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  1. 1. Pricing Options through the Trinomial Tree Ciaran Cox Mathematical Sciences, School of Information Systems, Computing and Mathematics, Brunel University Supervisor: Jacques-´Elie Furter 2014 December 15, 2014
  2. 2. Abstract I begin by a basic definition of option contracts and the pricing of these options through the Black-Scholes model, which is based upon the Geometric Brownian Motion (GBM). Using this model, one can solve for the implied volatility on an option through Newton’s iteration for find- ing a root of a function. Binomial and trinomial lattice methods are alternate ways of pricing options, but still assume that the stock price follows the GBM and constant volatility throughout the option. European, American, Barrier and double barrier knockout options are priced using the trinomial tree. Delta, Theta and Gamma Greeks are calculated through the Black-Scholes model, with a comparison of prices on the Delta and Gamma through the trinomial tree. Using these greeks, I move on to the delta-hedge rule and a delta tolerance applied practically to mar- ket data from the Bloomberg terminals, with comparison of different strike prices on different companies, concluding with a brief overview in to transaction costs.
  3. 3. Acknowledgements A special thanks to my supervisor Jacques-´Elie Furter for his increased support throughout my project. Also like to thank my mum, dad and my sister for inspirational motivation throughout my university studies, along with all my friends and a special thanks to Joel Johnson for excellent support during the project.
  4. 4. Contents 1 Introduction 6 1.1 Geometric Brownian Motion (GBM) . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Implied Volatility by the Black-Scholes Equation . . . . . . . . . . . . 10 2 Lattice models 11 2.1 Formulation of the Binomial Option Pricing Model by replicating portfolios . . 14 2.2 Binomial model to the Trinomial model . . . . . . . . . . . . . . . . . . . . . 15 3 Trinomial Tree 17 3.1 Pricing a European option through the Trinomial Tree . . . . . . . . . . . . . . 17 3.2 Pricing an American option through the Trinomial Tree . . . . . . . . . . . . . 19 3.3 Pricing a Barrier option through the Trinomial Tree . . . . . . . . . . . . . . . 20 3.4 Pricing a Double Barrier Knockout option through the Trinomial Tree . . . . . 23 4 The Greeks 25 4.1 Greeks via the Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Delta and Gamma through the Trinomial Tree . . . . . . . . . . . . . . . . . . 29 5 Hedging Strategies 30 5.1 Delta-hedging rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.1.1 Delta-hedge rule across different companies . . . . . . . . . . . . . . . 32 5.2 Rebalancing under delta tolerance . . . . . . . . . . . . . . . . . . . . . . . . 34 5.3 Delta-hedging including Transaction Costs . . . . . . . . . . . . . . . . . . . . 35 6 Conclusion 36 1
  5. 5. 7 Recommendations and Further Work 37 7.1 Implied Trinomial Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 7.2 Further Hedging Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 A my impvol.m Matlab file 39 B t treesize.m Matlab file 40 C tree barrier upcall.m Matlab file 41 D t double bar.m Matlab file 43 E g com.m Matlab file 44 F delta rebalance.m Matlab file 45 G delta rebalance tol.m Matlab file 47 H Background and Project Plan 49 Bibliography 57 Bibliography 59 2
  6. 6. List of Tables 3.1 Indicator Variables for Barrier Options . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Indicator Variables Double Barrier Knockout Option . . . . . . . . . . . . . . 23 5.1 Delta-hedging rule comparison of companies . . . . . . . . . . . . . . . . . . 32 3
  7. 7. List of Figures 2.1 Non-Recombining Binomial Tree and a Recombining Binomial Tree . . . . . . 12 2.2 Multi Step Binomial Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Binomial to Trinomial Formulation . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Implied Google Call Option prices through the trinomial tree compared with market prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1 European Call and Put Option Prices against Share price . . . . . . . . . . . . 18 3.2 Barrier Option Prices with varying barrier value . . . . . . . . . . . . . . . . . 21 3.3 Double Barrier Knockout Option . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1 Delta, Theta and Gamma Greeks via the Black-Scholes model varying time till maturity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Delta, Theta and Gamma Greeks via the Black-Scholes model varying Strike Price 27 4.3 Delta, Theta and Gamma Greeks via the Black-Scholes model up to maturity . 28 4.4 Delta and Gamma Greeks via the Trinomial tree and the Black-Scholes model . 29 5.1 Delta-hedge rule, rebalancing every day . . . . . . . . . . . . . . . . . . . . . 32 5.2 Delta-hedging rule while varying Delta tolerance . . . . . . . . . . . . . . . . 34 7.1 Trinomial tree and an Implied trinomial tree ([22]) . . . . . . . . . . . . . . . 37 A.1 my impvol.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 B.1 t treesize.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 C.1 tree barrier upcall.m (part a) . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 C.2 tree barrier upcall.m (part b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 D.1 t double bar.m, leaf vector discounted back through trinomial tree . . . . . . . 43 4
  8. 8. E.1 g com.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 F.1 delta rebalance.m (part a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 F.2 delta rebalance.m (part b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 G.1 delta rebalance tol.m (part a) . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 G.2 delta rebalance tol.m (part b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 H.1 Non-Recombining Binomial Tree(Left) and a Recombining Binomial Tree(Right) 52 H.2 Multi-Step Binomial Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 H.3 Trinomial Tree Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 H.4 Volatility Smile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5
  9. 9. Chapter 1 Introduction An option is a type of contract that gives the holder the right to, but not obligation to buy (call), or sell (put) an underlying asset or instrument at a specified strike price on or before a specified date. A European option can only be exercised at the maturity date while an American option can be exercised at any point up to and including maturity. Pricing of these options, has been around for a while, but in 1973 Fisher Black and Myron Scholes published a paper, ’The Pricing of Options and Corporate Liabilities’([4]). They had an idea to hedge the option by buying or selling the underlying asset in such a way to eliminate risk. With this they derived a stochastic partial differential equation which estimates the price of the option over time. The Black-Scholes equation led to a boom in finance and more specifically option trading around the world ([8]). 6
