loan (Rs) - 44.44-44.44 Value of the portfolio at maturity 25.00 0.00 Payoff of call option (Rs) 25.00 0.00 Payoffs of portfolio and call are same hence they are priced same. Value of call today = Value of the 5/9 share today -Value of the loan today = 5/9 100-44.44/1.06 = 55.55-41.92 = Rs 13.63 Spread of call option Spread of share price 25-0 = 125-80 = 25 45 = 5 9 15
Binomial Model for Put Pricing For pricing put we set up a portfolio of long one share and M puts with X = 100. When the price is Rs 125 and Rs 80 the value of portfolio is: 100 Binomial Put Pricing One Share & M Puts 125 125 80 80 + 20 x M Equating final values gives M = 2.25. Setting this portfolio of one share and 2.25 puts to yield a risk free return must give the value of the put, p (100 + 2.25 x p ) 1.06 = 125 or p = Rs 7.97 16
Equivalent Portfolio Approach Option delta helps in construction of riskless portfolio and therefore used in valuation of options. Under equivalent portfolio approach we proceeded as below: Calculated the option delta, Δ Set up the portfolio of 1 share and short Δ call or one share and long Δ puts, Found the values of portfolio at the end and its present value, Equated the present value with the cost of portfolio to solve for the value the option. 17
2-Period Binomial Model A two-period binomial model with 25% up and 20% down movement is as below: 100 Two Period Binomial Tree VALUES (Rs) Stock Call X = 100 156.25 56.25 + 25% 125 +25% - 20% 100.00 0.00-20 % + 25% 80-20 % T = 0 T = 1 T = 2 64.00 0.00 18
2-Period Binomial Model From the values at T = 2 we work backwards to value of the call using risk neutral probabilities and risk free rate of return. Risk neutral probabilities are 0.5778 and 0.4222 for up and down movements respectively. Value of call option for Upper Node at T = 1 Value of call at T =2: 0.5778 x 56.22 + 0.4222 x 0 = Rs 32.48 Value of call at T = 1: 32.48/1.06 = Rs 30.65 Value of call option for Upper Node at T = 1 Value of call at T = 2: 0.5778 x 0 + 0.4222 x 0 = 0 Value of call at T = 1: Zero Value of call option for Node at T = 0 = 1/1.06(.5778 x 30.65 + 0.4222 x 0) = 17.71/1.06 = Rs 16.71 19
Multi-Period Binomial Model Binomial model can be extended to multi-periods to broaden the range of price changes in a given period of time. For the price to converge to the original value the return relatives for the rise and the fall have to be reciprocal i.e. 1/1.25 = 0.80 (25% rise and 20% fall). Such selection of rise and fall would lead to recombining symmetrical trees. It makes computation easier. As we increase the number of trees by shortening the time interval the option price under binomial model moves closer and closer to analytical model such as Black Scholes. 20
Risk Neutrality & Binomial Model An American call is no greater in value than an equivalent European call as by exercising early one cashes in the intrinsic value only but loses the time value of the option. We may use multi-period binomial model to value options by: 1. Computing the risk neutral probabilities of up and down movement, then 2. Using the probabilities and starting backwards, we find the expected payoff of the option at preceding node, and 3. Discounting the payoff at risk free rate to find the value at that node. 4. Proceeding with all steps till node T = 0 is reached. 21
Valuing American Put For call (European or American) and European put, exercise of option before maturity does not make any difference in valuation. We may use multi-period model to value American put by 1. Computing the risk neutral probabilities of up and down movement, then 2. Using these probabilities and starting backwards, we find the expected payoff of the option at preceding node, and 3. Comparing risk-neutral values with the payoff if exercised at each node, with higher of the two retained, and 4. Discounting the payoff at risk free rate to find the value at that node 5. The process is repeated till the last node is reached. 22
