Implied Volatility Tree Construction - Coursework Example

MSc IN MATHEMATICAL TRADING AND FINANCE DERIVATIVES 2 (SMM615) al affiliation: Introduction Derman and Kani (1994) are the ones who introduced the implied volatility tree model of option pricing. The implied volatility tree is seen to be similar to implied binomial tree but tends to differ to some extent. Only European options price can be incorporated for the case of implied binomial tree, whereby this European option prices has different strike prices with same maturity. General binomial tree has the capacity of incorporating previous options that have maturities when compared to the implied tree’s maturity span. There is also a high possibility for implied volatility incorporating information on European options that have varying strikes as well as with maturities that is different When related to general binomial tree’s backward induction, implied volatility is known to begin from the initial node then extends forward. The center node at any tree step is decided first. All nodes’ transitional probabilities and prices that are above the center node could be iteratively solved with the use of specific European calls. On the other hand, all nodes below transitional probability and prices are solved in the same manner but with the use specific European puts. There is interpolation of these prices and calls from the current market traded option with the use of implied volatility surface that is considered as a tool of transformation. The major weakness of implied volatility surface is that is has no capacity of precluding bad transitional probabilities that are either less than one or greater than one. Therefore, the given nodal price, S that generates a bad probability is rewritten and set as, Si= (where si represents higher node price while si-1 represents a lower node price for the previous step).In case si is at the highest node at that given step for instance; si=Sn+1, therefore, si =s2n/Sn. In case Si is set at the lowest node, for instance, Si=SI, then Si=s21/S2 There were some improvements added by Barle and Cakici(1998) in order to increase Derman and Kani(1994) original tree stability through centering the tree with the use of forward price. Despite this, bad probability and arbitrage violations still take place .The implied volatility is seen to be extended by Chriss(1996) in order to be applied with American input options through iterative method application, referred to as the method of false-position. This method however involves computation. The non-standard options market-conform pricing is seen to be very challenging. Implied tree is considered to be the major approach of handling such cases and it’s the seminal binomial model extension that was developed by Cox et al.(1979).They are referred to as implied tree since they are inferred from the traded option’s market. In this regard, they are considered to be consistent volatility smile as well as seen as a generalized one-dimensional diffusion discretization. In this case, volatility parameter is permitted to act as a function of deterministic for both asset price and time. In order to construct a risk-neutral price process that is consistent, the implied tree will take the liquid standard option market prices as provided. Such options are then utilized to infer information regarding the process that generates data. The state space is required to be known as well as the transition probability in when we want to build an implied tree. The state space is the price of the asset at various nodes of the tree. The method of getting information from the data was suggested by Derman and Kani(1994) Constructing implied tree Induction is usually used in building an implied tree that has uniformly spaced levels(Δt apart).Let’s assume that the first n has already been constructed and it matches all options’ implied volatilities that has all strikes out to that given period of time. Figure 1 below indicates the tree nth level at the time tn , that has n implied tree nodes with their stock prices, S, that is already known. Figure 1. The implied tree nth level The compounded forward riskless interest rate is called at nth level r. Basically, forward riskless interest rate is known to be time dependent as well as it has the tendency of varying from one level to the other; the explicit level index should not be attached to it as well as other variables employed for the sake of notational simplicity. The nodes of the level (n+1)th at time tn+1 is the one to be determined. We have n+1=5 nodes that are to be fixed and n+1 corresponds to the stock prices, Si that is not known. Basing on figure 1, the ith node has been shown at level n, and it is represented by (n,i).This ith node a stock price that is known(Si) and it can either evolve to a down node with price SI as well as up node that has a price Si+1 at level n+1.In this case, the forward price corresponds to Si ;Fi=erΔtSi. The probability that makes a transition to up node is referred to as Pi .The Arrow-Debreu price is represented by λI and its at node(n,i). Forward induction is used to compute it; where the total is divided by all paths, that is, from the tree’s root to node (n,i) of the risklessly-dicounted product (transitional probabilities in each path at each node and it leads to node(n,i).At level n, all λI are known since the previous tree nodes with their transition probabilities have been implied out already to level n. The transition is defined by 2n+1 parameters from the tree nth level to the (n+1)th level that include transition probabilities, Pi, and the stock prices, Si. In this case, smile is used to illustrate how they are determined. The nodes at level (n+1) are implied through the tree where the theoretical values are calculated. In this case they are (2n) known quantities; one n represents