  10. 10. 1.1 Geometric Brownian Motion (GBM) A Geometric Brownian Motion is a continuous-time stochastic process where the logarithm of the randomly varying quantity, follows a Wiener process with drift ([9]). A stochastic process St is said to follow the GBM, if the process satisfies the following stochastic differential equation (SDE); dSt = µStdt +σStdWt, (1.1) where, • Wt is a Wiener process, • σ is the percentage volatility, • µ is the percentage drift, solving equation (1.1) under It¯o’s interpretation leads to ([10]), St = S0e(µ−σ2 2 )t+σWt . (1.2) The GBM assumes constant volatility when realistically in practice the volatility changes over time, maybe stochastic ([13]). Further Extensions of the GBM are ”Stochastic Volatility models in which the variance of a stochastic process is itself randomly distributed ([12]).” Below are some stochastic volatility models ([13]); • Heston Model. • Constant Elasticity of Variance Model, CEV Model. • Stochastic Alpha, Beta, Rho, (SABR Volatility model) 7
  11. 11. 1.2 Black-Scholes Model This model was first published by Fisher Black and Myron Scholes in their paper ”The Pricing of Options and Corporate Liabilities”. (1973,[4]). Pricing of European options is based upon there should not be any opportunities for risk-free profit in the market, (no-arbitrage principle). Black and Scholes showed that ”it is possible to create a hedged position, consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock” ([4]). The Black-Scholes model is based upon the following assumptions; • the stock price follows a Geometric Browian Motion, • buying and selling any amount of stock with no transaction costs incurred, • borrowing and lending takes place at the risk-free interest rate, • non-dividend paying stock. Under these assumptions the dynamic hedging strategy by Black and Scholes leads to the fol- lowing partial differential equation; ∂V ∂t + 1 2 σ2 S2 ∂2V ∂S2 +rs ∂V ∂S −rV = 0. (1.3) This equation is solved with boundary conditions depending on the characteristics of the options. For the standard European vanilla options we find explicit solutions leading to the Black-Scholes formulas. The values of the European vanilla call and put options, respectively, are: C(S,t) = N(d1)S−N(d2)Ke−r(T−t) , (1.4) P(S,t) = Ke−r(T−t) −S+C(S,t), = N(−d2)Ke−r(T−t) −N(−d1)S, (1.5) where d1 = 1 σ √ T −t [ln( S K )+(r + σ2 2 )(T −t)], (1.6) d2 = 1 σ √ T −t [ln( S K )+(r − σ2 2 )(T −t)], = d1 −σ √ T −t. (1.7) The quantities appearing in (1.4-1.7) are 8
  12. 12. • T −t being the time left till maturity, • S is the stock price, • K is the strike price, • r is the risk-free rate, • σ is the volatility of the underlying stock, • N(·) being the cumulative normal distribution. 9
  13. 13. 1.2.1 Implied Volatility by the Black-Scholes Equation The volatility of the market is a quantity difficult to measure. One possibility is to assume that the market satisfies the Black-Scholes model, and so the value of the traded options satisfy the Black-Scholes formulae (1.4,1.5). With all the other quantities known except for σ, we can set-up an equation in σ using (1.4) where C is the known traded value of the European vanilla call option. Solving for σ F(σ) = C −SN(d1(σ))+Ke−r(T−t) N(d2(σ)), (1.8) we find the Implied Volatility ([5]). Solving (1.8) can be done by using the following Newton’s iteration for finding the root of a function, σi = σi−1 − F(σi−1) F (σi−1) , (for i > 0), (1.9) ”In practice this iteration would be halted once |F(σi)| < ε for some user-specified tolerance ε.”[5] A function file my impvol.m was created in Matlab to solve for the implied volatility with inputs, Appendix A; • S - Stock price, • C - Call option price, • E - Exercise/Strike price, • r - Risk-free rate (annual), • T - Time till maturity (in years), • eps - convergence tolerance, • vol0 - Initial volatility (annual). 10
  14. 14. Chapter 2 Lattice models Another method for pricing a stock option is, a lattice model. Which divides time from now up to expiration into N discrete time points with each point going to two possible states (Binomial) or three possible states (Trinomial) all the way up to expiration date where the expected payoffs are calculated. Taking these payoffs and iteratively discounting the values back to the present by the continuously discounted risk-free interest rate, applying risk-neutral probabilities through a series of one-step trinomial trees until back with one node being the option price. The first lattice model was the Binomial Option Pricing Model (BOPM) By Cox, Ross and Rubinstein (CRR)(1979,[1]). From the source node the underlying asset either goes up with probability p or down with probability 1− p. This process repeats until you reach the expiration date with all the possible stock outcomes. The expected payoffs are then calculated at expiration by: Call Option = max{ST −K,0}, Put Option = max{K −ST ,0}. ST being the stock price at expiration (T) with K being the strike price on the option. Expected payoffs are then discounted back through the tree by risk-neutral probabilities until you are back at the source node and reach the option price. Figure 2.1 below shows the Binomial Tree. It is computationally efficient to have a recombining tree over a non-recombining tree, because if the tree recombines, there are only N +1 nodes at stage N, whereas there will be 2N nodes at stage N on a non-recombining tree. To make the tree recombine CRR ([1]) made ud = 1. There are three parameters in Binomial model u,d, and p, we therefore need three equations to solve uniquely for the parameters. First equation comes from matching the expectation of return on 11
  15. 15. Figure 2.1: Non-Recombining Binomial Tree and a Recombining Binomial Tree http://www.mathworks.co.uk/help/fininst/overview-of-interest-rate-tree-models.html the asset in a risk-neutral world. The second from matching the variance. We get pu+(1− p)d = er∆t , pu2 +(1− p)d2 −(er∆t ) 2 = σ2 ∆t. The third equation comes from making the tree recombine ([1]), u = 1 d . After some rearranging and solving for the three parameters in the three equations above, results showed[3]: p = er∆t −d u−d , u = eσ √ ∆t , d = e−σ √ ∆t , where σ is the assets volatility and r being the risk-free interest rate. When the Binomial tree has been created shown in Figure 2.2[17] with the expected future payoffs (leaf nodes), these need to be continuously discounted back to earlier nodes by the risk-free interest rate, taking into account the risk-neutral probabilities. The formula for calculating each node, Cn,j = e−r∆t (pCn+1,j+1 +(1− p)Cn+1,j−1). 12
  16. 16. Figure 2.2: Multi Step Binomial Tree Where Cn,j is the current option price for tier n with Cn+1,j+1 being the upper node and Cn+1,j−1 being the lower node at the next point in time. This process iterates through all time levels until you are back at the source node at the present with the option price. 13