Binomial Model Currency Options Binomial model can be extended to currency options with small modification in the risk free rate. While valuing currency option the interest rate must be considered net of foreign interest rate. If the domestic risk free interest rate is 10% p.a. and that in foreign currency is 4% p.a. then while computing the risk neutral probabilities the interest rate that must be used is 6% (10% - 4%); the differential of the two. 23
Binomial Model Index Options Binomial model can also be extended to index options with small modification in the risk free rate. While valuing index option the interest rate must be considered net of dividend yield on the index. If the risk free interest rate is 8% p.a. and the dividend yield on index is 3% p.a. then while computing the risk neutral probabilities the interest rate that must be used is 5% (8% - 3%); the differential of the two. 24
Index Option Value - Example The current value of NIFTY is 4,500. In a period of 3 months it can go up or down by 10%. If the risk free interest rate is 8% and dividend yield is 2% find the value of 3-m call and put with X = 4,600 using single stage binomial model one index point=re 1. Solution 3-m European call and put with X = 4,600 can be valued with risk neutral valuation with net interest rate of 6%. The risk neutral probabilities are: p 1- p r f t 0.06x3/12 e - (1- d) e (1-0.10) = = u + d 0.10 + 0.10 1.0151-0.90 = = 0. 5756 = 57. 56% 0.20 = 1-0.5756 = 0.4244 25
Index Option Value - Example Risk Neutral Valuation of Index Options (Figures in Rupees) T = 3 m Spot Call X = 4600 Put X = 4600 p= 0.5756 4,950 350-4,500 1- p = 0.4244 4,050-550 Value of call At T = 3 m = 350 x 0.5756 + 0 x 0.4244 = 201.45 At T = 0: = 201.45 x e -0.06 x3/12 = 201.45 x 0.9851 = Rs 198.45 Value of put At T = 3 m: =0 x 0.5756 + 550 x 0.4244 = 233.44 At T = 0: = 233.44 x e -0.06 x 3/12 = 233.44 x 0.9851 = Rs 229.96 26
Binomial Model in Practice The Binomial model seem to have following limitations: Too unrealistic for determination of fair value. The probability distribution seems far from reality. The primary determinant of the option value i.e. the volatility in the implied probability of the binary states. However, in actual practice they are overcome as follows: By adding binomial periods usually around 30 small intervals. With 30 intervals we have 31 different terminal prices. There would be 2 30 different paths to achieve these 31 terminal prices. 27
Binomial Model in Practice The probability of maximum price and minimum price is 1/2 30. The next level of prices would consist of 29 up moves and one down move. It is possible to achieve it in 30 different ways i.e. 30 C 1. The probability would be 30 C 1 /2 30. Similarly we may find the probability of each of the 31 possible terminal prices. Thus probability distribution would be far closer to reality than what one imagines in binary state. The way to overcome the objection of absence of volatility is to incorporate the stock volatility in the binary model. It is possible to equate the up or down move based on stock volatility, σ. 28
Factor Affecting Option Price Price of the Underlying Asset Exercise Price Time Left for Expiration Variability of Price Interest Rate Dividend during Option Period 29
Factors Affecting Option Value Price of the underlying, S: The payoff of call is Max (S-X, 0) and that of the put is Max (X S, 0) With increasing price of underlying asset the call option increases in value while put option decreases in value. Exercise Price, X: With payoff of call and put it can readily be seen that with increasing exercise price the value of call option decreases and put option increases. 30
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Factors Affecting Option Value Time left for expiration, T : With more time to maturity the values of call and put increase. As time elapses the values of options would fall down as there is lesser time available to pierce the strike price either way. Volatility of price of the underlying asset, σ : Options derive their value from the volatility in prices. If there is no volatility of prices the options to buy or sell them would be worthless. With increasing volatility of the price of underlying asset the both call and put options increase in value. 32