options and the other forwards and they expire at time tn+1.These theoretical values should match the market values that are interpolated. In this regard, the 2n equations are provided for the 2n+1 parameters. The remaining degree of freedom is used to make the tree center that coincide with standard Cox Ross-Rubinstein tree’s center and this tree is known to have a constant local volatility. In cases there are nodes numbers are odd at any level, the stock price central node is chosen and it is equivalent to the current spot; In case of even number, the natural logarithms’ average for the two stock prices of the central node is made and it is equal to the current spot price logarithm. The 2n equations are now derived and it is for the options and forward theoretical values. The implied tree is considered to be risk neutral and its known forward price include the stock at any node (n,i), with the expected value that is one period later. Such equation results to where Fi represents a known value.In this case, we have n forward equations and one is for each i. The n independent options values is expressed by the second equation set, whereby one is for each strike Si and it is equal to the stock price that is known at the nth level which is known to expire at the level;(n+1).The down and up nodes are split by the strike level Si as well as S1+1 and Si at the following level as indicated in figure 1.It is ensured that all the nodes below(above) and those down(up) it have a contribution to a call struck at Si. The derived below n equations for options together with the above equation as well as the choice regarding tree centering determines both the stock prices, Si at that level of the node and the transition probabilities Pi that results to (n+1)th level. Let P(Si,tn+1) and C(Si,tn+1) be the put struck today at Si and known market values that is known for a call respectively. In this case the expiry time is at tn+1 . The call stuck at K theoretical binomial value that has an expiration time at tn+1 is illustrated by the total divided by all nodes j at the (n+1)th level(discounted probability) that reaches each node(n+1,j) which is multiplied by the call payoff. In case the strike K is equals to si, there can be a separation of the transition in the money up node contributions, whereby, the equation can be rewritten based on the known Arrow-Debreu prices, the known forwards and the known stock prices, Such that The first term is based on the up node with the price (Si+1) that is unknown and the Pi that is unknown. The second term is considered to be the sum divided by the quantities that are already unknown Since it is known that both and Fi from smile, the equation; can be solved simultaneously and for Si+1.The transition probability Pi can be in terms of Si: Si+1= divided by ) The above equations can be used to determine iteratively Si+1 and Pi for all nodes that are above the tree center only if the Si at any initial node is known. In case we have an odd number node at (n+1)th level, then the initial Si for i=n/2+1 can be identified and it has a central node with the stock price chosen as the current spot value similarly to that of Cox-Ross-Rubinstein tree. Therefore, the stock price can be calculated at the node above from the equation Si+1= divided by and equation; ) to determine Pi. This process can now be repeated whereby the node is moved up one node at that given level, in this regard; the upper half of each level can be implied In case, there is even node value at the (n+1)th level .The first step will be identifying the initial Si+1 and Si, for i=(n+1)/2.In this case, the node is above and below the level center. The selected Cox-RossRubinstein centering condition is considered to be same as selecting the 2 central stock prices that will satisfy Si=S2/Si+1, in this case, S=si is the current spot price that corresponds to the Cox-Ross-Rubinstein-style central node of the earlier level. When this is substituted into equation Si+1= divided by ,the formula for the two upper central nodes with n odd at level n+1 is generated as below: The construction of the complete tree is illustrated based on the smile. In simple terms, the tree has been build for the levels which have been spaced one year apart for 5 years. This was done on a computer to obtain more closely spaced levels. It is assumed that the current index value is 100, while zero is taken to be its dividend yield. 3 percent per year is taken to be the compounded riskless interest rate for all maturities. It is assumed that the at –the-money European call annual implied volatility is 10 percent and this is for all the expectations. In this case, the implied volatility decreases (increases) linearly by 0.5% for every drop or rise in the strike, hence smile can be defined by this. Figure 2 below indicates the standard Cox-RossRubinstein binomial stock tree for 10 percent local volatility. The tree is seen to generate no smile as well as it is the Black-Scholes discrete binomial analogue equation. The implied volatilities can be converted using this equation and it can be converted to quoted options prices. The production of the up and downs moves is by factors exp(+/-α / 100).At every node, the transition probability is taken to be 0.625. The implied stock tree has been displayed in figure 3; also the Arrow-Debreu prices and the transition probability tree have been displayed. The node parameter has been shown how they are fixed in the model. It is first assumed that 3 percent interest rate implies that it is 1.03 times the stock price of the node of the forward price that is one year later for any given node. The current stock price is 100 at the 1st node and the Arrow-Debreu price λo= 1.000.To find the node A stock price of level two, the following is set as Si+1=Sa,S=100 and λo=1.000. Then In this case (100, 1) is the current value of one year call that has a strike 100. Ʃ should be set at zero since higher nodes than those with strike at level 0 above 100 never exist. Basing to smile, the call C (100, 1) should be valued at 10% implied volatility. Figure 2.Binomial stock tree with constant stock volatility of 10 % Figure 3.Implied tree, Arrow –Debreu tree and probability tree The C (100, 1) =6.38 has been used to make things simple and it is valued on the tree as seen in figure 3 above. When these values are inserted in the above equation we get SA= 110.52.As shown in figure 3, we see that the SB price that corresponds to the lower node B is provided by the selected centering condition SB =S2/SA =90.48.The transition probability in year zero at the node from the equation;) is given by: P=(103-90.48)/(110.52-90.48) =0.625 The Arrow-Debreu price that is at node A is provided by λa=(λoP)/1.03=(1.00x0.625)/1.03=0.607 as indicated below figure 3.In this case, the smile has implied the tree’s 2nd level For year two nodes, the central node is selected and it lies at 100.C being the next highest node, is determined using one year forward value FA=113.84 of SA=110.52 (the stock price at node A and struck at SA by the two year call C(SA,2).Since ,the nodes that has a higher stock value than the given node A in the year one does not exist, the S term is as well set as zero and the equation: Si+1= divided by ,gives C (SA, 2) value at 9.47% as the implied volatility and which corresponds to 110.52 as the strike is given as 3.92 in the binomial world. When these values are substituted to the above equation, the Sc=120.27 is generated. The transition probability equation gives PA = (113.84-100)/ (120.27-100) =0.682 The new Arrow-Derbreu price λc can be determined as well in the same way. It can as well be indicated that at node D, the stock price should be 79 and this makes the put price P (SB, 2) to have 10.4% as the implied volatility that is consistent with smile. At node A, the implied local volatility for one year is It is clearly evident that when the smile is fitted results to a greater local volatility that is one year out at a stock prices that are lower. In regard to figure 5,one more stock value has been shown how it is determined and node G has been set at year 5.Lets assume that the tree has been implied to year four already. Also the SF value has been found already at node F as 110.61 as indicated in figure 3 above. At node G, the stock price SG is provided by the equation Where SE=120.51, FE=120.52x1.03 =124.13;λE =0.329 The interpolated smile implied volatility is 8.86% at a strike of 120.51% corresponds to C(120.51,5)=6.24 value. In the above equation, the S value is provided by the call contribution from the node H that is above E in year 4. The higher nodes can be fixed as indicated above after the initial stock price of the node has been identified. Similarly, all the nodes can be fixed at this level (the nodes must be below the central node).The lower stock price of the node which is determined through the analogous formula as below Conclusion In conclusion, implied volatility tree construction is seen to be highly sensitive extrapolation and interpolation techniques, particularly when volatility smiles are observed to assume shapes that are different, the underlying volatility is large and the tree steps are large. Such are considered to be the weaknesses in the method of implied volatility method which seems to generate less accuracy in puts when compared to in calls in cases where the latter are applied to match the volatility smile. The implied tree model could as well be compared with volatility models like Cox and Ross(1976) traditional CEV model as well as the most recent Kernel model of Ait-Sahalia & LO(1998).On top of this, they could as well be tested against other trees that have stochastic volatility. In implied binomial tree method involves numerical procedures that are consistent with the effect of smile as well as the implied volatility term structure. The algorithm of implied binomial tree is the modification data of cox, Ross and Rubinstein (1976) method. In this case, the stock is seen to evolve along risk neutral binomial tree that has constant volatility. Generally, the objective of implied binomial tree is implied probability estimation or local volatility surface and state price densities estimation. In addition, the implied binomial tree could evaluate the distribution of the future stock prices basing on BS implied volatility surfaces that could be estimated form the prices of market European option. Only European options price can be incorporated for the case of implied binomial tree, whereby this European option prices has different strike prices with same maturity. General binomial tree has the capacity of incorporating options that have maturities that are earlier when compared to the implied tree’s maturity span. In order to construct a risk-neutral price process that is consistent, the implied tree will take the liquid standard option market prices as provided. Such options are then utilized to infer information regarding the process that generates data. The state space is required to be known as well as the transition probability in when we want to build an implied tree. References Ait-Sahalia, Y., & Lo, A. W. (1988). Non-parametric estimation of state-price densities implicit in financial asset prices. Journal of Finance, 53(2), 499–547. Barle, S., & Cakici, N. (1998). How to grow a smiling tree. Journal of Financial Engineering, 7(2), 127–146. Chriss, N. (1996). Transatlantic trees. RISK, 7, 45–48. Cox, J., Ross, S., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7, 229–263. Derman, E., & Kani, I. (1994). The volatility smiles and its implied tree.Goldman Sachs Quantitative Strategies Research Notes, January. Read More