One-period binomial option valuation

What you will be able to do

After this lesson, you should be able to:

construct the two-position portfolio that matches both state payoffs;
calculate the tree’s risk-neutral up-state weight;
show that replication and discounted risk-neutral expectation give the same value.

Two states, two traded positions

Let the underlying be worth $\explain{derivative-underlying-value}{S_0}$ at valuation-time and take exactly the local values binomial-up-underlying-value $\explain{binomial-up-underlying-value}{S^{\mathrm{up}}}$ or binomial-down-underlying-value $\explain{binomial-down-underlying-value}{S^{\mathrm{down}}}$ at the option-expiry-time $\explain{option-expiry-time}{T_{\mathrm{opt}}}$ . A claim pays local binomial-up-claim-payoff $\explain{binomial-up-claim-payoff}{X^{\mathrm{up}}}$ or binomial-down-claim-payoff $\explain{binomial-down-claim-payoff}{X^{\mathrm{down}}}$ .

The local binomial-hedge-units $\explain{binomial-hedge-units}{\Delta}$ is the number of units of the underlying that matches the difference between the two claim payoffs:

\explain{binomial-hedge-units}{\Delta} =\frac{\explain{binomial-up-claim-payoff}{X^{\mathrm{up}}} -\explain{binomial-down-claim-payoff}{X^{\mathrm{down}}}} {\explain{binomial-up-underlying-value}{S^{\mathrm{up}}} -\explain{binomial-down-underlying-value}{S^{\mathrm{down}}}}

(4.4.1)

The local binomial-risk-free-expiry-cash is then chosen so that the portfolio matches the claim payoff in the down state:

\explain{binomial-risk-free-expiry-cash}{M_{T_{\mathrm{opt}}}} =\explain{binomial-down-claim-payoff}{X^{\mathrm{down}}} -\explain{binomial-hedge-units}{\Delta}\, \explain{binomial-down-underlying-value}{S^{\mathrm{down}}}

(4.4.2)

So the local binomial-claim-value is the value of the replicating portfolio at time zero:

\explain{binomial-claim-value}{V_0} =\explain{binomial-hedge-units}{\Delta}\, \explain{derivative-underlying-value}{S_0} +\explain{discount-factor}{D(0,\explain{option-expiry-time}{T_{\mathrm{opt}}})} \explain{binomial-risk-free-expiry-cash}{M_{T_{\mathrm{opt}}}}

(4.4.3)

The same value with risk-neutral weights

The local binomial-risk-neutral-up-weight $\explain{binomial-risk-neutral-up-weight}{q}$ is chosen so that the model reproduces the current value of the underlying:

\explain{binomial-risk-neutral-up-weight}{q} =\frac{\explain{derivative-underlying-value}{S_0}/\explain{discount-factor}{D}(0,\explain{option-expiry-time}{T_{\mathrm{opt}}}) -\explain{binomial-down-underlying-value}{S^{\mathrm{down}}}} {\explain{binomial-up-underlying-value}{S^{\mathrm{up}}} -\explain{binomial-down-underlying-value}{S^{\mathrm{down}}}}

(4.4.4)

Under the risk-neutral-probability-measure $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ , the claim value is the discounted risk-neutral expected payoff:

\explain{binomial-claim-value}{V_0} =\explain{discount-factor}{D(0,\explain{option-expiry-time}{T_{\mathrm{opt}}})} \left(\explain{binomial-risk-neutral-up-weight}{q}\, \explain{binomial-up-claim-payoff}{X^{\mathrm{up}}} +\left(1-\explain{binomial-risk-neutral-up-weight}{q}\right) \explain{binomial-down-claim-payoff}{X^{\mathrm{down}}}\right)

(4.4.5)

Hull derives the one-step hedge and then shows that its no-arbitrage value equals the discounted risk-neutral expected payoff [1].

Worked call example

Let $\explain{derivative-underlying-value}{S_0}=100$ , $\explain{binomial-up-underlying-value}{S^{\mathrm{up}}}=120$ , $\explain{binomial-down-underlying-value}{S^{\mathrm{down}}}=80$ , $\explain{discount-factor}{D(0,\explain{option-expiry-time}{T_{\mathrm{opt}}})}=1/1.05$ , and let the option-strike-price be $\explain{option-strike-price}{K}=100$ . The call payoffs are $\explain{binomial-up-claim-payoff}{X^{\mathrm{up}}}=20$ and $\explain{binomial-down-claim-payoff}{X^{\mathrm{down}}}=0$ .

Then $\explain{binomial-hedge-units}{\Delta}=0.5$ , $\explain{binomial-risk-free-expiry-cash}{M_{T_{\mathrm{opt}}}}=-40$ , and $\explain{binomial-risk-neutral-up-weight}{q}=0.625$ . Both methods give the same claim value:

\explain{binomial-claim-value}{V_0} =0.5(100)+\frac{-40}{1.05} =\frac{0.625(20)+0.375(0)}{1.05} =11.9047619

(4.4.6)

The negative cash position is borrowing. It is not a negative option value.

