Numerical Methods for Option Pricing


 Philip Skinner
 1 years ago
 Views:
Transcription
1 Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly and obtain a closedform solution as in (8.18) or (8.24). However, for more complicated options, it may not be easy to do the integration and one has to resort to numerical means in finding the option prices. There are three common methods for evaluating option prices numerically: the binomial method, the Monte Carlo method, and the finite difference method. We begin with the binomial method. 1 Onestep Binomial Model Consider a very simple situation where a stock price is currently $20 and it is known that at the end of three months the stock price will be either $22 or $18. We suppose that the stock pays no dividends and that we are interested in valuing a European call option with exercise price $21 expiring in three months. This option will have one of the two values at the end of the three months. If the stock price turns out to be $22, the value of the option will be $1; if the stock price turns out to be $18, the value of the option will be zero, see Figure 1. p Stock price = $22 Option price = $1 Stock price = $20 1 p 3 months Figure 1. One step binomial method. Stock price = $18 Option price = $0 Consider a portfolio consisting of a long position in shares of the stock and a short position in one call option, i.e. Π(t) = V + S, where V is the option price. If the stock price moves up from 20 to 22, the value of the shares is 22 and the value of the option is 1, so that the total value of the portfolio is If the stock price moves down from 20 to 18, the value of the shares is 18 and the value of the option
2 98 MAT4210 Notes by R. Chan is zero, so that the total value of the portfolio is 18. The portfolio is riskless if the value of is chosen so that the final value of the portfolio is the same for both of the alternative stock prices. This means that 22 1 = 18, or = So the value of the portfolio at expiry is Π(T ) = = 4.5 = Riskless portfolio must, in the absence of arbitrage opportunities, earn the riskfree rate of interest. Suppose that in this case the riskfree rate is 12% per annum. It follows that the value of the portfolio today (where t = 0) must be the present value of 4.5, or Π(0) = 4.5e = The value of the stock price today is known to be 20. If the option price is denoted by V, the value of the portfolio today is also given by Π(0) = V = 5 V. It follows that 5 V = 4.367, or V = In general, if the stock price S moves up to u S or down to d S with u > 1 and d < 1, and the corresponding payoffs for the option are V u and V d respectively, then we have Π(T ) = Su V u = Sd V d, or see Figure 2. = V u V d Su Sd, (1) p Stock price = us Option price = V u Stock price = S e rt pv u + (1 p)v d 1 p Figure 2. One step binomial method. T Stock price = ds Option price = V d Denoting the riskfree interest rate by r, the present value of the portfolio must be Π(0) = (Su V u )e rt = (Sd V d )e rt. Since the cost of setting up the portfolio at t = 0 is (S V ), where V is the option price now, it follows that S V = Π(0) = (Su V u )e rt = (Sd V d )e rt.
3 Numerical Methods for Option Pricing 99 Solving for V and using (1), we obtain V = e rt (pv u + (1 p)v d ), (2) where Notice that from (2), we have p = ert d u d. (3) pv u + (1 p)v d = V e rt which is the expected option price at time T when there is no arbitrage. Since V u is what the option will be worth when the stock price goes to us, and V p is what the option will be worth when the stock price goes to ds, the variable p here can be interpreted as the probability of an up movement in the stock price, where as the quantity 1 p is then the probability of a down movement in the stock price. More precisely, if the probability of S going up to us is assumed to be p and there are no arbitrage, then the expected option price at time T is precisely given by the fair value pv u +(1 p)v d. Moreover, in this case, the expected stock price at time T is also given by E(S T ) = psu + (1 p)sd = ps(u d) + Sd. Using (3), it simplifies to E(S T ) = Se rt, (4) which means that the stock price grows on average at the riskfree rate (and hence is the option). Setting the probability of the up movement equal to p is therefore equivalent to assuming that the return on the stock equals the riskfree rate. This is the same idea as the riskneutrality we mentioned in 8.6, cf. (4) and (8.28). This riskneutrality interpretation can be used to find the value of the option faster. In the previous example, p must satisfy the equation 22p + 18(1 p) = E(S T ) = 20e , or p = 1 4 ( 20e ) = (5) Thus, the expected payoff for the option is = Discounting back to today at the riskfree rate, the value of the option today is e = Twostep Binomial Model We can extend the analysis above one step further to a twostep binomial model. Consider a call option with a strike price $21 and expiring half year from now. Assume that the current stock price is $20, and the riskfree interest rate is 12% per annum. We break the time interval of half a year into two timesteps, each of length 3 months. In each of the two timesteps, we assume the stock price can only go up 10% or down
4 100 MAT4210 Notes by R. Chan 10%. In Figure 3, the upper number of each node shows the stock price, which we can construct easily from the given information. For example, if the stock price is $22 at the 3month time, then it can only be $24.2 or $19.8 at the 6month time. Note that because $22 d = $19.8 = $18 u, we only have three nodes at the second timestep, and not four. Next we compute the probability of p for each timestep according to (5). Using the data, u = 1.1, d = 0.9, r = 0.12, timestep=0.25, we get p = Since we have the stock prices for the nodes D, E and F at the expiry date, we can obtain the payoffs of the option at these nodes. They are given by the lower numbers at the nodes. With the payoffs of the option given at nodes D and E, we can use (2) to compute the value of the option at node B: e ( ) = Similarly, using the value of the option at nodes E and F, we can find the values of the option at the node C (which obviously is equal to 0). Now we repeat the computation once again, using the value of the options at nodes B and C, we get the option price at node A: e ( ) = A B C months 3 months Figure 3. Twostep binomial method D E F 0.0 In general, if the stock prices move up or down by a factor u or d respectively in each timestep δt, then we have the binomial tree of Figure 4. As in the above example, we find V u = e rδt (pv uu + (1 p)v ud ), V d = e rδt (pv ud + (1 p)v dd ), where δt is the length of one timestep. Then V = e rδt (pv u + (1 p)v d ), or V = e 2rδt ( p 2 V uu + 2p(1 p)v ud + (1 p) 2 V dd ).
