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EBS – How to Price Options – Binomial & Black-Scholes Models Option Pricing Models There are two option pricing models; the binomial, a discrete time period model, and the Black-Scholes, which is a continuous time model. If you were given the same pricing information for each model, the answers generated would be very close together. Black Scholes uses continuous compounding, whereas the binomial uses annualised compounding. Both models will have the share price, exercise price, time to expiry and interest rate as pricing variables. They also both have volatility, but different wasy of representing it. With the binomial, volatility is captured through an upward price or a downward price that the underlying could move to. With the Black Scholes model volatility is represented by the standard deviation of returns on the underlying asset. The more it moves about, the higher the volatility. Binomial Pricing Model The formula for the binomial is as follows; Y = Cu – Cd S0(u – d) Z = uCd – dCu (u – d)(1+rf)t C = YS0 + Z Where Cu is the payoff from the option if the asset price goes up (ie, = uS0 –X, the up share price minus the exercise price). Cd is the payoff to the option if the price goes down. This is dS0 – X, which is usually a negative number and since you cannot have a negative payoff to an option, this would be fully written out as; (maximum: dS0 – X, 0). This will be 0. u and d are the upward and downward multipliers. u = up asset price/starting asset price, and d = down asset price/starting share price. For example if a question says, the current share price is 200p and can rise to 340p or fall to 140p, with an exercise price of 180p, time to expiry of one year and an interest rate of 5%, then the values for u, d, Cu and Cd will be; u = 340/200 = 1.7 d = 140/200 = 0.7 Cu = 340 – 180 = 160 Cd = 0 (max of: 140 – 180 or 0) Any binomial question you will be asked, you will have to work out the above before you can start on solving the Y and Z equations. 1
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EBS – How to Price Options – Binomial & Black-Scholes Models The option pricing model is really a two asset portfolio that replicates the price of the option and the Y and the Z are the two investments. Y represents the proportion of the underlying asset you are going to buy and have in the portfolio. Z represents the amount of borrowing required to allow you to purchase that proportion of the asset. In a binomial question you will not be given Cu and Cd, nor u and d. Using the formula from the text we can calculate the call option price; Y = Cu – Cd S0 (u – d) Y = 160 – 0 200*(1.7-0.7) Y = 160 200 Y = 0.8 This represents the proportion of the share that will be bought. The second part of the calculation is to work out the amount of borrowing needed. Z = uCd – dCu (u-d) (1 + rf)t Z = (1.7*0) – (0.7*160) (1.7 – 0.7) (1 + 0.05) Z = -112 1.05 Z = -106.67 Now we can price the call C = Y*S0 + Z C = (0.8*200) + (-106.67) C = 160 – 106.67 C = 53.33 One point to bear in mind with this example, is that it is a one year option. What do we do if the optio is for less than a year? The adjustment is only to the bottom line of the Z calculation, where we have (1+rf)t. If it is a six month option, this part of the equation becomes (1+rf)0.5, for three months it is (1+rf)0.25, and so on. 2
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EBS – How to Price Options – Binomial & Black-Scholes Models An alternative way of understanding the binomial model (this might help you understand the ideas behind the pricing of calls) We can value the call by examining two strategies. The first is to buy the call and the second is to buy a proportion of the share and to borrow a set amount of cash. In the example above, we need to buy 0.8 of the share and borrow 112. The cash flows from the second strategy will match the cash flows from buying the call. How do we get these figures? This is what is known as a replicating strategy, because the cash flows match we are duplicating the call purchase with this second strategy. The end period cash flows are laid out as follows: If stock = 340 If stock = 140 (a) Buy a call 