Factors Affecting Option Value Interest rate, r: The minimum call value is c S PV of X or S - Xe -rt The minimum put value is p Xe -rt S With increasing rate of interest the value of call declines and value of put increases. Expected dividend during the life of the option, D: Dividend accrue only to the holder of the asset and not to the person holding option. As dividend is paid the value of the asset declines by equivalent amount. With expected decline in the spot price the value of call must decrease and that of put must decline. 33
Black Scholes Option Pricing Model Derivation Assumptions Interpretation Applying BSM Call Pricing Put Pricing Dividend Paying Stock Options on Indices Options on Currencies 34
Black Scholes Model-Assumptions Black Scholes model is an analytical model for valuing options on non-dividend paying stocks with following assumptions: Stock returns have log normal distribution: The mean and standard deviation of ln r are proportional to time If sample mean and variance were m and σ 2 respectively, then Expected value E(Y T ) = m x T and Variance (VarY T ) = σ 2 x T Natural log of returns relative has normal distribution. 35
Black Scholes Model (BSM) BLACK SCHOLES MODEL (BSM) FOR OPTION PRICE ON NON-DIVIDEND PAYING STOCK c = S.N(d )- X.e 2 ln(s/x) + (r + σ /2)t where d1 = ; and σ t 2 ln(s/x) + (r - σ /2)t d2 = or d2 = d1 - σ t σ t c = Call Premium; p = Put Premium the underlying Asset X = Exercise Price; interest rate(annualised)in decimal form remaining for expiration of the option in years Annualised standard deviation of returns in decimal form )are cumulative normal distribution functions S = Spot Price of r = Risk free t = Time σ = N(d )and N(d 1 2 at d and 1 ln = Natural log 1 d 2 -rt.n(d 2 ) respectively p = X.e -rt.n(-d )- S.N(-d ) 2 1 36
Applying BSM Call Option Given the following information: Current market price: Rs 50, Volatility: 30% p a, Risk free interest rate: 10%. Find the value of 3 m call with X = 40. What are the intrinsic and time values? Solution We find d 1 and d 2 for S = 50, X = 40, r = 0.10, t = 0.25, σ = 0.30. ln(50/ 40) + (0.10 + 0.30x0.30/ 2)x0.25 1 = = 1.7293 d2 = d 0.30x 0.25 = 1.7293 0.15 = 1.1593 0.30x 0.25 d 1 Using Excel function N(d 1 ) = 0.9581 and N(d 2 ) = 0.9429 c = S N(d 1 ) Xe -rt N(d 2 ) = 50 x 0.9581 40 e -0.1x0.25 x 0.9429 = 47.9050 39.0124 x 0.9429 = Rs. 11.12 The intrinsic value of the call is Rs 10 (50 40) and remaining value of (11.12 10.00) = Rs. 1.12 represents the time value. 37
Interpreting BSM The first term SN(d 1 ) can be said to be the cash inflow and the second term of Xe-rt N(d 2 ) can be said to be potential cash outflow of the exercise price. The term N(d 1 ) refers to the delta of the option which represents the fraction of stock bought for each call written in the hedged portfolio. Hence the first term is the value of the fraction of stock owned and is an asset. The expression Xe -rt reflects the present value of the exercise price. N(d 2 ) can be interpreted as probability of call becoming in the money. And if it does it entails cash outflow of X, the present value of which is Xe -rt. 38
Applying BSM Put Option Given the following information: Current market price: Rs 50, Volatility: 30% p a, Risk free interest rate: 10%. Find the value of 3 m put with X = 40. What are the intrinsic and time values? Solution We find d 1 and d 2 for S = 50, X = 40, r = 0.10, t = 0.25, σ = 0.30. ln(50/ 40) + (0.10 + 0.30x0.30/ 2)x0.25 1 = = 1.7293 d2 = d 0.30x 0.25 = 1.7293 0.15 = 1.1593 0.30x 0.25 d 1 Using Excel function N(-d 1 ) = 0.0419 and N(-d 2 ) = 0.0571 p = Xe -rt N(d 2 ) - S N(d 1 ) = 40 e -0.1x0.25 x 0.0571-50 x 0.0419 = 2.2267 2.0950 = Rs. 0.13 The intrinsic value of the put is zero (S > X) and entire value of Rs. 0.13 represents the time value. 39
