3.3.6 Binomial Trees

1. One-step Binomial Trees

Definition of Binomial Model
Binomial model: after one period, the value of the underlying asset will either go up to SuS_uSu or go down SdS_dSd.

No-arbitrage Argument

Law of one price: Assets that produce identical(完全相同的) future cash flows regardless of future everts should have the same price.
Construct a no uncertainty portfolio: using option and stock to construct a portfolio which has certain value after one period. If there is no arbitrage opportunity, the option price can be derived from the cost of this portfolio.
Short 1 call option and long Δ\DeltaΔ stocks.

The stock is currently trading at $50. The stock price will either go up to $75 or go down to $25 after one year. The risk-free rate is 2%. Please calculate the value of a 1-year European call option with an exercise price of $60 using one-step binomial tree.

The value of the portfolio after 1 year.
Δ∗Su−Cu=Δ∗Sd−Cd→Δ=0.3\Delta*S_u-C_u=\Delta*S_d-C_d \to \Delta=0.3Δ∗Su−Cu=Δ∗Sd−Cd→Δ=0.3
Regardless of whether the stock price moves up or down, the value of the portfolio is $7.5 without uncertainty.
Cost of portfolio: 0.3×50−call=7.5×e−0.020.3 \times 50-call=7.5\times e^{-0.02}0.3×50−call=7.5×e−0.02
call=7.6485call=7.6485call=7.6485
Calculata Δ\DeltaΔ of a stock option
Δ×Su−Cu=Δ×Sd−Cd\Delta \times S_u-C_u=\Delta \times S_d-C_dΔ×Su−Cu=Δ×Sd−Cd
Δ=Cu−CdSu−Sd\Delta=\frac{C_u-C_d}{S_u-S_d}Δ=Su−SdCu−Cd

Risk Neutral Valuation

In a risk neutral world, all individuals are indifferent to risk. Investors requires no compensation for risk and expected return on all securities is the risk-free interest rate.
The option should be valued based on risk neutrality.
In a risk-neutral world, the expected return on all securities is the risk-free interest rate, and the discount rate is the risk-free interest rate.

u=eσtu=e^{\sigma\sqrt{t}}u=eσt
d=1u=e−σtd=\frac{1}{u}=e^{-\sigma\sqrt{t}}d=u1=e−σt
The higher the standard deviation, the greater the dispersion between stock prices in up and down states.
πu×S0×u+(1−πu)×S0×d=S0ert\pi_u\times S_0 \times u+(1-\pi_u)\times S_0 \times d=S_0e^{rt}πu×S0×u+(1−πu)×S0×d=S0ert
πu=ert−du−d\pi_u=\frac{e^{rt}-d}{u-d}πu=u−dert−d

Risk neutral valuation-Process
The valuation process of one-step binomial tree as follow:

Calculate uuu and ddd, then construct the whole binomial tree.
Calculate the payoff of the option at the maturity node.
Compute the risk-neutral up and down probability, then calculate the expected value of option in one period.
Use risk-free rate to discount the expected value to present.

2. Two-step Binomial Trees

Two-Step Binomial Model
The basic valuation process of a two-step European option is similar with one-step binomial model, but with more steps.

Two-Step Binomial Model - European
An analyst is using two-step binomial model to calculate the price of a 2-year European put option with strike price of $75. The continuously compounded risk-free rate is 5%. The stock pays no dividend and is trading at $70. The volatility of the stock price is 20%. What will be the European put option price?

Firstly, calculate uuu and ddd, then construct the whole binomial trees:
u=eσT=e0.02∗1=1.2214u=e^{\sigma \sqrt{T}} = e^{0.02 * \sqrt{1}} =1.2214u=eσT=e0.02∗1=1.2214
d=1u=e−σT=0.8187d=\frac{1}{u}=e^{-\sigma \sqrt{T}}=0.8187d=u1=e−σT=0.8187
Secondly, calculate the payoff of the option at the maturity node:
Thirdly, compute the risk-neutral up and down probability, and then calculate the expected value in one period.
πu=ert−du−d=e5%×1−0.81871.2214−0.8187=57.7529%\pi_u=\frac{e^{rt}-d}{u-d}=\frac{e^{5\%\times1}-0.8187}{1.2214-0.8187}=57.7529\%πu=u−dert−d=1.2214−0.8187e5%×1−0.8187=57.7529%
πd=1−πu=1−57.7528%=42.2471%\pi_d=1-\pi_u=1-57.7528\%=42.2471\%πd=1−πu=1−57.7528%=42.2471%
At last, use risk-free rate to discount the expected value to present.
Node2=(0×57.7529%+5×42.2471%)e−5%=2.0093Node2=(0\times 57.7529\%+5\times 42.2471\%)e^{-5\%}=2.0093Node2=(0×57.7529%+5×42.2471%)e−5%=2.0093
Node3=(5×57.7529%+28.0811×42.2471%)e−5%=14.0317Node3=(5\times 57.7529\%+28.0811\times 42.2471\%)e^{-5\%}=14.0317Node3=(5×57.7529%+28.0811×42.2471%)e−5%=14.0317
Node1=(2.0093×57.7529%+14.0317×42.2471%)e−5%=6.7427Node1=(2.0093\times 57.7529\%+14.0317\times 42.2471\%)e^{-5\%}=6.7427Node1=(2.0093×57.7529%+14.0317×42.2471%)e−5%=6.7427
The European Put Option price is 6.74276.74276.7427

Two-Step Binomial Model - American

We need to determine if the option will be exercised at each node including Node 1.

Node1=(2.0093×57.7529%+17.691×42.2471%)e−5%=8.2133Node1=(2.0093\times 57.7529\%+17.691\times 42.2471\%)e^{-5\%}=8.2133Node1=(2.0093×57.7529%+17.691×42.2471%)e−5%=8.2133
Node1 should also be checked. In this case, if there is an early exercise in Node 1, payoff will be 5 which is less than 8.2133. Therefore, the option will not be exercised early in Node 1 and value of this American put option is 8.2133.

As time periods are added

Suppose that a binomial tree with nnn steps in its life T, if nnn approaches infinity, the length of each step approaches to
zero. A continuous binomial tree will be achieved and this is one of the ways that derive the Black-Scholes-Merton model.

Options on Other Assets

Options on stock indices with continuous dividend yield q

πu×S0×u+(1−πu)×S0×d=S0e(r−q)t\pi_u\times S_0 \times u+(1-\pi_u)\times S_0 \times d=S_0e^{(r-q)t}πu×S0×u+(1−πu)×S0×d=S0e(r−q)t
πu=e(r−q)t−du−d\pi_u=\frac{e^{(r-q)t}-d}{u-d}πu=u−de(r−q)t−d

Options on currencies with the domestic risk-free rate RDCR_{DC}RDC and foreign risk-free rate RFCR_{FC}RFC

πu=e(rDC−rFC)t−du−d\pi_u=\frac{e^{(r_{DC}-r_{FC})t}-d}{u-d}πu=u−de(rDC−rFC)t−d

Option on futures
πu=1−du−d\pi_u=\frac{1-d}{u-d}πu=u−d1−d