  17. 17. 2.1 Formulation of the Binomial Option Pricing Model by replicating portfolios Let a portfolio contain ∆ shares of the stock and an amount B invested in risk-free bonds with a present value of ∆s+B. We want the option payoff equal to the portfolio payoff ([6]). Value of replicating portfolio at time h with stock price Sh is ∆Sh + erhB. At the two possible states, Su = uS and Sd = dS the replicating portfolio must satisfy ([7]): (∆uSeδh )+(Berh ) = Cu, (∆dSeδh )+(Berh ) = Cd, with δ being the dividend yield, Cu being the upper option node and Cd being the lower option node. Then solving for ∆ and B: Cu −∆uSeδh = Cd −∆dSeδh , ∆ = eδh ( Cu −Cd uS−dS ), (2.1) Berh = Cu −∆uSeδh , (2.2) substituting (2.1) into (2.2) yields Berh = Cu − uS(Cu −Cd) uS−dS , B = e−rh ( uCd −dCu u−d . The cost of creating the option is the cash flow required to buy the shares and bonds ([7]): ∆S+B = e−rh [ uCd −dCu u−d +e(r−δ)hCu −Cd u−d ], = e−rh [Cu e(r−δ)h −d u−d +Cd u−e(r−δ)h u−d ], = e−rh [Cu p+Cd(1− p)], arriving at the required equation. 14
  18. 18. 2.2 Binomial model to the Trinomial model One step in the trinomial tree can be seen as a combination of two steps of the binomial model, either two up or down jumps or one of each. Taking equations from the binomial model and applying two steps, Figure 2.3 shows the probabilities and the jump sizes for the Trinomial model. Figure 2.3: Binomial to Trinomial Formulation Option prices were calculated using the trinomial tree with different values for the jump sizes and probabilities depending on either CRR, RB and JR, and the equal probability tree for Google between 23/04/2013 up to 20/12/2013 on a Call option with a strike price=1000. The volatility used is the average of all the implied volatilities over the time of the contract at each discrete daily time step. Plotted below in figure 2.4 are the trinomial trees calculated prices compared to the market prices in the first row and the corresponding difference between the two on the second row. By figure 2.4, we can see that the equal-probability tree is closer to the market price than the CRR and the RB and JR tree with an average difference of 2, compared to an average of 14 for the CRR and RB and JR. Also on the difference plots of CRR and RB and JR, we notice the 15
  19. 19. Figure 2.4: Implied Google Call Option prices through the trinomial tree compared with market prices closer the option reaches maturity, the difference is approximately increasing. This could be due to the increasing volatility over the time of the contract and only calculating the prices with a constant volatility. 16
  20. 20. Chapter 3 Trinomial Tree 3.1 Pricing a European option through the Trinomial Tree Solving the trinomial tree with the initial stock price (S0), number of time steps up to maturity (n), strike price on the option (K), t treesize.m computes the option price with the choice of a equal-probability tree, RB and JR tree and finally the CRR tree, Appendix (B). The payoffs on the trinomial tree will be the leaf nodes comprising of 2n+1 elements as shown below; max(S0un −K,0) max(S0un−1 −K,0) ... max(S0u−K,0) max(S0mn −K,0) max(S0d −K,0) ... max(S0dn−1 −K,0) max(S0dn −K,0) , The values of u,m and d are pre-calculated along with the probabilities depending on what trinomial tree is being solved. To discount these option payoffs back to present, two vectors A and B with lengths 2n + 1 and 2n − 1 respectively are created with assigning the payoffs to vector A, then dynamically iterating back through n time steps with the calculation of each of the nodes at each time level, while simultaneously making the vector shorter until we are back at the present with one node being the option price. For each time level the vector B is computed 17
  21. 21. by; Bi−1 = er·dt (puAi−1 + pmAi + pdAi+1) dt = T/n, pu + pm + pd = 1, with k representing the time level and n being the number of iterations back to the present. Clearing vector A with length 2(n−k)+1 and setting this vector equal to vector B. The current vector B is not needed, so is cleared and a new vector B is created two elements shorter than the previous vector A. These elements of B are computed again one time step closer to the present being the option price at k equal to n. Plotted below in Figure 3.1 are the option prices, against share price with a strike price of 25, σ being 0.25, time till maturity being a year with a risk-free interest rate of 0.005. Figure 3.1: European Call and Put Option Prices against Share price When the stock price is equal to strike price, Figure 3.1 shows the option price for a call and a put option are the same, also known as ’at the money’. When S > K the call option becomes ’in the money’ with the put option becoming ’out of the money’, therefore the price of a call option is increasing as the share price becomes deeper in the money. Vice versa for S < K as the put option is now ’in the money’ and a call option being ’out of the money’. 18
  22. 22. 3.2 Pricing an American option through the Trinomial Tree American option contracts can be exercised at any point up to and including maturity, therefore the iteration sequence back to the present needs to be modified by re-defining the payoff at every node of the tree ([14]); Calloptionpayof f = max(Si,j −K,0), Putoptionpayof f = max(K −Si,j,0), where, Si,j = S0uNu dNd mNm with Nu +Nd +Nm = n, n being the time level in between the present and maturity. Each node for each time level is computed by, Cn,j = max(optionpayof f,e−r∆t [puCn+1,j+1 + pmCn+1,j + pdCn+1,j−1]. To solve for these changes we modify the matlab m file t treesize.m (Appendix B) by creating a vector C the same size as vector A after transferring the values from vector B in the iteration. The initial payoffs are calculated the same way, and then computing the stock prices for the corresponding point in time by the up, middle and down jumps applied to the initial stock price. This vector’s length will change with the point of time by 2(n−k)+1, with k representing the point in time. max(S0un−k −K,0) max(S0un−k−1 −K,0) ... max(S0u−K,0) max(S0mn−k −K,0) max(S0d −K,0) ... max(S0dn−k−1 −K,0) max(S0dn−k −K,0) , (3.1) the elements of vector A will be, Ai = max(Ci,Ai). The process repeats again one step closer to the present. 19
  23. 23. 3.3 Pricing a Barrier option through the Trinomial Tree A barrier option is a path-dependent option whose price is equal to that of a European option, depending if the price crosses or doesn’t cross a barrier up to maturity, otherwise the payoff is equal to zero. This is represented by an indicator variable ’I’ taking values 1 or 0 and multiplying this variable by the payoff function for a European option. To use the trinomial tree to price such an option, we simulate 2n+1 runs through the Geometric Brownian Motion (GBM), matching the required leaf nodes on the trinomial tree; Si = Si−1e∆t(µ−σ2 2 )+σ √ ∆tε(i−1) ,i ≥ 1. ∆t = T n Once the runs have been simulated the maximum or minimum of each run is taken depending on whether the option is an up version or a down version with B > S0 or B < S0 respectively, with the indicator variables. Once the indicator variables have been configured for each simulated Table 3.1: Indicator Variables for Barrier Options Up and In max0<S≤ST >B I=1 max0<S≤ST <B I=0 Up and Out max0<S≤ST >B I=0 max0<S≤ST <B I=1 Down and In min0<S≤ST >B I=0 min0<S≤ST <B I=1 Down and Out min0<S≤ST >B I=1 min0<S≤ST <B I=0 run, the payoffs of each run are computed by, callpayof fs = I·max(ST −K,0), putpayof fs = I·max(K −ST ,0), 20