BSM with Dividend Black Scholes model can be adapted for dividend paying stocks. a) modifying the spot value by the present value of dividend, and b) modifying the growth m = r + σ 2 /2 to r q + σ 2 /2. Adjusting the spot price with dividend would result in substitution of S with S.e -qt. MERTON MODEL FOR CALL OPTION PRICE ON DIVIDEND PAYING STOCK -qt -rt c = S.e.N(d )- X.e.N(d ) p = X.e -rt 1.N(-d )- S.e 2.N(-d ) 2 ln(s/x) + (r - q + σ /2)t where d1 = ; and σ t 2 ln(s/x) + (r - q - σ /2)t d2 = or d2 = d1 - σ σ t q = Dividend Yield (Continuous compounding basis) -qt 2 1 t 40
BSM Index Options NIFTY is at 4,500. If the risk free interest rate is 8% and dividend yield on NIFTY is at 3% what would be the value of 3-m call option with X = 4,600? The volatility of NIFTY is at 25% p.a. Solution We find d 1 and d 2 for S = 4500, X = 4600, r = 0.08, q = 0.03, t = 0.25 years and σ = 0.25. ln(4500/4600)+(0.08-0.03 + 0.25 x 0.25/2)x 0.25 = = -0.0133 0.25x 0.25 d1 2 1 Using Excel function N(d 1 ) = 0.4947 and N(d 2 ) = 0.4450 c = S e -qt N(d 1 ) Xe -rt N(d 2 ) = 4466.37 x 0.4947 4600 e -0.08x0.25 x 0.4450 = 2209.52 2006.47= Rs 203.05. d = d - 0.25 x 0.25 = -0.1383 41
BSM for Currency Options Black Scholes model applies to options on currencies with foreign risk free interest rate, r f replacing the dividend yield in the valuation formula of options on indices. where c = S.e p = X.e r f -r t f -rt d 1.N(d )- X.e 1.N(-d 2 -rt )- S.e.N(d -r t ln(s/x) + (r - rf = σ t f.n(-d ) + σ /2)t ; and 2 ln(s/x) + (r - rf - σ /2)t d2 = or d2 = d σ t = Risk free interest rate in foreign currency 2 ) 1 2 1 - σ t 42
Valuing Currency Options Consider 3-m ATM European call and put on with spot of Rs 75/. The risk free interest rates in rupee and euro are 8% and 4% respectively. The volatility of exchange rate for euro is 15%. S = 75, X = 75, r = 8%, r f = 4%, T = 0.25 years, and σ = 0.15. ln(75/75)+(0.08-0.04 + 0.15 x 0.15/2)x 0.25 d1 = = 0.1708 d2 = d1-0.15 x 0.15x 0.25 0.25 = 0.0958 We get N(d 1 ) = 0.5678 N(d 2 ) = 0.5382 N(-d 1 ) = 0.4322 N(-d 2 ) = 0.4618 Call price, c = S e -r f t N(d 1 ) Xe -rt N(d 2 ) = 75.00 x e -0.04 x 0.25 x 0.5678 75 e -0.08 x 0.25 x 0.5382 = 74.25 x 0.5678 73.51 x 0.5382 = 42.16 39.17 = Rs 2.59 Put price, p = Xe -rt N(-d 2 ) - S e -r f t N(-d 1 ) = 75 e -0.08 x 0.25 x 0.4618-75.00 x e -0.04 x 0.25 x 0.4322 = 73.51 x 0.4618-74.25 x 0.4322 = 33.99 32.09 = Rs 1.90 43
Volatility Call Option Put Option Effects of Dividend Currency Options Options on Indices Chapter 8 Options - Basics 44
Volatility Volatility in spot prices is a major driver of option premium. More volatility makes option more valuable irrespective of it being a call or a put. Volatility of return over time interval T is proportional to standard deviation of returns measured over a small interval multiplied by square root of number of time intervals. Annual volatility is equal to σ daily x 250 if we assume 250 number of trading days in a year. 45
Measuring Volatility Volatility of the prices is the non-observed parameter in option price determination. It needs to be measured. The historical volatility can be calculated in following manner: Feed the price data (daily, weekly or monthly) for sufficient number of periods. Find out the relative return for a period by dividing the price of the period by that of the preceding period. Find out the natural log of the relative return. Find out the mean of natural log returns and the its variance and find annual volatility σ annual = σ period. T where T = Nos. of period per annum. 46
Implied Volatility The volatility used in the Black Scholes model is an unobservable statistic. Using historical volatility in the Black Scholes formula for option prices may not match with the market prices. The reason for divergence between the theoretical price as per Black Scholes model and the actual market price may lie in the different estimate of volatility used by the market. The fact that options are traded the price of option implies a volatility in the prices of the underlying asset, which may be different than the measured volatility based on historical data. The volatility that matches the Black Scholes price to the actual market price is called implied volatility. 47