The power of deterministic option-implied trees in pricing European options

ABSTRACT

The aims of the current article are threefold. First, to investigate the power of deterministic option-implied trees, constructed either by forward or by backward induction, in pricing European options, in order to assess the proper representation of the smile. Second, to investigate and contrast the power of deterministic option-implied trees during tranquil and volatile market conditions. Last, to assess the correctness of the representation of the smile in different parts of the risk-neutral distribution. Three main results are obtained. First, the pricing performance of the Enhanced Derman and Kani model (EDK), based on forward induction, is superior to that of the Rubinstein model, based on backward induction. Second, the EDK model produces better results (smaller errors) on the left tail of the distribution, i.e. it is better in pricing out-of-the-money put options. Third, it performs better in turmoil periods where correct pricing a challenge, and accuracy is of greater importance than in tranquil periods. Diebold and Mariano test of equal predictive accuracy confirms the superiority of the EDK model in both sub-periods.

KEYWORDS:

• Option pricing
• option-implied trees
• Derman and Kani
• Rubinstein
• smile

JEL CLASSIFICATION:

• G13
• G14
• C63

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix A. Description of the implied tree models

Below, we briefly describe the Enhanced Derman and Kani and the Rubinstein models. These models will be used in the empirical section to estimate the underlying asset risk-neutral distribution and price FTSE MIB-index options (MIBO).

A1. The Enhanced Derman and Kani (EDK) implied tree model

Derman and Kani (1994) construct an implied tree using forward induction for the purpose of pricing other options consistently with the options used for the derivation of the implied tree. They build a recombining binomial tree which uses as inputs the market prices of European-style index options across all strikes and expirations. Their model has uniformly spaced levels apart. Let j=0,…,n be the number of levels of the tree, that are spaced by . As the tree recombines, i=1,…,j+1 is the number of nodes at level j. Let us assume that the tree has already been constructed up to the current level j-1. We wish to derive the stock prices for the next period j. The known stock price prevailing over this time period can evolve in two states in the next period : the up one, and the down one, . The risk-neutral probability of an up-jump is . This probability is defined as:, where F_i,j is the forward price. Arrow-Debreu prices, defined as the prices of a security that pays one euro in state (i,j) and zero in all the other states, and can be used to discount the payoff of the option in each state, are derived as the sum over all paths leading to node of the product of the risk-neutral probabilities discounted at the risk-free rate at each node. If the level is odd, the centering condition is given by equation (A1), whereas if the level is even, the two central nodes will have to satisfy equation (A2):

(A1) $S_{\frac{j}{2}+1,j}=S_{0,0}$(A1)

(A2) $S_{j/2,j}=\frac{S_{0,0}}{S_{\frac{j}{2}+1,j}}$(A2)

The centering condition allows the tree to grow around the current spot price S₀. It ensures that the final nodes of the tree are sufficiently dispersed around the current spot price. Let and be the price of a call and a put with strike and maturity, respectively. These prices are computed using Black-Scholes formulae with constant volatility obtained from the smile function. In the upper part of the tree, the recursive formula used to compute, given, can be described as (A3). Similarly, in the lower part of the tree the recursive formula to compute, given, can be described as (A4):

(A3) $S_{i+1,j}=\frac{S_{i,j}\left[e^{r\mathrm{\Delta }t}{C}_{i,j-1}-{\mathrm{\Sigma }}_C\right]-{\lambda }_{i,j-1}{S}_{i,j-1}\left(F_{i,j-1}-S_{i,j}\right)}{\left[e^{r\mathrm{\Delta }t}{C}_{i,j-1}-{\mathrm{\Sigma }}_C\right]-{\lambda }_{i,j-1}\left(F_{i,j-1}-S_{i,j}\right)}$(A3)

(A4) $S_{\left(i,j\right)}=\frac{S_{i+1,j}\left[e^{r\mathrm{\Delta }t}{P}_{i,j-1}-{\mathrm{\Sigma }}_P\right]-{\lambda }_{i,j-1}{S}_{i,j-1}\left(F_{i,j-1}-S_{i,j}\right)}{\left[e^{r\mathrm{\Delta }t}{P}_{i,j-1}-{\mathrm{\Sigma }}_P\right]-{\lambda }_{i,j-1}\left(F_{i,j-1}-S_{i,j}\right)}$(A4)

where is the risk-free rate, and is the forward value of .