  24. 24. ST being the final value of the corresponding simulated run. The payoffs being 2n+1 in length these are simply plugged into the initial vector A from the Trinomial tree and iteratively dis- counted back to the source node giving the final option price. Matlab file tree barrier upcall.m is shown in Appendix (C). Plotted below in Figure 3.2 are the option prices of the different barrier options while varying the barrier value, with inputs, • S0 = 50, • n = 250, • K = 50, • σ = 0.25, • T = 1 • r = 0.005, • dB = 0.1,m = 250. Figure 3.2: Barrier Option Prices with varying barrier value Two separate barrier vectors were used in Figure 3.2, one for the up version (top row of Figure 3.2) with the first barrier value being 50, increasing by 0.1 up to 75. The second still starting at 50 but decreasing by 0.1 down to 25 (bottom row of figure 3.2). One can see that for a up and out 21
  25. 25. option the price of the option is increasing as the barrier value increases, due to less simulated runs crossing the barrier due to the volatility remaining constant. If the volatility was to increase proportional to the increase in the barrier value, the option price would expect to approximately maintain the same price. For the up and in barrier option, the price is decreasing due to the same reasons. The put options on all barrier options increase or decrease more dramatically, than the call option of the barrier values. This could be the time value of money in the call’s favour against the put, therefore incorporating higher charges on put options as barrier value varies. For the down and in, as the barrier moves further away from the initial stock price the option becomes cheaper, again due to the constant volatility. Vice versa for the down and out barrier option. 22
  26. 26. 3.4 Pricing a Double Barrier Knockout option through the Trinomial Tree Extending the single barrier option to a double barrier can be beneficial for informed investors betting on the price-movements of the security while still maintaining the same strike price on a cheaper option[14]. A double barrier knockout option payoff is equal to that of a European option payoff, if the maximum and the minimum of the underlying asset over the life of the contract is between the two barriers. A slight modification on the calculation of the indicator variables is needed to incorporate these double barriers. The payoffs are computed the same way Table 3.2: Indicator Variables Double Barrier Knockout Option Double Barrier Knockout Option max<uB and min>lB I=1 else I=0 as before and then discounted back through the trinomial tree to the source node giving the price of the option contract. Plotted below in Figure 3.3 are the option prices against the difference between the two barriers on the knockout option with inputs, • S0 = 50, • n = 250, • K = 50, • σ = 0.2, • T = 1, • r = 0.005, • dB = 0.1 up to 25 away from S0 in both directions. Figure 3.3 shows as the difference between the two barriers is increasing from the initial stock price, the option price is increasing. This would be expected as the maximum and minimum of each of the simulated runs up to maturity, are not breaching either of the two barriers giving the European option payoff. Matlab file t double bar.m computes the price of a double barrier knockout, Appendix (D). 23
  27. 27. Figure 3.3: Double Barrier Knockout Option 24
  28. 28. Chapter 4 The Greeks The Greeks are partial derivatives with respect to the underlying parameter to see the sensitivity of small changes in that parameter. Delta measures the rate of change of the option price with respect to the underlying security ([15]), ∆ = ∂V ∂S . Delta being between 0 and 1 for long position and 0 and -1 for a short position, signifying the amount of stock to hold with respect to number of option contracts in the portfolio. This idea is known as the Delta-Hedging rule. Delta can also be seen as the probability of that option being ’in the money’ at maturity ([16]). Theta measures the sensitivity of the value of option price given the passage of time, commonly divided by the number of days in a year ([15]),θ = ∂V ∂t . Gamma measures the rate of change in the delta with respect to the underlying security, therefore being a second order derivative, Γ = ∂∆ ∂S = ∂2V ∂S2 ([15]). Gamma is commonly used as an extension of the delta hedging rule allowing for a wider range of movements in the underlying security, known as Delta-Gamma-Hedging rule. 25
  29. 29. 4.1 Greeks via the Black-Scholes Model The solution of the Black-Scholes model for a call option at a point in time till maturity (t) for an underlying security (x) given by [18](p159-160), c(t,x) = xN(d+(T −t,x))−Ke−r(T−t) N(d−(T −t,x)), d±(τ,x) = 1 σ √ τ [log( x K )+(r ± σ2 2 )τ], N(y) = 1 √ 2π y −∞ e−Z2 2 dZ = 1 √ 2π ∞ −y e−Z2 2 dZ. Taking partial derivatives of the above equation to show the value of the required Greek under the input parameters of current stock price (x), time till expiration (τ), strike price (K), risk-free interest rate (r) and the stocks volatility (σ). Delta Cx(t,x) = N(d+(T −t,x)), Theta Ct(t,x) = −rKe−r(T−t) N(d−(T −t,x))− σx 2 √ T −t N (d+(T −t,x)), Gamma Cxx(t,x) = N (d+(T −t,x)) ∂ ∂x d+(T −t,x), = 1 σx √ T −t N (d+(T −t,x)). Plotting the above equations in figure 4.1 for share prices between 0 up to 50 in increments of 0.1 and the time till maturity of a year in increments of 0.2. Strike price of 30 with a risk-free interest rate of 0.005 and a σ of 0.2. More than half of the shares are shown to be purchased when the share price crosses the strike price, with holding all the shares with a delta of 1 when the option contract is ’deep in the money’. Reflecting a less riskier portfolio and incurring a cheaper cost of buying shares if the share price holds around the strike price, reflecting an uncertainty of the option maintain- ing ’in the money’. Theta showing the option looses more value per the passage of time the closer the option reaches maturity around the share price equalling the strike price. While loos- ing less value as time reaches maturity with the option being ’in the money’ or ’out of the money’. Gamma showing the rate of range of Delta being the greatest nearer maturity concen- trated around the strike price on the option. This is seen with Delta’s biggest change when the 26
  30. 30. Figure 4.1: Delta, Theta and Gamma Greeks via the Black-Scholes model varying time till maturity share price crosses the strike price. Plotted in figure 4.2 are the same three Greeks but taking a range of strike price values from 5 in increments of 5 up to 45 with all other variables kept constant. Figure 4.2: Delta, Theta and Gamma Greeks via the Black-Scholes model varying Strike Price 27