160 0 The first part of the table represents the outcome from the call purchase if the share goes up or down. Below is the outcome from the alternative strategy of stock purchase and borrowing. (b) Proportion of stock to buy 0.8 272 112 Borrow at 5% 106.67 -112 -112 Total from stock and borrowing strategy 160 0 The buying stock and borrowing strategy produces exactly the same outcome as the call option, so the net of the buy stock and borrowing today, must equal the call price. How do we know how much stock to buy? This is the delta of the option, it is the ratio of the spread of the option prices divided by the spread of the stock prices. The option will either be 160 or 0, so the spread of the option price is 160. The spread of the stock price is between 140 and 340, ie, 200. So the delta, or the proportion to buy is 0.8 (this is the same as Y in the original calculation). How do we work out the 106.67 amount of borrowing? Well buying 0.8 of the stock produces outcomes that are exactly 112 more than the payoffs to the option (160 and 0). To duplicate the call by a stock purchase and borrowing, we need to borrow enough so that we have to pay back 112 of interest and principal at expiry. The borrowing is simply 112 present valued at 5%, which equals 106.67. The value of the call is: Proportion of stock 0.8 * 200 = 160 Borrow 106.67 = -106.67 Call price = 53.33 Which is the same answer as we calculated earlier with the Y, Z and C equations. 3
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EBS – How to Price Options – Binomial & Black-Scholes Models Pricing the Put option The put is priced using a method called the put-call parity theorem. This says that the call plus the present value of the exercise price is equal to the put plus the stock price. This can be written as; Call + PV Exercise Price = Put + Stock Price So to calculate the put price, rearrange the formula to solve for the put; Put = Call + PV Exercise Price – Stock Price The Put price for the worked example can be calculated; Put = C + PVX – S0 Put = 53.33 + 180/1.05 – 200 Put = 24.76 4
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EBS – How to Price Options – Binomial & Black-Scholes Models Binomial Practice questions 1. The share price is currently $140, the exercise price is $120, the interest rate is 10% and the share price can either rise to 180 or fall to 90 at the end of the one year option life. What is the call option price and what is the put price? 2. An option has an exercise price of 410p and the current stock price is 400p. Over the next year the price will either rise to 500p or fall to 350p The interest rate is 5%. What are the prices of the call option and the put option? 3. Berio is a Spanish exporter, you have to price an option on the company’s shares. The share price just now is €60. The options you are interested in have an exercise price of €55. You expect the share price to be either €80 or €45 in six months time. The interest rate is 8%. What are the prices of the call and put options? 4 Durbin is an agricultural company. The stock in the company is expected to be either 950p or 780p in three months time. The current share price is 854p and the interest rate is 5%.. What are the call and put prices if the exercise price is 800p? 5. Ocean Transport and Trading’s shares are trading at 250p. What is the price of a nine month call option, if the exercise price is 260p, the interest rate is 6% and the share price will either be 300p or 220p at the end of that period? 5
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EBS – How to Price Options – Binomial & Black-Scholes Models Binomial Solutions Q.1 Exercise pr (X) = 120 1 yr up price = (S0 * u) 180 1 yr down price = (S0 * d) 90 Risk free rate (1 + rf)^t = 1.10 one year Share price So = 140 these > Upward multiplier 'u' = 1.286 have to be > Downwrd mult. 'd' = 0.643 worked > Cu, (up price - ex.price) 60 0 maximum of (up price - X, or 0) out > Cd, (down price - ex pr) -30 0 maximum of (down price - X, or 0) Y = (Cu - Cd) / So*(u - d) 0.667 Z = (uCd - dCu) / [(u - d)*(1+rf)] Z-top line = -38.571 Z-bottom line = 0.707 Z = -54.545 C = Y*So + Z = 38.788 Put price: Put = Call + PV Ex price - Underlying share price Put = 38.788 + (120/1.10) - 140 Put = 7.879 Q.2 Exercise pr (X) = 410 1 yr up price = (S0 * u) 500 1 yr down price = (S0 * d) 350 Risk free rate (1 + rf)^t = 1.05 one year Share price So = 400 these > Upward multiplier 'u' = 1.250 have to be > Downwrd mult. 'd' = 0.875 worked > Cu, (up price - ex.price) 90 0 maximum of (up price - X, or 0) out > Cd, (down price - ex pr) -60 0 maximum of (down price - X, or 0) Y = (Cu - Cd) / So*(u - d) 0.600 Z = (uCd - dCu) / [(u - d)*(1+rf)] Z-top line = -78.750 Z-bottom line = 0.394 Z = -200.000 C = Y*So + Z = 40.000 6