These equations can be used only if the level is odd. If level is even, we combine equations (A2) and (A3) to obtain (A5):

(A5) $S_{i+1,j}=\frac{S_{0,0}\left[e^{r\mathrm{\Delta }t}{C}_{i,j-1}+{\lambda }_{i,j-1}{S}_{0,0}-{\mathrm{\Sigma }}_C\right]}{{\lambda }_{i,j-1}{F}_{i,j-1}-e^{r\mathrm{\Delta }t}{C}_{i,j-1}+{\mathrm{\Sigma }}_C}$(A5)

The transition probability of an up move is computed as equation (A6):

(A6) $q_{i,j-1}=\frac{F_{i,j-1}-S_{i,j}}{S_{i+1,j}-S_{i,j}}$(A6)

The main problem in the derivation of the implied tree is the presence of risk-free arbitrage opportunities, represented by a risk-neutral probability falling outside the (0, 1) interval. The Derman and Kani (1994) implied tree, even with the Barle and Cakici (1998) modifications, is not free from arbitrage, in particular at the boundary of the tree. Moreover, this tree may become numerically unstable when the number of steps becomes large. As a result, we adopt the Enhanced Derman and Kani (EDK) model to ensure the absence of no-arbitrage violations in the Derman and Kani implied tree. The EDK model provides no-arbitrage checks and proposes no-arbitrage replacements for all the nodes in the tree (for further details see Moriggia et al. (2009)). This model enhances the Derman and Kani model to determine option prices and implied moments free of arbitrage.

A2. Rubinstein’s implied binomial tree model

Rubinstein (1994) proposes an implied binomial tree using backward induction in order to incorporate the smile into an option pricing model. Rubinstein’s procedure can be divided into two steps. In the first step the risk-neutral probability distribution of the underlying asset at the end of the tree is estimated. In the second step, the tree is derived using a backward technique with a simple three-step algorithm. Rubinstein’s model consists of minimizing the square difference between prior and posterior risk-neutral probabilities, under certain constraints. Let us define and, respectively, as the posterior and the prior probability of arriving at node (i,n) at expiry date . The posterior () can be derived as the solution of the following optimization problem where we minimize the sum of squared deviations between prior and posterior probabilities:

(A7) $\mathrm{m}\mathrm{i}\mathrm{n}\sum _{i=0}^n{\left(Q_{i,n}-Q_{i,n}^{\prime }\right)}^2$(A7)

subject to:

(A8) $\sum _{i=0}^n{Q}_{i,n}=1and{Q}_{i,n}>0fori=0,\dots ,n,C_k^b\le C_k\le C_k^a,S^b\le S\le S^a$(A8)

where and are, respectively, the option bid and ask price quotes observed for the European call with strike with, expiring at, and and are the bid and ask prices of the underlying asset, is the price of a call with maturity and strike price :

(A9) $C_k=e^{-rn}\sum _{i=0}^n{Q}_{i,n}{\left(S_{i,n}-K_k\right)}^{+}$(A9)

and is the value of underlying asset at time :

(A10) $S_{0,0}=e^{-rn}\sum _{i=0}^n{Q}_{i,n}{S}_{i,n}$(A10)

The posterior implied risk-neutral probabilities are called nodal probabilities since is the probability to reach node at expiry regardless of the path to reach that node. The rather arbitrary and restrictive assumption of equal path probabilities makes it possible to build the tree in a simple way with a three-step procedure. First, calculate the nodal probabilities at the preceding nodes as described by equation (A11). Second, compute the probability of an upward move over the next time interval as described by equation (A12). This process rules out negative probabilities. Finally, compute the stock price at the preceding level with the risk-neutral valuation formula as described by equation (A13). The implied tree is derived by repeating this simple algorithm up to the first node,

(A11) $Q_{i-1,n-1}=\left(1-w_{i,j}\right)\ast Q_{i-1,n}+w_{i,j}\ast Q_{i,n}$(A11)

(A12) $q_{i-1,n-1}=w_{i,j}\ast \frac{Q_{i,n}}{Q_{i-1,n-1}}$(A12)

(A13) $S_{i-1,n-1}=e^{-r\mathrm{\Delta }t}\left[\left(1-q_{i-1,n-1}\right)\ast S_{i-1,n}+q_{i-1,n-1}\ast S_{i,n}\right]$(A13)

Despite its simplicity and the absence (by construction) of arbitrage opportunities, the Rubinstein model presents some drawbacks: it requires a costly optimization routine, with different possible outcomes depending on the choice of the prior distribution and it is suitable only for European-type options. However, given the characteristics of our data-set (consisting of European-type options), and the aim of our exercise (pricing plain-vanilla options) the model is consistent with the purpose.

Notes

¹ Forward induction option implied trees are derived by starting from current time t and estimating the future evolution of the stock price up to a future date T. Backward induction implied trees are derived by estimating the risk-neutral distribution at a future date T and deriving the tree backwardly up to current time t.

² See also Zhang et al. (2012) for extensions in a fuzzy setting of jump diffusion models.

Additional information

Funding

This work was supported by the Università Degli Studi di Modena e Reggio Emila [FAR2019]. The first author is also a Fellow at The Wharton Financial Institution Center, University of Pennsylvania, Philadelphia, PA, US, and a visiting professor and Dean’s Fellow at the Hebrew University, Jerusalem.