5 Numerical Methods for Option Pricing 101 S V p 1 p Su V u Sd V d p 1 p p 1 p Su 2 V uu Sud V ud Sd 2 δt δt V dd Figure 4. Twostep binomial method in the general case. 3 The Binomial Tree The binomial models described above can be generalized to an Mstep binomial model. We first construct a binomial tree of possible asset prices, and then evaluate the option prices backward one step at a time as we did in the 2step binomial method. Assume that at time t = 0 we know the asset price S0 0 = S. Then at the next timestep δt there are two possible asset prices, S0 1 = ds0 0 and S1 1 = us0 0. Given u, d, r and δt, we can compute p as in (3): p = erδt d u d. (6) At the following time, 2δt, there are three possible asset prices, S0 2 = d 2 S0, 0 S1 2 = uds0 0 and S2 2 = u2 S0 0, and so on, see Figure 5. In general, at the mth time mδt there are (m + 1) possible asset prices, S m n = d m n u n S 0 0, n = 0,..., m. Thus, at the final time Mδt there are M + 1 possible asset prices. Example 1. If we set u = 1/d, then the possible stock prices can be simplified further to: Sn M = dm 2n S0 0 = u2n M S0 0, n = 0,..., M. Next we apply the payoff function for the option on S M n. For a put we have, V M n = max(e S M n, 0), n = 0,..., M, where E is the exercise price and Vn M denotes the nth possible value of the put at timestep M where the asset value is Sn M. For a call we have V M n = max(s M n E, 0), n = 0,..., M. We can find the expected value of the option at the timestep prior to expiry, (M 1)δt, and for possible asset price Sn M 1, n = 0,..., M 1, since we know that
6 102 MAT4210 Notes by R. Chan S0 0 V 0 0 p 1 p δt S1 1 V 1 1 S0 1 V 1 0 p 1 p p 1 p δt S2 2 V 2 2 S1 2 V 2 1 S0 2 V 2 0 Figure 5. mstep binomial method. the probability of an asset priced at Sn M 1 moving to Sn+1 M during a timestep is p, and the probability of it moving to Sn M is (1 p). Using the riskneutral argument we can calculate the value of the option at each possible asset price for timestep (M 1). Similarly, this allows us to find the value of the option at timestep (M 2), and so on, until t = 0. This gives us the value of the option at the current time. For a European option e rδt Vn m = pvn+1 m+1 m+1 + (1 p)vn, n = 0,..., m, or V m n = e rδt ( pv m+1 n+1 ) m+1 + (1 p)vn, n = 0,..., m. 4 Other Options by Binomial Methods To evaluate the option prices of other options, say American options, we use the same idea. First, similar to the European options, we construct a binomial tree of possible asset prices. Consider the situation at timestep m and at asset price Sn m+1. The option can be exercised prior to expiry to yield a profit determined by the payoff function. For a put Vn m = max(e Sm n, 0), n = 0,..., m, and for a call V m n = max(s m n E, 0), n = 0,..., m. If the option is retained, its value is, as in the European case, Vn m = e rδt ( pvn+1 m+1 ) m+1 + (1 p)vn, n = 0,..., m. The value of the American option is the maximum of two choices: the choice of exercising the option and the choice of keeping the option, i.e. for a put and for a call, n = 0,..., m. V m n = max ( max(e S m n, 0), e rδt (pv m+1 n+1 V m n = max ( max(s m n E, 0), e rδt (pv m+1 n+1 m+1 + (1 p)vn ) ) m+1 + (1 p)vn ) )
7 Numerical Methods for Option Pricing 103 Example 2. Let us compute an American put by a 2step binomial method. We will use the same data as given as in 1, i.e. S 0 = 20, E = 21, T = 0.5, δt = 0.25, and u = 1.1 and d = 0.9. Thus by (3), p = The binomial tree is given in Figure 6. The option prices at the expiry nodes D, E, and F are clear. Next consider node B. If we exercise, we get 0. If we do not exercise, then the option price is the same as the European one: e ( ) = Clearly, in this case, we do not exercise, and the option price is We illustrate this in Figure 6 by underlying the maximum of the two values: 0 and It is similar for node C. If we exercise, we get 3. If we do not exercise, then the option price is the same as the European one: e ( ) = Clearly, in this case, we exercise, and the option price is 3. We illustrate this in Figure 6 by underlying the maximum of the two values: 3 and Finally at t = 0, the option price is given by e ( ) = A {0,0.4160} 18 B C D E months {3,2.4444} months 16.2 F 4.8 Figure 6. Twostep binomial method for an American option. What if the underlying stock pays dividend? We can use the same idea to compute the option price. Just remember that the stock price will drop across the exdividend date t d, but the option price will be continuous across t d, see Proposition 4.7. Example 3. Let us compute the same American put as in Example 2 by a 2step binomial method, but we assume that the stock pays dividend at the end of the first 3 months and the dividend yield is 5%. Since all other parameters are unchanged, we still have p = The binomial tree is given in Figure 7. Since the stock pays dividend at the end of 3 months, we break this particular exdividend date into two parts, one is t d, with