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EBS – How to Price Options – Binomial & Black-Scholes Models Put price: Put = Call + PV Ex price - Underlying share price Put = 40.000 + (410/1.05) - 400 Put = 30.476 Q.3 Exercise pr (X) = 55 1 period up price = (S0 * u) 80 1 period down price = (S0 * d) 45 Risk free rate (1 + rf)^t = 1.08 1.0392 six months Share price So = 60 these > Upward multiplier 'u' = 1.333 have to be > Downwrd mult. 'd' = 0.750 worked > Cu, (up price - ex.price) 25 0 maximum of (up price - X, or 0) out > Cd, (down price - ex pr) -10 0 maximum of (down price - X, or 0) Y = (Cu - Cd) / So*(u - d) 0.714 Z = (uCd - dCu) / [(u - d)*(1+rf)] Z-top line = -18.750 Z-bottom line = 0.606 Z = -30.929 C = Y*So + Z = 11.928 Put price: Put = Call + PV Ex price - Underlying share price Put = 11.928 + (55/1.08^0.5) - 60 Put = 4.851 Q.4 Exercise pr (X) = 800 1 period up price = (S0 * u) 950 1 period down price = (S0 * d) 780 Risk free rate (1 + rf)^t = 1.05 1.0123 three months Share price So = 854 these > Upward multiplier 'u' = 1.112 have to be > Downwrd mult. 'd' = 0.913 worked > Cu, (up price - ex.price) 150 0 maximum of (up price - X, or 0) out > Cd, (down price - ex pr) -20 0 maximum of (down price - X, or 0) Y = (Cu - Cd) / So*(u - d) 0.882 Z = (uCd - dCu) / [(u - d)*(1+rf)] Z-top line = -137.002 Z-bottom line = 0.202 Z = -679.892 7
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EBS – How to Price Options – Binomial & Black-Scholes Models C = Y*So + Z = 73.638 Put price: Put = Call + PV Ex price - Underlying share price Put = 73.638 + (800/1.05^0.25) - 854 Put = 9.939 Q.5 Exercise pr (X) = 260 1 period up price = (S0 * u) 300 1 period down price = (S0 * d) 220 Risk free rate (1 + rf)^t = 1.06 1.04467 nine months Share price So = 250 these > Upward multiplier 'u' = 1.200 have to be > Downwrd mult. 'd' = 0.880 worked > Cu, (up price - ex.price) 40 0 maximum of (up price - X, or 0) out > Cd, (down price - ex pr) -40 0 maximum of (down price - X, or 0) Y = (Cu - Cd) / So*(u - d) 0.500 Z = (uCd - dCu) / [(u - d)*(1+rf)] Z-top line = -35.200 Z-bottom line = 0.334 Z = -105.296 C = Y*So + Z = 19.704 Put price: Put = Call + PV Ex price - Underlying share price Put = 19.704 + (260/1.06^0.75) - 250 Put = 18.586 8
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EBS – How to Price Options – Binomial & Black-Scholes Models Option Pricing using the Black-Scholes formula In the text in Section 12.2.5 there is a worked example of a Black Scholes option price calculation. The share price is £1.50, the exercise price is £1.25, the risk free rate 10%, the volatility 69% and the time one year. The text shows how to calculate the option price. The text does not show how to accurately get the N(d) values from the tables. This is shown below. In this file there are two worked examples of Black Scholes option pricing; one is an option on a share, the other the real option example from the text. Worked example 1 The share price is 120p, the exercise price is 100p, the risk free rate of interest is 3%, the volatility is 25% , what are the prices of a one year call option and put option, using the Black-Scholes option pricing method? S0 = 120 X = 100 rf = 3% σ = 25% T = 1 The call option will be priced at expiry using S0 – X. If the option is exercised before the expiry date, this would be used to give the exercise value of the option. For pricing an option with time left until expiry we can use the Black – Scholes Option Pricing (BSOP) formula. This is just a slightly more elaborate version of S0 – X. BSOP is basically a speeded up version of S0 – X, that builds in volatility and time to