  31. 31. Can see that the option contract looses more value when crossing the strike price with the pas- sage of time (Theta), however is proportional to the current share price. Also delta clearly show- ing the greatest change when the share price goes through the strike price, shown by Gamma. Figure 4.3 shows how the Greeks change up to maturity with all other variables kept constant, with either the option being ’out of the money’, ’at the money’ or ’in the money’. Time till maturity of a year, risk-free interest rate of 0.05, and a σ of 0.1 were used for figure 4.3. Figure 4.3: Delta, Theta and Gamma Greeks via the Black-Scholes model up to maturity Keeping all other variables constant, one can see that the delta decreases closer the option reach- ing maturity. This is expected due to less time for the share price to vary, possibly coming ’out of the money’. A lower probability of the option maturing ’in the money’ being represented by the Delta. Option contracts still with a significant time till maturity loose more value over the passage of time when they are ’out of the money’ than option contracts being ’in the money’. However gradually gets reversed closer the option reaches maturity ending with ’in the money’ options loosing greater value than ’out of the money’ options. Gamma, again showing the rate of change of Delta with respect to the underlying asset, has the greatest change with ’out of the money’ options linked to a decreased probability (Delta) of the option maturing ’in the money’ the closer the option reaches maturity. 28
  32. 32. 4.2 Delta and Gamma through the Trinomial Tree Creating a function and denoting my trinomial tree by f(S,n,K,σ,T,r), Delta and Gamma Greek’s are calculated through the tree by[14](p8-9), ∆ = f(S+dS,n,K,σ,T,r)− f(S,n,K,σ,T,r) dS , Γ = f(S+dS,n,K,σ,T,r)−2 f(S,n,K,σ,T,r)+ f(S−dS,n,K,σ,T,r) dS2 , dS = Sσ √ T, • dS is chosen such that the amount is proportional to the volatility and the current share price, while taking into consideration time till maturity. Plotted in Figure 4.4 are the Delta and Gamma values against share price, via the trinomial tree and the Black-Scholes model. From the figure we can clearly see the trinomial tree approxi- mately follows the same values as the Black-Scholes model. This would be expected as both methods are built on the assumption the stock price follows the Geometric Brownian Motion (GBM), with assumed constant volatility up to maturity. Figure 4.4: Delta and Gamma Greeks via the Trinomial tree and the Black-Scholes model Done using matlab file g com.m, Appendix (E). 29
  33. 33. Chapter 5 Hedging Strategies Investors like to diversify their risk against stock movements by going short on European call options, while at the same time being long on the underlying asset. Or long on European put option, while going short on the underlying asset. The amount of stock held is equal to the Delta of the option multiplied by the number of option contracts purchased in the portfolio, along with the multiple of lot size, (number of share’s the option contract gives right to buy/sell at strike price at maturity). A portfolio with this characteristic is known to be delta-neutral, the share price will vary leading up to maturity and in turn the Delta will change value. To keep the portfolio delta-neutral, the underlying asset needs to be bought or sold appropriately on the change of the delta leading up to maturity. Rebalancing the portfolio keeps the portfolio more risk averse to small changes in the stock price. Ideally the number of rebalances would be continuous, called self-financing portfolio but in practice is impossible due to transaction costs. A high Gamma showing a high rate of change of Delta, indicates the portfolio becomes more riskier the longer the time interval becomes between the portfolio rebalancing. The stock price moving from S to S indicates the option price to move from C to C , however moves to C , the difference between C −C is the hedging error[19](p361). Fixing this error will allow for larger price jumps in the share price, making the portfolio less riskier than just the delta-hedging rule, extending on to the delta-gamma-hedging rule. 30
  34. 34. 5.1 Delta-hedging rule To begin the delta-hedging rule, an initial cost is incurred of setting up the portfolios positions. Going long on the shares with a short European call option (lot size being a 100 shares), therefore borrowing the initial cost minus the cost of the option contracts, C0 = ∆0N100S0, B0 = C0 −N f0. C and B being the cost an amount borrowed respectively, N the number of option contracts and f being the price of the call option with ∆ being calculated via the trinomial tree. At each rebalancing point (i) the cumulative cost and borrowed money being ([21]), Ci = Ci−1e r 252 +N100(∆i −∆i−1)Si, Bi = N100∆iSi −N fi. At maturity the option can be exercised if ST ≥K giving the replication cost, repcost = CT −N100K, leading on to the gain after taking in to account the initial price of the option contracts, netgain = f0Ne rx 252 −repcost, x being number days between initial purchase of option contract and maturity. Call Option price data and stock price data was taken from the Bloomberg Terminals for Mi- crosoft (MSFT) between 19/08/2013 up to 20/12/2013 with a strike price of 34, the delta-hedge rule was applied to the data while rebalancing every day. The volatility used for the calculations was the average of all the Implied Volatilities leading up to maturity. 10 option contracts were purchased with a lot size of a 100 and a risk-free interest rate of 0.005, plotted in Figure 5.1 is the cumulative cost, delta, stock price and the amount of money needed to borrow up to maturity on the contract. Matlab file delta rebalance.m was used for calculations, Appendix (F). The option matured in the money with a final stock price of 36.8 making the European call op- tion ’in the money’, therefore exercisable with a strike price of 34. The cumulative cost of the hedge was 34011 resulting in a replication cost of 11.0826. Giving final net value of -2.77. Such a small loss in size, in comparison to the cost showing the delta-hedge rule eliminates more risk, but in turn giving a lower return. 31
  35. 35. Figure 5.1: Delta-hedge rule, rebalancing every day 5.1.1 Delta-hedge rule across different companies The same process was run again on Microsft (MSFT), Google (GOOG) and Apple (AAPL) each with a variety of three strike prices, 10 option contracts with a risk-free interest rate of 0.005. Following table shows key results along with net loss/gain. Table 5.1: Delta-hedging rule comparison of companies MSFT AAPL ST 36.8 549.02 K 34 35 36 450 500 550 CT 34,027 34,723 29,446 513,770 525,280 245,330 Rep Cost 26.7219 -277.419 6,553.5 63,774 2528.1 -257,030 Net Value -18.4073 285.5298 6557.2 -63,085 -24,924 257,200 Net Value/CT -0.054 0.822 22.27 -12.28 -4.74 104.83 Table 5.1 showing the delta-hedge rule eliminating a lot of potential loss when ST hasn’t crossed the strike price, while taking a nice return on options maturing ’in the money’. Seen with MSFT with strike price of 36 and GOOG with a strike price of 1100, taking returns of 22.27% and 32