8 104 MAT4210 Notes by R. Chan nodes B and C, and the other part is t + d, with nodes D and E. The stock price at nodes D and E are obtained by reducing the stock prices at nodes B and C by 5%, e.g. at Node D, the stock price is 20.9 = After the exdividend date, the stock prices will move according to a normal binomial tree, i.e. it goes up or down according to u or d. With this, we construct the stock prices at nodes F, G and H. Once we have the stock prices at the terminal nodes, then the remaining computation is pretty much the same as in previous examples. We first compute the option prices at the expiry nodes. Next consider node D. If we exercise, we get 0.1. If we do not exercise, then the option price is the same as the European one: e ( ) = Clearly, in this case, we do exercise, and the option price is It is similar for node E where we would exercise to get 3.9. At t d, the option price will be the same as t+ d. Of course we can do the same thing asking if we should hold or exercise the option at t d. For American put, we can easily see that we will not exercise at t d, for if we do, we get higher return if we exercise at t + d, see nodes C and E. Once we get the option prices at nodes at t d (in our case, they are nodes B and C), we can continue to backcompute the option price until we get back to t = 0. In this example, at t = 0, the option price is given by e ( ) = A B D {0,0.7390} {0.1,0.7390} C E {3,3.9} {3.9,3.2793} months t d t + d 3 months Figure 7. Twostep binomial method for an American option F G H 5.61 As expected, the put option price is higher at t = 0 than the one where the stock does not pay dividend (see Example 2 where V = ). The price is higher at t = 0 in anticipation that the stock will pay dividend before the expiration time T which will drive down the stock price. 5 Determination of the Parameters In 1 4, we assume we know u and d, and then we calculate p using (6) and then V accordingly. In general, the parameters p, u and d are unknown, and have to be
9 Numerical Methods for Option Pricing 105 chosen so that they give the correct values for the mean and variance of stock price changes during a time interval of length δt. Under the riskneutral assumption, the expected return from a stock is the riskfree interest rate, r. Hence at the end of the time interval δt, the expected value of the stock E(S δt ) is Se rδt, see (4) or Corollary 8.5. It follows that Se rδt = E(S δt ) = psu + (1 p)sd, (7) or It is nothing but just (3). By Corollary 8.5 again, Since Var(S δt ) = E(S 2 δt ) (E(S δt)) 2, and it follows that or e rδt = pu + (1 p)d. (8) Var(S δt ) = S 2 e 2rδt (e σ2 δt 1). E(S 2 δt) = p(us) 2 + (1 p)(ds) 2, S 2 e 2rδt (e σ2 δt 1) = pu 2 S 2 + (1 p)d 2 S 2 S 2 (pu + (1 p)d) 2, e 2rδt+σ2 δt = pu 2 + (1 p)d 2. (9) Equations (8) and (9) impose two conditions on p, u and d. We still need one more equation to determine them. One can equate the third moment of S δt, but it will lead to extremely difficult expressions. A popular choice for the third condition is Solving (8), (9) and (10), we obtain u = 1 d. (10) where d = A A 2 1, u = A + A 2 1, p = erδt d u d, (11) A = 1 2 ( e rδt + e (r+σ2 )δt ). (12) Another popular choice is to set p = 1/2. (13) This together with (8) and (9) leads to ( ) ( ) d = e rδt 1 e σ2 δt 1, u = e rδt 1 + e σ2 δt 1. Assumptions (10) and (13) may give different pricing of the options, but one can show that if δt 0, then the two pricings should be the same.
10 106 MAT4210 Notes by R. Chan M = (T t)/δt T t P DE Table 1. Binomial Method (u = 1/d) for European put with E = 10, S = 5, r = 0.06, σ = 0.3. Example 4. Consider a European put with E = 10, S = 5, r = 0.06 and σ = 0.3. Table 1 shows the comparison of the binomial method with u = 1/d and the PDE solution of the BlackScholes equation for various expiry time with different timesteps. We see that the results are more accurate if δt is smaller, i.e. M is getting larger. We should note that this is the same with solving PDE using numerical methods. Example 5. Consider an American put with E = 10, S = 9, r = 0.06 and σ = 0.3. Table 2 shows the comparison of the binomial method with u = 1/d and the PDE solution to the BlackScholes equation for various expiry time with different timesteps. Again we see that the results are more accurate if δt is smaller. M = (T t)/δt T t P DE Table 2. Binomial Method (u = 1/d) for American put with E = 10, S = 9, r = 0.06, σ = 0.3. The binomial method is extremely memory efficient for options with only one underlying asset: although there are O(M 2 ) nodes, the memory required grows only linearly with the number of timesteps, i.e. it is of order O(M) only. The reason is that we can reuse the memory in the (m + 1)st timestep for the mth timestep. Thus the largest memory requirement is at the last timestep M, where we require 2(M +1) memory for the stock prices and the option prices. Since there are O(M 2 ) nodes, and each node requires O(1) calculations, the execution time grows quadratically with the number of timesteps, i.e. of order O(M 2 ). For an option whose price depends on d stock prices, the binomial method will require O(M d ) memory locations and O(M d+1 ) calculations. Thus if d is large, it is not an efficient method or one needs a very fast computer to compute such options. It is interesting to note that about 10% of the world top 500 fastest computers are installed in financial institutions across the world to compute option prices. 6 Monte Carlo Method Monte Carlo method is another name for simulation method. Here we try to simulate the stock price at the expiry date T according to the lognormal process (7.7). If we know the stock price S(T ) at T, then we know the option price c(s, T ) at T. By discounting that back to the current time t, we know the option price c(s, t) at t.