expiry as key variables. The formula represents a two asset portfolio; an amount of the underlying asset that you will buy (this is the S0 part of the formula), and an amount of borrowing taken out (this is the present value of the exercise price, X, element). The N(d0) figures in the formula represent the probabilities of getting a value less than d (you don’t need to know about the derivation of the formula, this is just some background info that might be helpful in understanding the formual). The Black Scholes formula is; C = S0N(d1) – Xe-rfTN(d2) Where d1 = ln(S0/X) + rfT + 0.5 σ√T σ√T d2 = d1 - σ√T d1 = ln(120/100) + 0.03*1 + 0.5 * 0.25 * √1 0.25 * √1 9
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EBS – How to Price Options – Binomial & Black-Scholes Models d1 = 0.18232 + 0.03 + 0.125 0.25 d1 = 0.84928 + 0.125 d1 = 0.97428 d2 = d1 - σ√T d2 = 0.97428 – 0.25 d2 = 0.72428 We need to find N(d1) and N(d2) now. These are the probabilities of getting a number less than d1 or d2 from th e Normal Distribution tables (Table 5 & 6 from the appendix). For d1 we can’t look up 0.97428 directly, but we can look up 0.97 and 0.98 and our value is in between the two values. 0.97 = 0.8340 0.98 = 0.8365 N(0.97428) = N(0.97) + 0.428(N(0.98) – N(0.97)) N(0.97428) = 0.8340 + 0.428(0.8365 – 0.8340) N(0.97428) = 0.8340 + 0.00107 N(0.97428) = 0.83507 For d2, we repeat the process; d2 = 0.72428 From the tables; 0.72 = 0.7642 0.73 = 0.7673 N(0.72428 = N(0.72) + 0.428(N(0.73) –N(0.72)) N(0.72428 = 0.7642 + 0.428(0.7673 – 0.7642) N(0.72428 = 0.7642 + 0.428(0.0031) N(0.72428 = 0.7642 + 0.0013268 N(0.72428 = 0.765527 Now we can price the option; 10
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EBS – How to Price Options – Binomial & Black-Scholes Models C = S0N(d1) – Xe-rfTN(d2) C = (120*0.83507) – 100e-(0.03*1)*0.765527 C = 100.2084 – 100*0.970446*0.765527 C = 100.2084 – 97.0446*0.765527 C = 100.2084 – 74.2902 C = 25.918 Having worked out the call price, you can calculate the put price using the put-call parity relationship; Put = Call + PV X – S0 Put = 25.918 + 97.0446 – 120 Put = 2.963 (Using excel will generate a more accurate answer; using excel the call is 25.912 and the put is 2.957.) Text example – real option Using the other example from the text, the WalkPhone/cId project in Section 12.2.9, the option can be priced as follows; The Black Scholes formula is; C = S0N(d1) – Xe-rfTN(d2) Where d1 = ln(S0/X) + rfT + 0.5 σ√T σ√T d2 = d1 - σ√T In this example; S0 is 4150.8 (this is the present value today of the future project cash flows’s t4 present value of 8048, discounted by 18%). X is 9075 (this is the initial investment required in the future for the project. So the project at t4 has a negative NPV: 9075 – 8048) T is 4 (the company has the exclusive right to undertake the project for four years, then the option lapses – other companies could enter this market) 11
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EBS – How to Price Options – Binomial & Black-Scholes Models rf is 10% (this is the rate at which you would discount the exercise price. The present value of the project cash flows at time t4 will be discounted by the WACC, not the rf) σ is 23.471% (this is the value for volatility. The more volatile the underlying asset the more valuable the option) The text example does not give the volatility to begin with, but approaches the problem from the point of view of trying to generate an option value of at least 272, to offset the negative - 272 NPV from the original project. The values are inputed into the B-S equation; d1 = ln(4150.8/9075) + 0.10*4 + 0.5 * 0.23471 *√4 0.23471 *√4 d1 = -0.78222 + 0.4 + 0.23471 0.46942 d1 = -0.38222 + 0.23471 0.46942 d1 = -0.814244 + 0.23471 d1 = -0.579534 d2 = d1 - σ√T d2 = -0.579534 – 0.46942 d2 = -1.048954 Now we work out N(d1) and N(d2), using the Normal Distribution tables ( Tables 5 & 6 in the text appendix). For N(d1) first; N(-0.579534) = N(-0.57) – 0.9534(N(-0.57) – N(-0.58)) N(-0.579534) = 0.2843 – 0.9534(0.2843 – 0.2810) N(-0.579534) = 0.2843 – 0.9534(0.0033) N(-0.579534) = 0.2843 – 0.003146 N(-0.579534) = 0.281154 N(d2) is worked out in the same way; N(-1.048954) = N(-1.04) – 0.8954(N(-.1.04)- N(-1.05)) N(-1.048954) = 0.1492 – 0.8954(0.1492 – 0.1469) 12
