## Trinomial tree

The trinomial tree is a lattice based computational model used in financial mathematics to price options. It was developed by Phelim Boyle in 1986. It is an extension of the binomial options pricing model, and is conceptually similar.It can also be shown that the approach is equivalent to the explicit finite difference method for option pricing.

 Formula and algorithm guide Under the trinomial method, the underlying stock price is modeled as a recombining tree, where, at each node the price has three possible paths: an up, down and stable or middle path.These values are found by multiplying the value at the current node by the appropriate factor u, d or m, where $$u = e^{\alpha \sqrt {2 \Delta t}}$$ $$d = e^{- \alpha \sqrt {2 \Delta t}} = \frac 1u(the \space structure \space is \space recombining)$$ $$m = 1$$ and the corresponding probabilities are: $$P_u = \bigg ( \frac {e^ {(r-q) \Delta t/2} -e^{- \sigma \sqrt {\Delta t/2} } } {e^{\sigma \sqrt {\Delta t/2}} -e^{- \sigma \sqrt {\Delta t/2} }} \bigg ) ^ 2$$ $$P_d = \bigg ( \frac {e^ {\sigma \sqrt {\Delta t/2}} -e^{{(r-q) \Delta t/2} } } {e^{\sigma \sqrt {\Delta t/2}} -e^{- \sigma \sqrt {\Delta t/2} }} \bigg ) ^ 2$$ $$P_m = p-{(P_u+P_d)} .$$ In the above formulae: Δt is the length of time per step in the tree and is simply time to maturity divided by the number of time steps; r is the risk-free interest rate over this maturity; σ is the corresponding volatility of the underlying; q is its corresponding dividend yield. As with the binomial model, these factors and probabilities are specified so as to ensure that the price of the underlying evolves as a martingale, while the moments - considering node spacing and probabilities - are matched to those of the log normal distribution (and with increasing accuracy for smaller time-steps). Note that for Pu, Pd or Pm to be in the interval (0,1) the following condition on Δt has to be satisfied $$\Delta t < 2 \frac {\sigma^2 }{(r-q)^2}$$ Once the tree of prices has been calculated, the option price is found at each node largely as for the binomial model, by working backwards from the final nodes to today. The difference being that the option value at each non-final node is determined based on the three - as opposed to two - later nodes and their corresponding probabilities. The model is best understood visually For the Algorithm guide click here
 Variables K(29.00) = strike price S(30.00) = spot price T(40) = Time in year (days/365) D(2.50) = Dividend per share v(30.0) = volatility in % r(5.00) = risk-free interest rate in % td(25) = Time to Dividend Payment n(6) = steps PutCall(P) = P for put and C for call OpStyle(E) = E for European option and A American option

 Strike price ($): Underlying asset price ($): Days to expiration (In Day): Dividend (%): Enter an amount (\$.cc) for discrete dividend, or an annual yield (eg 3.5 = 3.5% pa) Volatility (%): Interest rate (%): Days to ex-dividend: Enter days for discrete dividend; leave blank or zero for yield No. Tree Steps (1- 150): (max. 15 displayed) Option type : Call Put Exercise style: American European

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