11 Numerical Methods for Option Pricing 107 Clearly, if we only simulate one price at T, the option price will not be accurate at all. We need to simulate many many stock prices at T to get enough samples of c(s, T ) at T. Then we take the average of these samples to get a good estimate of the expected value of c(s, T ). By discounting back to t, then we get the expected value of c(s, t). By (7.7) and the fact that X T t N (0, T t), we have S T = S t e (µ σ2 /2)(T t)+σε T t, where S t is the current asset price and ε is a random variable drawn from a standardized normal distribution N (0, 1). However, in a riskneutral world (or when we are only interested in computing the option prices), µ is taken as the interest rate r, i.e., S T = S t e (r σ2 /2)(T t)+σε T t, (14) see (8.40). Using this the stock prices can be simulated between t and T. In general, we fix a timestep δt = (T t)/m and the number of paths N, which usually goes from 10,000 to 1,000,000, depending on the accuracy we want. Then we generate the jth path from t to T : S j i = Sj i 1 e(r σ2 /2)δt+σε j i δt, i = 1,..., M, j = 1,..., N, (15) where ε j i are random numbers distributed as N (0, 1). In Figure 8, we show ten simulated stock price paths using (15), with S t = 100, t = 0, T = 50, and δt = 1. Note that for each j = 1,..., N, S j M is one sample of the stock price at time T, from which one can calculate the payoff of the option Figure 8. Ten simulated stock prices. Note that if we are only interested in evaluating European options, there is no need to know what the stock prices are in between t and T ; we only need to know the stock price at T. We can modify (15) to get it directly: S j T = Sj t e (r σ2 /2)(T t)+σε j T t, j = 1,..., N,
12 108 MAT4210 Notes by R. Chan i.e. for each ε j sampled, we can generate one sample of S j T. From each of them, a payoff for the option can be calculated at expiry. For example, if it is a European call option, we can compute V j = max(s j T E, 0), j = 1,..., N. Then the expected value of the payoff can be estimated as the arithmetic average of these payoffs, i.e. E(V T ) = 1 N N V j. j=1 The current value of the option can be calculated by discounting the expected payoff value, i.e., V = e r(t t) E(V T ). Example 6. Consider a European put with E = 10, S 0 = 5, r = 0.1 and σ = 0.4. For computing European options, one can use (14) to generate S T directly from S 0, and there is no need to partition the time interval [0, T ]. Table 3 shows the comparison of the Monte Carlo method and the PDE solution of the BlackScholes equation for different expiry time and numbers of paths. We see that the results are more accurate when we increase the number of paths N. T t P DE N = 10 N = 10 2 N = 10 3 N = 10 4 N = Table 3. Monte Carlo method for European put with E = 10, S 0 = 5, r = 0.1 and σ = 0.4. The number of simulation trials carried out depends on the accuracy required. If K independent trials are carried out as described above, it is usual to calculate the standard deviation as well as the mean of the option prices given by the simulation trials. Denote their mean by µ and the standard deviation by σ. The Central Limit Theorem states that the option price V should be distributed as N ( µ, σ2 K ). Hence a 95% confidence interval for V is given by µ 1.96 σ K < V < µ σ K, or { Prob µ V < 1.96 σ } = 0.95, K see (6.7). This shows that our uncertainty about the value of the option is inversely proportional to the square root of the number of trials K. That would mean that to double the accuracy of the simulation, we must quadruple the number of trials; and to increase the accuracy by a factor of 10, the number of trials must be increased by a factor of 100. Example 7. Consider the European put in Example 6. Now we study how the number of trials K affects the accuracy of simulation. Here we focus on the case with N = 1000
13 Numerical Methods for Option Pricing 109 and T t = First we perform K = 10 independent trials. The mean and the standard deviation of the 10 discounted payoffs are given by µ = and σ = , with a 95% confidence interval for the option price V : < V < Note that the simulation s estimate of the option value is accurate to 2 decimal places, compared to the PDE solution V = in Table 3. To increase the accuracy by a factor of 10, i.e. to increase the significant digit by 1, the number of trials must be increased by a factor of 100, i.e., K = Thus, we then carry out K = 1000 independent trials and obtain the mean and the standard deviation of the discounted payoffs: µ = and σ = , with a 95% confidence interval for the option price V : < V < Monte Carlo simulation tends to be numerically more efficient than other procedures when there are three or more underlying assets that the option price is dependent on. This is because the time taken to carry out a Monte Carlo simulation increases approximately linearly with the number of unknown variables, i.e. it is of order O(MN). In contrast, the time taken for most other procedures increases exponentially with the number of unknown variables. For example, the cost of the binomial method and the finite difference method both increase like O(M d+1 ) where d is number of underlying assets. Monte Carlo simulation also has the advantage that it provides a standard error for the estimates that are made. It is an approach that can accommodate complex payoffs and complex stochastic processes. It can be used when the payoff depends on some function of the whole path followed by a variable, not just its terminal value. For example, we can use it to compute barrier options easily. The limitations of the Monte Carlo simulation are that it converges slowly, and it is used normally for European options, for it is very difficult to use for American options. In fact, in computing American options, one may have to generate and remember all the intermediate stock prices in all the paths, i.e. S j i for 1 i M and 1 j N in (15). For these prices are to be used later to compute the price of exercising the options prematurely and the price to hold onto the options till the next timestep. The price to hold onto the options till the next timestep is called the continuation price, which is quite difficult to estimate accurately from the sampled prices. 7 Finite Difference Approximation Besides the BlackScholes equation, in practice, there are many other options that can be modeled by partial differential equations too. Unfortunately not many of them have closedform solutions as in (8.18) or (8.24). In this section, we discuss how to solve partial differential equations by finite difference approach. The main idea is to approximate the differential operators in the differential equation by difference operators.