From one period to many periods

Repeating this valuation one period at a time, from expiry backward, is backward induction on a multi-period tree. The later bond-option lessons need a discount factor at each node and an additional state for survival and default. An option with a knockout triggered by default cannot be valued from the distribution at expiry alone, because the option can terminate before expiry.

Knowledge check 4.4.1 One-period binomial option valuation

Model boundary and review note

This complete two-state model has one non-income-paying traded underlying, one risk-free account, and deterministic discounting. It does not estimate real-world probabilities or volatility. It does not model multiple time steps, stochastic rates, default, or transaction and funding effects.

The two valuation methods are implemented together in a pure tested domain module. The hand-calculated answer and implementation both remain subject to independent human quantitative review.

References

Hull, Options, Futures, and Other Derivatives (8th ed., 2012). Ch. 12 §§12.1-12.2, printed pp. 253-259, one-step binomial replication and risk-neutral valuation. draft ↩

Notation used on this page (19)

V_0

Binomial claim valuedraft

Time-zero value obtained identically from the replicating portfolio and risk-neutral expected payoff.

Units: stated currency at valuation time

X^{\mathrm{down}}

Binomial down claim payoffdraft

Signed expiry payoff of the claim when the underlying takes its down-state value.

Units: stated currency at expiry

S^{\mathrm{down}}

Binomial down underlying valuedraft

The lower of the two possible underlying values at the one-period expiry.

Units: stated currency per unit of underlying at expiry

\Delta

Binomial hedge unitsdraft

Signed units of underlying in the portfolio that replicates the two claim payoffs.

Units: units of underlying per claim

M_{T_{\mathrm{opt}}}

Binomial risk-free expiry cashdraft

Signed expiry cash amount held with the underlying units to reproduce both claim payoffs.

Units: stated currency at expiry

q

Binomial risk-neutral up weightdraft

Model-implied pricing probability assigned to the up state by the no-arbitrage tree. It is dimensionless and lies between zero and one when the tree is arbitrage-consistent.

Units: probability between zero and one

X^{\mathrm{up}}

Binomial up claim payoffdraft

Signed expiry payoff of the claim when the underlying takes its up-state value.

Units: stated currency at expiry

S^{\mathrm{up}}

Binomial up underlying valuedraft

One of the two possible underlying values at the one-period expiry.

Units: stated currency per unit of underlying at expiry

A(0,t)

Accumulation factordraft

Grows one current unit over a stated future horizon under the selected rate model.

Units: dimensionless currency-units per current currency-unit

m

Compounding frequencydraft

Number of equal compounding periods per year under the stated rate convention, a positive integer fixed by the model convention.

Units: compounding periods per year

S_t

Derivative underlying valuedraft

Value at model time t of one unit of the asset or claim named as the derivative's underlying; a positive quoted value, while a position in the underlying carries its own signed quantity.

Units: stated currency per unit of underlying at model time

D(0,t)

Discount factordraft

Converts one deterministic future unit into value at valuation time.

Units: current currency-units per future currency-unit

\mathbb{E}

Expectationdraft

Operator returning the average of a random quantity, each outcome weighted by its probability under a stated measure; a superscript names that measure when more than one is in play.

\explain{expectation}{\mathbb{E}}[\explain{expectation.random-variable}{X}]=\int_{\explain{expectation.sample-space}{\Omega}} \explain{expectation.random-variable}{X}(\explain{expectation.outcome}{\omega})\,d\explain{expectation.probability-measure}{\mathbb{P}}(\explain{expectation.outcome}{\omega})

Symbols: $X$ random variable; $\Omega$ sample space; $\omega$ outcome; $\mathbb{P}$ probability measure(dimensionless probability weights between zero and one)

Units: value units

j^{(m)}

Nominal annual ratedraft

Annualized rate quote that must be paired with its compounding frequency.

Units: decimal rate per year

T_{\mathrm{opt}}

Option expiry timedraft

Future model time when a European option's exercise decision and payoff are determined; a contract date shared by holder and writer, distinct from a bond maturity.

Units: model-years from the stated valuation time

K

Option strike pricedraft

Contractual price per unit of underlying used to determine the option's exercise payoff; a positive contractual amount, with payoff signs depending on call or put and holder or writer perspective.

Units: expiry-time currency per unit of underlying

r_m

Periodic ratedraft

Rate applied once in each compounding period under the stated convention, derived from the stated nominal annual quote in this model.

Units: decimal per compounding period

\mathbb{Q}

Risk-neutral probability measuredraft

Gives model pricing weights under which discounted traded prices satisfy the martingale condition; it prices payoffs relative to a stated numeraire and is not a forecast of actual event frequencies.

Units: dimensionless probability weights between zero and one

0

Valuation timedraft

Common origin from which later model times and present values are measured.

Units: years from the valuation date

One-period binomial option valuation

What you will be able to do

Two states, two traded positions

The same value with risk-neutral weights

Worked call example

From one period to many periods

Knowledge check 4.4.1 One-period binomial option valuation

DirectQuestion 1Not answered

TransferQuestion 2Not answered

DirectQuestion 3Not answered

TransferQuestion 4Not answered

Model boundary and review note

References