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EBS – How to Price Options – Binomial & Black-Scholes Models N(-1.048954) = 0.1492 – 0.8954(0.0023) N(-1.048954) = 0.1492 – 0.002059 N(-1.048954) = 0.147141 Now we can price the real option; C = S0N(d1) – Xe-rfTN(d2) C = (4150.8 * 0.281154)- 9075e-(0.10*4) *0.147141 C = 1167.01 – 9075*0.67032 * 0.147141 C = 1167.01 – 6083.15*0.147141 C = 1167.01 – 895.08 C = 271.93 Using excel this works out at 272.02 (The volatility measure in the text is down as 24.1%, which actually generates a slightly higher value than 272 at 289. Rounding off in the early stages of the calculation can cause the final answer to drift, so it is important to be as accurate as possible when calculating the d1 and d2 values.) So the real option is valued at 272 and when added to the original project, we get a zero overall NPV. A zero NPV still means that you are meeting the required rate of return, so on that basis the project would be attractive. If the volatility were 40% rather than 23.471%, the real option would be worth 790.74. This transforms the project into a big winner with an NPV of over 500. If the volatility were 30% the real option would be worth 467, giving an NPV of nearly 200. 13
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EBS – How to Price Options – Binomial & Black-Scholes Models Black-Scholes - Practice questions 1. The share price is 225p, the exercise price is 200p, the risk free rate of interest is 5%, the volatility is 30% , what are the prices of a one year call option and put option, using the Black-Scholes option pricing method? 2. If the share price is 330p and the exercise price is 350p, the risk free rate of interest 4% and the volatility 40%, what are the six month call and put prices using the Black-Scholes option pricing model? 3. Using the data from Q.2, but changing the volatility to 80%, holding everything else the same, what are the call and put prices using the Black-Scholes model? 4. Using the data from Q.2, but changing the time to 1 year (volatility going back to 40%), what are the call and put prices using Black-Scholes? 5. Using the data from Q.2, but changing the interest rate from 4% to 8% (time is 6 months and volatility is 40%), what are the call and put prices using Black-Scholes? 6. Compiling all the prices from Q.35 to 38 and putting them in a table, what comments would you make on the prices that you have generated? 14
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EBS – How to Price Options – Binomial & Black-Scholes Models Black-Scholes - Practice questions - Solutions 1. The share price is 225p, the exercise price is 200p, the risk free rate of interest is 5%, the volatility is 30% , what are the prices of a one year call option and put option, using the Black-Scholes option pricing method? The Black Scholes formula is; C = S0N(d1) – Xe-rfTN(d2) Where d1 = ln(S0/X) + rfT + 0.5 σ√T σ√T d2 = d1 - σ√T d1 = ln(225/200) + 0.05*1 + 0.5 * 0.30 * √1 0.30 * √1 d1 = 0.11778 + 0.05 + 0.15 0.30 d1 = 0.559277 + 0.15 d1 = 0.709277 d2 = d1 - σ√T d2 = 0.709277 – 0.30 d2 = 0.409277 We need to find N(d1) and N(d2) now. These are the probabilities of getting a number less than d1 or d2 from th e Normal Distribution tables (Table 5 & 6 from the appendix). For d1 we can’t look up 0.709277 directly, but we can look up 0.70 and 0.71 and our value is in between the two values. 0.70 = 0.7580 0.71 = 0.7611 N(0.709277) = N(0.70) + 0.9277(N(0.71) – N(0.70)) N(0.709277) = 0.7580 + 0.9277(0.7611 – 0.7580) N(0.709277) = 0.75800 + 0.002876 N(0.709277) = 0.760876 For d2, we repeat the process; 15
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EBS – How to Price Options – Binomial & Black-Scholes Models d2 = 0.409277 From the tables; 0.40 = 0.6554 0.41 = 0.6591 N(0.409277) = N(0.40) + 0.9277(N(0.41) –N(0.40)) N(0.409277) = 0.6554 + 0.9277(0.6591 – 0.6554) N(0.409277) = 0.6554 + 0.9277(0.0037) N(0.409277) = 0.6554 + 0.003422 N(0.409277) = 0.658832 Now we can price the option; C = S0N(d1) – Xe-rfTN(d2) C = (225*0.760876) – 200e-(0.053*1)*0.658832 C = 171.197 – 200*0.951229*0.658832 C = 171.197 – 190.24588*0.658832 C = 171.197 – 125.34 C = 45.8569 (Using Excel the call value works out at 45.8678) Having worked out the call price, you can calculate the put price using the put-call parity relationship; Put = Call + PV X – S0 Put = 45.8569 + 190.24588 - 225 Put = 11.103 (Using excel will generate a more accurate answer; using excel the call is 25.912 and the put is 2.957.) 16