14 110 MAT4210 Notes by R. Chan Given a function u(x), using Taylor s expansion, we have u(x + δx) = u(x) + δx u (x) + δx2 2 u (x) + δx3 6 u (x) + O(δx 4 ), (16) u(x δx) = u(x) δx u (x) + δx2 2 u (x) δx3 6 u (x) + O(δx 4 ). (17) If δx is small, we can approximate the derivative u (x) at x by the backward difference : du(x) u(x) u(x δx) = + O(δx), (18) dx δx or more accurately, by the first order central difference : du(x) dx u(x + δx) u(x δx) = + O(δx 2 ). (19) 2δx From (16) and (17), we can also approximate the second order derivative u (x) at x by the second order central difference : d 2 u(x) dx 2 = u(x + δx) 2u(x) + u(x δx) (δx) 2 + O(δx 2 ). (20) Suppose we are to solve the BlackScholes equation for a European option V (S, t), V (S, t) t σ2 S 2 2 V (S, t) V (S, t) S 2 + rs rv (S, t) = 0, (21) S with given final condition V (S, T ) (for 0 < S < ) and boundary conditions V (0, t) and V (, t) (for 0 < t < T ). Since it is impossible to work with S = numerically, we first replace the boundary S = by S = S max where S max is a sufficiently large stock price. Then we replace the condition V (, t) by V (S max, t), see Figure 9. Next we partition the solution domain [0, S max ] [0, T ] by grid lines. More precisely, we partition [0, S max ] into N equal subintervals, each of length δs; and [0, T ] into M equal subintervals, each of length δt, see Figure 9. Let S j = jδs and t i = iδt for 0 j N and 0 i M. Note that for 0 i M and 0 j N, V (S 0, t i ), V (S N, t i ) and V (S j, t M ) are given boundary and final conditions. They are known numbers. Our problem is to find V (S j, t i ) for 0 < j < N and 0 i < M, which are the option prices inside the solution domain. Our idea of finding them is to replace the derivatives in (21) by the finite differences in (18) (20). This reduces the partial differential equation to a difference equation. We then solve the difference equation in a timemarching manner, marching back from time T to 0, at one δt a time. Let us illustrate that by computing V (S j, t M 1 ) for 1 j N. Let us consider (21) at the point (S j, t M ) and apply a backward difference (18) at t M and central differences (19) and (20) at S j. Then we have V (S j, t M ) V (S j, t M 1 ) + 1 δt 2 σ2 Sj 2 V (S j+1, t M ) V (S j 1, t M ) + rs j 2δS V (S j+1, t M ) 2V (S j, t M ) + V (S j 1, t M ) δs 2 rv (S j, t M ) = 0, 0 < j < N. (22) Notice that in (22), all the quantities are known from the boundary and final conditions except V (S j, t M 1 ). Hence we can compute V (S j, t M 1 ) for all 0 < j < N.
15 Numerical Methods for Option Pricing 111 Figure 9. Solution domain of European options and the grids. Once we have obtained all the option prices at the time step t M 1, we can march one δt backward in time to obtain V (S j, t M 2 ) for all 0 < j < N. For general 0 < i M, applying differencing on (21) at (S j, t i ) will lead to V (S j, t i ) V (S j, t i 1 ) + 1 δt 2 σ2 Sj 2 V (S j+1, t i ) V (S j 1, t i ) + rs j 2δS V (S j+1, t i ) 2V (S j, t i ) + V (S j 1, t i ) δs 2 rv (S j, t i ) = 0, 0 < j < N. (23) With V (S j, t i ) known for all S j, we can compute V (S j, t i 1 ) by (23). By repeating the process M times, we can obtain the option prices V (S j, t 0 ) = V (S j, 0) at the current time for all S j. Obviously, the accuracy of the solution depends on the size of δt and δs. From (18) (20), we see that the error in approximating the partial differential equation (21) by the finite difference equation (23) is O(δt) and O(δS 2 ). In order to balance the two errors, one would choose δt = O(δS 2 ). That would imply that δt will be a very small number and hence M will be very big (of O(N 2 )). That would mean that we will have to do many timemarching steps. Unfortunately, there is no way to overcome this. In fact, one can show that α δt/(δs) 2 has to be less than a certain constant α 0 in order that V (S j, t i ) of (23) converges to the true solution of (21) when δt and δs go to 0. If α α 0, then V (S j, t i ) actually diverges. For some problems, the condition α < α 0 will restrict δt to a very small number, and hence the method may be too slow to find the solution. There are better finite difference schemes that can circumvent this problem but they will require more computational costs. Example 8. Consider a European put option with the exercise price E = 10 and T = 1/2 year. Assume r = 0.05 and σ = 0.2. Using the method, we can obtain the
16 112 MAT4210 Notes by R. Chan approximate option prices for different asset values S 0, which are shown in Table 7. Here we choose S max = 25, and the number of intervals for [0, T ] is M = 60, i.e., every three days. The stepsize δs is determined by α. The exact solution is obtained by the closedform formula (8.24). We see that the method diverges when α > 0.1, and the solution becomes oscillatory, see Figure 10. S 0 α = 0.02 α = 0.1 α = 0.13 Exact Table 4. E = 10, r = 0.05, σ = 0.2, M = 60, 6 months to expiry 10 5 Option Price S 0 Figure 10. Option price is oscillatory when α > α 0. The finite difference method is extremely efficient for options with only one underlying asset: its computational cost is of O(NM). However, it also has the curse of dimension in that the cost is O(MN d ) for an option whose price depends on d stocks. The timemarching scheme (23) for computing European option prices can be adapted easily to compute the American option prices. The main idea is that once V (S j, t i ) is computed by (23), we have to compare it with the exercise price at (S j, t i )
17 Numerical Methods for Option Pricing 113 to determine the true value of the option at (S j, t i ). For example, if we are computing the call option price, and c h (S j, t i ) is the price computed by (23), then c(s j, t i ) = max{c h (S j, t i ), S j E}. Example 9. Consider the American put option with parameters the same as in Example 8. Using the above method, we compute the values of the put option for asset prices from 0 to 16. The results are shown in Table 5. S 0 American (α = 0.1) European (α = 0.1) Table 5. E = 10, r = 0.05, σ = 0.2, M = 60, 6 months to expiry
18 114 MAT4210 Notes by R. Chan
Lecture 11: RiskNeutral Valuation Steven Skiena. skiena
Lecture 11: RiskNeutral Valuation Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena RiskNeutral Probabilities We can
More informationBinomial lattice model for stock prices
Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }
More informationLecture 11. Sergei Fedotov. 20912  Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 7
Lecture 11 Sergei Fedotov 20912  Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 7 Lecture 11 1 American Put Option Pricing on Binomial Tree 2 Replicating
More informationBinomial trees and risk neutral valuation
Binomial trees and risk neutral valuation Moty Katzman September 19, 2014 Derivatives in a simple world A derivative is an asset whose value depends on the value of another asset. Call/Put European/American
More informationLecture 10. Sergei Fedotov. 20912  Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 7
Lecture 10 Sergei Fedotov 20912  Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 7 Lecture 10 1 Binomial Model for Stock Price 2 Option Pricing on Binomial
More informationLecture 12: The BlackScholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The BlackScholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The BlackScholesMerton Model
More informationOptions and Derivative Pricing. U. NaikNimbalkar, Department of Statistics, Savitribai Phule Pune University.