  36. 36. GOOG ST 1100.6 K 900 1000 1100 CT 922,310 998,240 329,040 Rep cost 22,310 -1759.2 -770,960 Net value -21,994 1836.3 770,980 Net value/CT -2.38 0.184 234.311 234.311% respectively. Compared with loss return of 0.054% and 2.38% for MSFT and GOOG respectively for the lower strike prices. Signifying option contracts with higher strike price in the future become cheaper, reflecting a lower probability that the contract will mature ’in the money’. This is shown by a small difference between the replication cost and net value for the higher strike prices. 33
  37. 37. 5.2 Rebalancing under delta tolerance Instead of rebalancing every data point (daily), modifying the delta rebalance.m (Appendix F), with an additional input for delta tolerance. Only rebalancing if the absolute value of the change between the delta of the previous rebalance, and the current delta is greater than the delta toler- ance. If the tolerance is not met, leave the holding of shares the same. Doing this will reduce the amount of transaction costs incorporated over the life of the option, however may not give a higher gain due to the increased volatility closer to the option reaching maturity. More fre- quent rebalancing would be required to hedge more of the investors risk, this could be done by the delta tolerance decreasing closer the option reaches maturity. Better still make the decrease proportional to the change of the implied volatility over time. Plotted below in Figure 5.2 is the net gain of the hedging rule against delta tolerance being constant throughout the option. Gain was calculated via Matlab file delta rebalance tol.m, Appendix (G). One can see the net gain diminishing, as the tolerance increases. Showing that a portfolio with Figure 5.2: Delta-hedging rule while varying Delta tolerance more frequent rebalancing is more ideal, however transaction costs were not incorporated in Figure 5.2. 34
  38. 38. 5.3 Delta-hedging including Transaction Costs Buying and selling stocks on the market to rebalance the portfolio incurs transaction costs. Either a fixed charge per share, a percentage of shares bought or sold, or just a flat fee regardless of the number of shares ([20]). At each rebalancing point, additional charges are included in the cumulative cost of the delta-hedge, Ci = Ci−1e r 252 +N100(∆i −∆i−1)Si + pSiN100(∆i −∆i−1), where (p) is the percentage charge of the transaction, only being applied to number of shares purchased keeping the portfolio delta-neutral. More additional costs occur in practice, the dif- ference between buying and selling from the broker, known as the bid-ask spread. Stamp duty, tax and other over night financing costs occur with the holding of your securities. More so- phisticated pricing techniques of these options are required to give a more accurate and realistic option price, with additional extension on to allowing the volatility to vary up to maturity on the option contract. 35
  39. 39. Chapter 6 Conclusion The equal probability trinomial tree being the most accurate against market prices, even with assumed constant volatility with an average difference of 2 between the trinomial tree option price and the market prices. Using the equal probability tree throughout for further calculations due to the increased accuracy compared to CRR and JR. Using this trinomial tree for the calcu- lation of the delta and gamma of an option, allows us to rebalance an option contract with the underlying asset to minimize the risk to the market. This is known as the delta-hedge rule where rebalancing is carried out on the portfolio to eliminate risk, however continuous rebalancing is infeasible due to additional transaction costs. Further extension of the delta-hedging rule would be the delta-gamma hedging rule, rebalancing the holding of the traded option with respect to the delta on the underlying asset. Therefore a delta-gamma hedging rule allows for a larger price movement in the underlying asset between rebalancing points. Extending this again by a delta-gamma-vega hedging rule, incorporating an additional option in the portfolio taking ad- vantage of the volatility between rebalancing points. Additionally transaction cost are incurred from the broker, bid-ask spread. Buying and selling of the underlying asset are not of the same value. So two share price vectors will need to be included in the calculations, one for buying the underlying asset and one for selling. Doing this will include the brokers transaction cost as well as adding the fixed charge percentage on the rebalancing transaction. 36
  40. 40. Chapter 7 Recommendations and Further Work 7.1 Implied Trinomial Tree The trinomial tree assumes constant volatility throughout, an extension of this being the implied trinomial tree. Where the implied volatilities are computed through the market prices, and the volatility smile is interpolated across the tree varying the size of the jumps and time between the jumps. Figure 7.1 shows a constant trinomial tree and an implied trinomial tree. Figure 7.1: Trinomial tree and an Implied trinomial tree ([22]) 37
  41. 41. 7.2 Further Hedging Techniques Delta-Gamma hedging rule rebalances the holding of option contracts between the delta-rebalance points. ”What is required is a position in an instrument such as an option that is not linearly de- pendent on the underlying asset” ([19],p363). Letting Γ being the Gamma of a delta-neutral portfolio and Γτ be the Gamma of a traded option, then the the overall Gamma of the portfolio with wτ holding of the option contract being ([19],p363), wτΓτ +Γ (7.1) holding − Γ Γτ of the option contract will in turn make the portfolio gamma-neutral, but the port- folio may not be delta-neutral anymore, so a rebalancing of the underlying asset is needed. Vega is an another partial derivative of the Black-Scholes equation with respect to volatility, ν = ∂V ∂σ ([15]). Having a holding of − ν ντ in a traded option will make the portfolio Vega neu- tral, a portfolio cant be gamma and Vega neutral unless another traded option is bought into the portfolio ([19],p365). Solving simultaneously the amount of options to hold for the Gamma and Vega making the respective partial derivative equal to zero on the portfolio. Correspondingly buying or selling the underlying asset to maintain delta neutrality, in turn made the portfolio delta-gamma-vega neutral. 38
  42. 42. Appendix A my impvol.m Matlab file Figure A.1: my impvol.m 39
  43. 43. Appendix B t treesize.m Matlab file Figure B.1: t treesize.m 40
  44. 44. Appendix C tree barrier upcall.m Matlab file Figure C.1: tree barrier upcall.m (part a) 41
  45. 45. Figure C.2: tree barrier upcall.m (part b) 42
  46. 46. Appendix D t double bar.m Matlab file Figure D.1: t double bar.m, leaf vector discounted back through trinomial tree 43
  47. 47. Appendix E g com.m Matlab file Figure E.1: g com.m 44
  48. 48. Appendix F delta rebalance.m Matlab file Figure F.1: delta rebalance.m (part a) 45
  49. 49. Figure F.2: delta rebalance.m (part b) 46
  50. 50. Appendix G delta rebalance tol.m Matlab file Figure G.1: delta rebalance tol.m (part a) 47
  51. 51. Figure G.2: delta rebalance tol.m (part b) 48