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EBS – How to Price Options – Binomial & Black-Scholes Models 2. If the share price is 330p and the exercise price is 350p, the risk free rate of interest 4% and the volatility 40%, what are the six month call and put prices using the Black-Scholes option pricing model? Using the same method as in Q.1; d1 = 0.0041 d2 = -0.2787 N(d1) = 0.50164 N(d2) = 0.3922 PVX (this is Xe-rfT) = 343.0695 Call = 31.667 Put = 44.736 3. Using the data from Q.2, but changing the volatility to 80%, holding everything else the same, what are the call and put prices using the Black-Scholes model? Using the same method as in Q.1; d1 = 0.2142 d2 = -0.3515 N(d1) =0.58480 N(d2) = 0.36261 PVX (this is Xe-rfT) = 343.0695 Call = 68.58 Put = 81.65 4. Using the data from Q.2, but changing the time to 1 year (volatility going back to 40%), what are the call and put prices using Black-Scholes? Using the same method as in Q.1; d1 = 0.1529 d2 = -0.2471 N(d1) = 0.56076 N(d2) = 0.40241 PVX (this is Xe-rfT) = 343.0695 Call = 49.728 Put = 56.00 5. Using the data from Q.2 but changing the interest rate from 4% to 8% (time is 6 months and volatility is 40%), what are the call and put prices using Black-Scholes? 17
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EBS – How to Price Options – Binomial & Black-Scholes Models Using the same method as in Q.1; d1 = 0.0748 d2 = -0.2080 N(d1) =0.52982 N(d2) = 0.41760 PVX (this is Xe-rfT) = 343.0695 Call = 34.41 Put =40.686 6. Compiling all the prices from Q.2 to 5 and putting them in a table, what comments would you make on the prices that you have generated? See next page 18
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EBS – How to Price Options – Binomial & Black-Scholes Models Black Scholes – option pricing – Solution Q.6 A table has been created below, which shows the option price for changes in one variable at a time. The changed variable is shown in bold. The call price is in the row under the table. The line under the call price shows the variability in the option price to the change in the variable. From the table, it can be seen that volatility is the most sensitive variable, followed by time to expiry. The interest rate is seen to be quite insensitive. On the right hand side of the table, option prices for different exercise prices and share prices are shown. From this it can be seen that time value (the excess of the option value over the S0 – X exercise value, is the greatest when the share price is at the money, ie, S0 = X). When the share price is at the money, all the option price is time value, as the S0 – X is 0. The aim of the table is to help you understand how the Black-Scholes option variables affect the option price, you would not be asked to do this in an exam, if there was a Black-Scholes calculation, you would just price one call option, given the relevant variables. You could also be asked to price the put using the put-call parity relationship. Q.2 Q.3 Q.4 Q.5 Change exercise Change share original vol*2 vol*0.5 T*2 T*0.5 rf * 2 rf * 0.5 price price So 330 330 330 330 330 330 330 330 330 500 250 350 X 350 350 350 350 350 350 350 400 300 350 350 350 T 0.5 0.5 0.5 1 0.25 0.5 0.5 0.5 0.5 0.5 0.5 0.5 rf 4.0% 4% 4% 4% 4% 8% 2% 4% 4% 4% 4% 4% vol 40% 80% 20% 40% 40% 40% 40% 40% 40% 40% 40% 40% Call option price 31.67 68.58 13.15 49.73 19.48 34.41 30.34 16.81 55.92 161.88 5.44 42.53 % change in option price 116.5% -58.5% 57.0% -38.5% 8.7% -4.2% S0 - X = S0 - X = S0 - X = S0 - X = S0 - X = S0 - X = 0 30 150 0 0 Time value - Volatility is the most sensitive variable = 16.81 25.92 11.88 5.44 42.53 - Time is a sensitive variable also, but deep in at the money the time value is - the interest rate is a very insensitive variable money highest far out of at the money the money 19
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