Options and Derivative Pricing U. NaikNimbalkar, Department of Statistics, Savitribai Phule Pune University. email: uvnaik@gmail.com The slides are based on the following: References 1. J. Hull. Options,
More informationLecture 6: Option Pricing Using a Onestep Binomial Tree. Friday, September 14, 12
Lecture 6: Option Pricing Using a Onestep Binomial Tree An oversimplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More information10 Binomial Trees. 10.1 Onestep model. 1. Model structure. ECG590I Asset Pricing. Lecture 10: Binomial Trees 1
ECG590I Asset Pricing. Lecture 10: Binomial Trees 1 10 Binomial Trees 10.1 Onestep model 1. Model structure ECG590I Asset Pricing. Lecture 10: Binomial Trees 2 There is only one time interval (t 0, t
More informationIntroduction to Binomial Trees
11 C H A P T E R Introduction to Binomial Trees A useful and very popular technique for pricing an option involves constructing a binomial tree. This is a diagram that represents di erent possible paths
More informationJorge Cruz Lopez  Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12.
Week 9 Binomial Trees : Hull, Ch. 12. 1 Binomial Trees Objective: To explain how the binomial model can be used to price options. 2 Binomial Trees 1. Introduction. 2. One Step Binomial Model. 3. Risk Neutral
More informationThe BlackScholes pricing formulas
The BlackScholes pricing formulas Moty Katzman September 19, 2014 The BlackScholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
More information7: The CRR Market Model
Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The CoxRossRubinstein
More informationOne Period Binomial Model
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing
More informationExample 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r).
Chapter 4 PutCall Parity 1 Bull and Bear Financial analysts use words such as bull and bear to describe the trend in stock markets. Generally speaking, a bull market is characterized by rising prices.
More information1. Assume that a (European) call option exists on this stock having on exercise price of $155.
MØA 155 PROBLEM SET: Binomial Option Pricing Exercise 1. Call option [4] A stock s current price is $16, and there are two possible prices that may occur next period: $15 or $175. The interest rate on
More informationCS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options
CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common
More informationEC3070 FINANCIAL DERIVATIVES
BINOMIAL OPTION PRICING MODEL A OneStep Binomial Model The Binomial Option Pricing Model is a simple device that is used for determining the price c τ 0 that should be attributed initially to a call option
More informationLecture 9. Sergei Fedotov. 20912  Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 8
Lecture 9 Sergei Fedotov 20912  Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 8 Lecture 9 1 RiskNeutral Valuation 2 RiskNeutral World 3 TwoSteps Binomial
More informationEuropean Options Pricing Using Monte Carlo Simulation
European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical
More informationFinancial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan
Financial Modeling An introduction to financial modelling and financial options Conall O Sullivan Banking and Finance UCD Smurfit School of Business 31 May / UCD Maths Summer School Outline Introduction
More informationChapter 13 The BlackScholesMerton Model
Chapter 13 The BlackScholesMerton Model March 3, 009 13.1. The BlackScholes option pricing model assumes that the probability distribution of the stock price in one year(or at any other future time)
More information1.1 Some General Relations (for the no dividend case)
1 American Options Most traded stock options and futures options are of Americantype while most index options are of Europeantype. The central issue is when to exercise? From the holder point of view,
More information1 Geometric Brownian motion
Copyright c 006 by Karl Sigman Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM
More informationChapter 2: Binomial Methods and the BlackScholes Formula
Chapter 2: Binomial Methods and the BlackScholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a calloption C t = C(t), where the
More informationInstitutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)
Copyright 2003 Pearson Education, Inc. Slide 081 Institutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared
More informationThe Binomial Option Pricing Model André Farber
1 Solvay Business School Université Libre de Bruxelles The Binomial Option Pricing Model André Farber January 2002 Consider a nondividend paying stock whose price is initially S 0. Divide time into small
More informationChapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.
Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of riskadjusted discount rate. Part D Introduction to derivatives. Forwards
More informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationLectures. Sergei Fedotov. 20912  Introduction to Financial Mathematics. No tutorials in the first week
Lectures Sergei Fedotov 20912  Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 1 1 Introduction Elementary economics
More informationPricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation
Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation Yoon W. Kwon CIMS 1, Math. Finance Suzanne A. Lewis CIMS, Math. Finance May 9, 000 1 Courant Institue of Mathematical Science,
More informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
More informationLecture 11: The Greeks and Risk Management
Lecture 11: The Greeks and Risk Management This lecture studies market risk management from the perspective of an options trader. First, we show how to describe the risk characteristics of derivatives.