  52. 52. Appendix H Background and Project Plan Ciaran Cox (1115773) Jacques Furter Pricing options with trinomial trees Aims and Objectives • To understand the pricing of options and implement algorithms in Matlab programming. • To compute prices of barrier options by trinomial trees and compare the price with the Black-Scholes equation from Mathematical Finance (MA3667) module assignment. • To understand the concepts of an Implied Trinomial Tree (ITT). • Use trinomial trees to calculate the option greeks. Project Plan I will begin my project with a brief history of option pricing and some of the key breakthroughs in mathematical finance, along with the definition of an option contract along with its features and properties. Then I will talk about the binomial option pricing model by Cox,Ross and Ru- binstein (CRR)(1979) and its variant by Rendleman-Barter (RB) and Jarrow-Rudd (JR)(1979). Then extending the idea of a binomial model to a trinomial model by Boyle (1986) and imple- ment algorithms in Matlab to formulate a trinomial tree and calculate the price of barrier options. I will also mention the trinomial tree with a diffusion parameter by Kamrad and Ritchken (1991). Black-Scholes equation by Black and Scholes (1973) will be covered and compared with trino- mial trees. Also comparing the price a barrier option valued by the trinomial tree along with 49
  53. 53. the price of the same option calculated by the Black-Scholes equation from the Mathematical Finance (MA3667) module. The family of option greeks will be covered and a few of them calculated through the trinomial tree. I will then extend from the Black-Scholes equation and trinomial tree which assumes constant volatility to the Implied Trinomial Tree by Derman, Kani, and Chriss (1996). Relevant option price data will also need to be collected either via the Bloomberg terminals or Datastream termi- nals for the calculation of the implied volatility at different time points throughout the implied trinomial tree, not all option price data will be available therefore interpolation will be required to match the volatilities to the volatility smile. Gant Chart showing the project layout is in Appendix 1. Background Introduction An option is a type of contract that gives the holder the right to, but not obligation to buy (call) or sell (put) an underlying asset or instrument at a specified strike price on or before a specified date. • European option can only be exercised at the expiration date. • American option can be exercised at any time between the purchase date and the expiration date. • Bermuda option can only be exercised at certain times leading up to expiration date, which are discussed in the option contract. Pricing of these options has been around for a while, but in 1973, Fisher Black and Myron Scholes published a paper, ’The Pricing of Options and Corporate Liabilities’.[1] They had an idea to hedge the option by buying or seeling the underlying asset in such a way to eliminate risk. With this they derived a stochastic partial differential equation which estimates the price of the option over time.The Black-Scholes equation let to a boom in finance and more specifically option trading around the world.[8] Overiew 50
  54. 54. In the following I cover a brief overview of where I’m taking my project and what areas of option pricing I’ll be covering. Firstly looking at the binomial option pricing model which was the first of its kind by Cox, Ross and Rubinstein (CRR)(1979)[1], followed by the formulation of the model by replicating portfolios. With an extension of the model to 3 states (trinomial) first introduced by Boyle (1986)[5]. Concluding on Implied Trinomial Trees which is a further extension by allowing for changing volatility over the time period of the asset by matching the implied volatilities with the volatility smile.[10] Binomial Option Pricing Model Another method for a pricing a stock option is a lattice model, which divides time from now up to expiration into N discrete time periods with each point going to 2 possible states (Binomial) or 3 possible states (Trinomial) all the way up to expiration date where the expected payoff’s are calculated. Then discounting yourself back through the tree until you reach the option price at the source node. The first lattice model was the Binomial Option Pricing Model (BOPM) By Cox, Ross and Rubinstein (CRR)(1979)[1]. From the source node the underlying asset either goes up with probability p or down with probability 1− p. This process repeats until you reach the expiration date with all the possible stock outcomes. The expected payoffs are then calculated at expiration by: CallOption = max{ST −K,0}, PutOption = max{K −ST ,0}. ST being the stock price at expiration (T) with K being the strike price on the option. Expected payoffs are then discounted back through the tree by risk-neutral probabilities until your back at the source node and reached the option price. Figure 1 below shows the Binomial Tree. Figure 1 shows a non-recombining tree and a recombining tree. It is computationally efficient to have a recombining tree over a non-recombining tree, because if the tree recombines, there are only N + 1 nodes at stage N, whereas there will be 2N nodes at stage N on a non-recombining tree. To make the tree recombine CRR[1] made ud = 1, making an up jump followed by a down jump equal to your original position. There is 3 parameters in Binomial model u,d, and p, we therefore need 3 equations to solve for the parameters. First equation comes from matching the expectation of return on the asset in a risk-neutral world. The Second from matching the 51
  55. 55. Figure H.1: Non-Recombining Binomial Tree(Left) and a Recombining Binomial Tree(Right) http://www.mathworks.co.uk/help/fininst/overview-of-interest-rate-tree-models.html variance. pu+(1− p)d = er∆t , pu2 +(1− p)d2 −(er∆t ) 2 = σ2 ∆t. The third equation comes from making the tree recombine,(CRR)(1979)[1] u = 1 d . After some rearranging and solving for the 3 parameters in the 3 equations above, results showed[3]: p = er∆t −d u−d , u = eσ √ ∆t , d = e−σ √ ∆t , where σ is the assets volatility and r being the risk-free interest rate. When the Binomial tree has been created (Figure 2) with the expected future payoffs (leaf nodes), these need to be continuously discounted back to earlier nodes by the risk-free interest rate, taking into account the risk-neutral probabilities. The formula is Cn,j = e−r∆t (pCn+1,j+1 +(1− p)Cn+1,j−1). Where Cn,j is the current option price for tier n with Cn+1,j+1 being the upper node and Cn+1,j−1 being the lower node at the next point in time. This process repeats until your back at the source node with the option price. 52
  56. 56. Figure H.2: Multi-Step Binomial Tree http://investexcel.net/binomial-option-pricing-excel/ Formulation of the Binomial Option Pricing Model by replicating portfolios Let a portfolio contain ∆ shares of the stock and an ammount B invested in risk-free bonds with a present value of ∆s+B. We want the option payoff = portfolio payoff.[6] Value of replicating portfolio at time h with stock price Sh is ∆Sh + erhB. At Sh = uS and Sh = dS the replicating portfolio must satisfy[7]: (∆uSeδh )+(Berh ) = Cu, (∆dSeδh )+(Berh ) = Cd, with δ being the dividend yield, Cu being the upper option node and Cd being the lower option node. Then solving for ∆ and B: Cu −∆uSeδh = Cd −∆dSeδh , ∆ = eδh ( Cu −Cd uS−dS ), (H.1) Berh = Cu −∆uSeδh , (H.2) substituting eq(1) into eq(2) yields: Berh = Cu − uS(Cu −Cd) uS−dS , B = e−rh ( uCd −dCu u−d . 53