More informationNumerical methods for American options
Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment
More informationExam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D
Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS
More informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of inthemoney options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationDoes BlackScholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does BlackScholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial
More informationLecture 21 Options Pricing
Lecture 21 Options Pricing Readings BM, chapter 20 Reader, Lecture 21 M. Spiegel and R. Stanton, 2000 1 Outline Last lecture: Examples of options Derivatives and risk (mis)management Replication and Putcall
More information第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model
1 第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model Outline 有 关 股 价 的 假 设 The BS Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American
More information1 The BlackScholes model: extensions and hedging
1 The BlackScholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes
More informationCall Price as a Function of the Stock Price
Call Price as a Function of the Stock Price Intuitively, the call price should be an increasing function of the stock price. This relationship allows one to develop a theory of option pricing, derived
More informationA SNOWBALL CURRENCY OPTION
J. KSIAM Vol.15, No.1, 31 41, 011 A SNOWBALL CURRENCY OPTION GYOOCHEOL SHIM 1 1 GRADUATE DEPARTMENT OF FINANCIAL ENGINEERING, AJOU UNIVERSITY, SOUTH KOREA Email address: gshim@ajou.ac.kr ABSTRACT. I introduce
More informationDERIVATIVE SECURITIES Lecture 2: Binomial Option Pricing and Call Options
DERIVATIVE SECURITIES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis review of pricing formulas assets versus futures practical issues call options
More informationPricing Options with Discrete Dividends by High Order Finite Differences and Grid Stretching
Pricing Options with Discrete Dividends by High Order Finite Differences and Grid Stretching Kees Oosterlee Numerical analysis group, Delft University of Technology Joint work with Coen Leentvaar, Ariel
More informationOPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options
OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis binomial model replicating portfolio single period artificial (riskneutral)
More informationBINOMIAL OPTION PRICING
Darden Graduate School of Business Administration University of Virginia BINOMIAL OPTION PRICING Binomial option pricing is a simple but powerful technique that can be used to solve many complex optionpricing
More informationMartingale Pricing Applied to Options, Forwards and Futures
IEOR E4706: Financial Engineering: DiscreteTime Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the
More informationLecture 3.1: Option Pricing Models: The Binomial Model
Important Concepts Lecture 3.1: Option Pricing Models: The Binomial Model The concept of an option pricing model The one and two period binomial option pricing models Explanation of the establishment and
More informationOptions 1 OPTIONS. Introduction
Options 1 OPTIONS Introduction A derivative is a financial instrument whose value is derived from the value of some underlying asset. A call option gives one the right to buy an asset at the exercise or
More informationValuing Stock Options: The BlackScholesMerton Model. Chapter 13
Valuing Stock Options: The BlackScholesMerton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The BlackScholesMerton Random Walk Assumption
More informationOn BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationTwoState Option Pricing
Rendleman and Bartter [1] present a simple twostate model of option pricing. The states of the world evolve like the branches of a tree. Given the current state, there are two possible states next period.
More informationFinite Differences Schemes for Pricing of European and American Options
Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the BlackScholes
More informationLecture 5: Put  Call Parity
Lecture 5: Put  Call Parity Reading: J.C.Hull, Chapter 9 Reminder: basic assumptions 1. There are no arbitrage opportunities, i.e. no party can get a riskless profit. 2. Borrowing and lending are possible
More informationOther variables as arguments besides S. Want those other variables to be observables.
Valuation of options before expiration Need to distinguish between American and European options. Consider European options with time t until expiration. Value now of receiving c T at expiration? (Value
More informationOption pricing. Vinod Kothari
Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate
More informationACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs)
Problem ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 0,, 2, 8. October 2, 200 (Thurs) (i) The current exchange rate is 0.0$/. (ii) A fouryear dollardenominated European put option
More informationMore Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options
More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path
More informationTABLE OF CONTENTS. A. PutCall Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13
TABLE OF CONTENTS 1. McDonald 9: "Parity and Other Option Relationships" A. PutCall Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:
More informationPart V: Option Pricing Basics
erivatives & Risk Management First Week: Part A: Option Fundamentals payoffs market microstructure Next 2 Weeks: Part B: Option Pricing fundamentals: intrinsic vs. time value, putcall parity introduction
More informationLECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS
LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS Robert S. Pindyck Massachusetts Institute of Technology Cambridge, MA 02142 Robert Pindyck (MIT) LECTURES ON REAL OPTIONS PART II August, 2008 1 / 50
More informationBlackScholes Option Pricing Model
BlackScholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,
More informationLecture 6 BlackScholes PDE
Lecture 6 BlackScholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the riskneutral measure Q by If the contingent
More informationFinancial Modeling. Class #06B. Financial Modeling MSS 2012 1
Financial Modeling Class #06B Financial Modeling MSS 2012 1 Class Overview Equity options We will cover three methods of determining an option s price 1. BlackScholesMerton formula 2. Binomial trees
More informationBlackScholes Equation for Option Pricing
BlackScholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there
More informationAdditional questions for chapter 4
Additional questions for chapter 4 1. A stock price is currently $ 1. Over the next two sixmonth periods it is expected to go up by 1% or go down by 1%. The riskfree interest rate is 8% per annum with
More informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
More informationFinancial Options: Pricing and Hedging
Financial Options: Pricing and Hedging Diagrams Debt Equity Value of Firm s Assets T Value of Firm s Assets T Valuation of distressed debt and equitylinked securities requires an understanding of financial
More informationA Comparison of Option Pricing Models
A Comparison of Option Pricing Models Ekrem Kilic 11.01.2005 Abstract Modeling a nonlinear pay o generating instrument is a challenging work. The models that are commonly used for pricing derivative might
More informationS 1 S 2. Options and Other Derivatives
Options and Other Derivatives The OnePeriod Model The previous chapter introduced the following two methods: Replicate the option payoffs with known securities, and calculate the price of the replicating
More informationwhere N is the standard normal distribution function,
The BlackScholesMerton formula (Hull 13.5 13.8) Assume S t is a geometric Brownian motion w/drift. Want market value at t = 0 of call option. European call option with expiration at time T. Payout at
More informationEXP 481  Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0
EXP 481  Capital Markets Option Pricing imple arbitrage relations Payoffs to call options Blackcholes model PutCall Parity Implied Volatility Options: Definitions A call option gives the buyer the
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More informationPricing Options: Pricing Options: The Binomial Way FINC 456. The important slide. Pricing options really boils down to three key concepts
Pricing Options: The Binomial Way FINC 456 Pricing Options: The important slide Pricing options really boils down to three key concepts Two portfolios that have the same payoff cost the same. Why? A perfectly
More informationOptions Pricing. This is sometimes referred to as the intrinsic value of the option.
Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the PutCall Parity Relationship. I. Preliminary Material Recall the payoff
More informationChapter 7: Option pricing foundations Exercises  solutions
Chapter 7: Option pricing foundations Exercises  solutions 1. (a) We use the putcall parity: Share + Put = Call + PV(X) or Share + Put  Call = 97.70 + 4.16 23.20 = 78.66 and P V (X) = 80 e 0.0315 =
More informationBINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract
BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple
More informationPractice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set?
Derivatives (3 credits) Professor Michel Robe Practice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set? To help students with the material, eight practice sets with solutions
More informationOverview. Option Basics. Options and Derivatives. Professor Lasse H. Pedersen. Option basics and option strategies
Options and Derivatives Professor Lasse H. Pedersen Prof. Lasse H. Pedersen 1 Overview Option basics and option strategies Noarbitrage bounds on option prices Binomial option pricing BlackScholesMerton
More informationBUS 316 NOTES AND ANSWERS BINOMIAL OPTION PRICING
BUS 316 NOTES AND ANSWERS BINOMIAL OPTION PRICING 3. Suppose there are only two possible future states of the world. In state 1 the stock price rises by 50%. In state 2, the stock price drops by 25%. The
More informationLogNormal stockprice models in Exams MFE/3 and C/4
Making sense of... LogNormal stockprice models in Exams MFE/3 and C/4 James W. Daniel Austin Actuarial Seminars http://www.actuarialseminars.com June 26, 2008 c Copyright 2007 by James W. Daniel; reproduction
More informationOn MarketMaking and DeltaHedging
On MarketMaking and DeltaHedging 1 Market Makers 2 MarketMaking and BondPricing On MarketMaking and DeltaHedging 1 Market Makers 2 MarketMaking and BondPricing What to market makers do? Provide
More informationPricing Discrete Barrier Options
Pricing Discrete Barrier Options Barrier options whose barrier is monitored only at discrete times are called discrete barrier options. They are more common than the continuously monitored versions. The
More informationOption Basics. c 2012 Prof. YuhDauh Lyuu, National Taiwan University Page 153
Option Basics c 2012 Prof. YuhDauh Lyuu, National Taiwan University Page 153 The shift toward options as the center of gravity of finance [... ] Merton H. Miller (1923 2000) c 2012 Prof. YuhDauh Lyuu,
More informationLecture 3: Put Options and DistributionFree Results
OPTIONS and FUTURES Lecture 3: Put Options and DistributionFree Results Philip H. Dybvig Washington University in Saint Louis put options binomial valuation what are distributionfree results? option
More informationPathdependent options
Chapter 5 Pathdependent options The contracts we have seen so far are the most basic and important derivative products. In this chapter, we shall discuss some complex contracts, including barrier options,
More informationAmerican and European. Put Option
American and European Put Option Analytical Finance I Kinda Sumlaji 1 Table of Contents: 1. Introduction... 3 2. Option Style... 4 3. Put Option 4 3.1 Definition 4 3.2 Payoff at Maturity... 4 3.3 Example
More informationChapter 1: The binomial asset pricing model
Chapter 1: The binomial asset pricing model Simone Calogero April 17, 2015 Contents 1 The binomial model 1 2 1+1 dimensional stock markets 4 3 Arbitrage portfolio 8 4 Implementation of the binomial model
More informationResearch on Option Trading Strategies
Research on Option Trading Strategies An Interactive Qualifying Project Report: Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE In partial fulfillment of the requirements for the Degree
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the
More informationFinance 400 A. Penati  G. Pennacchi. Option Pricing
Finance 400 A. Penati  G. Pennacchi Option Pricing Earlier we derived general pricing relationships for contingent claims in terms of an equilibrium stochastic discount factor or in terms of elementary
More informationGenerating Random Numbers Variance Reduction QuasiMonte Carlo. Simulation Methods. Leonid Kogan. MIT, Sloan. 15.450, Fall 2010
Simulation Methods Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Simulation Methods 15.450, Fall 2010 1 / 35 Outline 1 Generating Random Numbers 2 Variance Reduction 3 QuasiMonte
More informationTwoState Model of Option Pricing
Rendleman and Bartter [1] put forward a simple twostate model of option pricing. As in the BlackScholes model, to buy the stock and to sell the call in the hedge ratio obtains a riskfree portfolio.
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationHedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging
Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in
More informationMATH3075/3975 Financial Mathematics
MATH3075/3975 Financial Mathematics Week 11: Solutions Exercise 1 We consider the BlackScholes model M = B, S with the initial stock price S 0 = 9, the continuously compounded interest rate r = 0.01 per
More information金融隨機計算 : 第一章. BlackScholesMerton Theory of Derivative Pricing and Hedging. CH Han Dept of Quantitative Finance, Natl. TsingHua Univ.
金融隨機計算 : 第一章 BlackScholesMerton Theory of Derivative Pricing and Hedging CH Han Dept of Quantitative Finance, Natl. TsingHua Univ. Derivative Contracts Derivatives, also called contingent claims, are
More informationLecture. S t = S t δ[s t ].
Lecture In real life the vast majority of all traded options are written on stocks having at least one dividend left before the date of expiration of the option. Thus the study of dividends is important
More information