  57. 57. The cost of creating the option is the cash flow required to buy the shares and bonds[7]: ∆S+B = e−rh [ uCd −dCu u−d +e(r−δ)hCu −Cd u−d ], = e−rh [Cu e(r−δ)h −d u−d +Cd u−e(r−δ)h u−d ], = e−rh [Cu p+Cd(1− p)]. Trinomial Model The Trinomial model is an extension of the Binomial model. but taking an additional path at each node of the stock price staying the same. This was first introduced by Boyle (1986) [5]. The foundations of the model are similar in the fact that the first two moments are matched but with the first two moments of the Geometric Brownian Motion (GBM)[14], which behaves similar to stock price movements. E[S(ti+1)|S(ti)] = er∆t S(ti), Var[S(ti+1)|S(ti)] = ∆tS(ti)2 σ2 , ud = 1. The last constraint is needed to make the tree recombine, solving for the above equations yields[14]: u = eσ √ 2∆t , v = e−σ √ 2∆t , with the transition probabilities being: pu = ( e r∆t 2 −e −σ ∆t 2 e σ ∆t 2 −e −σ ∆t 2 )2 , pd = ( e σ ∆t 2 −e r∆t 2 e σ ∆t 2 −e −σ ∆t 2 )2 , pm = 1− pu − pd. The same discounting method is used from the Binomial model just extended to the trinomial model: Cn,j = e−r∆t (puCn+1,j+1 + pmCn+1,j + pdCn+1,j−1). 54
  58. 58. Figure H.3: Trinomial Tree Example http: //www.24-something.com/2011/03/07/how-to-create-trinomial-option-pricing-trees-with-excel-applescripts/ This process keeps repeating until back at the source node just like the Binomial Model. Figure 3 below shows an example of a trinomial tree with the blue being the stocks price with the option price beneath. Implied Trinomial Trees (ITT) Implied trees allows for changing volatility between nodes by extracting an implied evolution for the stock prices in equilibrium from market prices of liquid standard options on the underly- ing stock.[2] Making implied trees an extension to the Black-Scholes equation which assumes volatility is constant. A couple of new concepts are needed for the calculation of ITT, Arrow- Debreu prices and the volatility smile. Arrow-Debreu prices are the sum of the product of the risklessly discounted transition proba- bilities over all paths starting in the root of the tree and leading to node (n,i), with n being the nth time level and i being the highest node on that level. [10] The Volatility Smile is the plot of implied volatility against varying strike prices as shown in figure 4 below. ‘In the money’ meaning the option is worth something, ‘at the money’ being the option is at the strike price and ‘out of the money’ meaning the option is worthless. There is also a reverse skew and forward skew also known as the volatility smirk. In the reverse skew pattern the implied volatility’s are higher at lower strike prices than the implied volatility at higher strike prices. More frequently appears for longer term equity options and index options. [11] In the forward skew pattern, the implied volatility for lower strike prices are lower than the implied volatility at higher strike prices, commonly seen for options in the commodities market. 55
  59. 59. Figure H.4: Volatility Smile http://www.investopedia.com/terms/v/volatilitysmile.asp [11] The Implied trinomial tree desires the following properties to model the underlying price cor- rectly [10]. 1. Reproduces correctly the volatility smile. 2. Is risk neutral. 3. Uses transition probabilities from the interval (0,1). The study of implied trinomial trees is currently a work in progress. 56
  60. 60. Bibliography [1] Black, Fischer; Myron Scholes (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81 (3): 637654. doi:10.1086/260062. [1] (Black and Scholes’ original paper.) [2] MacKenzie, Donald (2006). An Engine, Not a Camera: How Financial Models Shape Markets. Cambridge, MA: MIT Press. ISBN 0-262-13460-8. [3] John C. Cox, Stephen A. Ross, and Mark Rubinstein. 1979. Option Pricing: A Simplified Approach. Journal of Financial Economics 7: 229-263. [4] http://www.goddardconsulting.ca/option-pricing-binomial-index.html [5] P. Boyle, Option Valuation Using a Three-Jump Process, International Options Journal 3, 7-12 (1986). [6] Professor P.A.Spindt Binomial Option Pricing [7] https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd= 3&ved=0CDgQFjAC&url=http%3A%2F%2Fwww2.fiu.edu%2F~dupoyetb%2FAdvanced_ Risk_Mgt%2Flectures%2Fweek%25201.ppt&ei=IPWDUu-gMMXIhAev8oCQDg&usg= AFQjCNEOr7-1rmnXwdsiVze2JeAMszww2A&bvm=bv.56343320,d.ZG4&cad=rja [8] P. Clifford, O. Zaboronski. Pricing Options Using Trinomial Trees (2008) [9] E. Derman, I. Kani, N.Chriss. Implied Trinomial Trees of the Volatility Smile (1996) [10] P.Cizek, K.Komorad Implied Trinomial Trees SFB 649 Discussion Paper (2005-007) [11] http://www.theoptionsguide.com/volatility-smile.aspx 57
  61. 61. Appendix 1: Gant Chart for Pricing Options with Trinomial Trees Major Project 58
  62. 62. Bibliography [1] John C. Cox, Stephen A. Ross, and Mark Rubinstein. 1979. Option Pricing: A Simplified Approach. Journal of Financial Economics 7: 229-263. [2] E. Derman, I. Kani, N.Chriss. Implied Trinomial Trees of the Volatility Smile (1996) [3] http://www.goddardconsulting.ca/option-pricing-binomial-index.html [4] F.Black and M.Scholes. The Pricing of Options and Corporate Liabilities The Journal of Political Economy, Vol. 81, No. 3 (May - Jun., 1973), pp. 637-654 [5] P.Date and S.Virmani MA3667:Mathematical and Computational Finance Assignment [6] Professor P.A.Spindt Binomial Option Pricing [7] https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd= 3&ved=0CDgQFjAC&url=http%3A%2F%2Fwww2.fiu.edu%2F~dupoyetb%2FAdvanced_ Risk_Mgt%2Flectures%2Fweek%25201.ppt&ei=IPWDUu-gMMXIhAev8oCQDg&usg= AFQjCNEOr7-1rmnXwdsiVze2JeAMszww2A&bvm=bv.56343320,d.ZG4&cad=rja [8] MacKenzie, Donald (2006). An Engine, Not a Camera: How Financial Models Shape Markets. Cambridge, MA: MIT Press. ISBN 0-262-13460-8. [9] Ross, Sheldon.M (2007). ”10.3.2”. Introduction to Probability Models [10] http://en.wikipedia.org/wiki/Geometric_Brownian_motion [11] Wilmott, Paul (2006). ”16.4”. Paul Wilmott on Quantitative Finance (2 ed.). [12] Gatheral, J, (2006). The volatility surface: a practitioners guide, Wiley. [13] http://en.wikipedia.org/wiki/Stochastic_volatility 59
  63. 63. [14] P. Clifford, O. Zaboronski. Pricing Options Using Trinomial Trees (2008) [15] Haug, Espen Gaardner (2007). The Complete Guide to Option Pricing Formulas. McGraw- Hill Professional. ISBN 9780071389976. ”ISBN 0-07-138997-0” [16] http://en.wikipedia.org/wiki/Greeks_(finance) [17] http://investexcel.net/binomial-option-pricing-excel/ [18] Shreve, Steven.E Stochastic Calculus for Finance 2, Continious Time models [19] John.C.Hull Options, Futures, and other derivatives, (7th ed.) [20] Chi Lee Option Pricing in the Presence of Transaction Costs [21] Prof. Yuh-Dauh Lyuu, National Taiwan University (2007) Delta Hedge [22] E. Derman, I. Kani and N. Chriss Implied Trinomial Trees of the Volatility Smile (1996). 60

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