CAPM Pricing for a Binomial Call

Article Quant Q&A · Author: Mango

Summary

The document shows how to price a call in a one-period binomial model by applying CAPM, then compares the result with the standard replicating-portfolio calculation. It specifies a stock’s up and down outcomes, probabilities, and risk-free rate, computes the stock’s expected excess return and variance, and derives the option’s covariance with the stock. That covariance gives the call’s beta, which is inserted into CAPM to solve for its initial price.

For the stated inputs and strike, the CAPM calculation matches the price obtained by choosing cash and stock positions that reproduce the call’s two possible payoffs. This illustrates a connection between replication and equilibrium pricing in a simple setting. The example assumes a single period, two possible stock outcomes, known probabilities, and a risk-free asset. It does not establish that CAPM generally prices options correctly; the calculation relies on this specific model and its assumptions.

Key ideas

A one-period binomial call can be priced by matching its payoff in both stock outcomes with cash and stock.
The stock is treated as the market portfolio when calculating the call’s CAPM beta.
The option beta is derived from its covariance with the stock divided by the stock variance.
In the numerical example, CAPM and replication produce the same call price.
The illustration depends on restrictive single-period binomial assumptions.

Tags

Options Derivatives pricing Statistics

Full text

# Using CAPM to find the price of an option


# Using CAPM to find the price of an option












I was reading a textbook about finding the price of an option on a one-period binomial model. The textbook way of doing it is to replicate the option with cash and stock for $t=T$, and then calculate the portfolio value at $t=0$.

For example: $r=0.1, s=10, u=1.2, d=0.7, p_u=0.2,p_d=0.8$

we want to find the price of a call option with strike $K=9$

we can replicate this option with $x$ units of cash and $y$ units of stock. we get:

$x(1+r)+yS_u=(S_u-K)^+, x(1+r)+yS_d=(S_d-K)^+$

plug in numbers, we get:

$1.1x+12y=3, 1.1x+7y=0$

solution is: $y=\frac{3}{5}, x=\frac{21}{5.5}$

portfolio value at $t=0$ is $x+ys=\frac{12}{5.5}$

if we use CAPM to solve this problem, we firstly find the market return (treat the stock as the market) $\mu_m$ and market variance $\sigma_m^2$

Let the call option price at $t=0$ be $c$

$\mu_m=s(p_uu+p_dd)-s=10(1.2*0.2+0.8*0.7)-10=-2$

$\sigma_m^2=p_u(s(u-1)-\mu_m)^2+p_d(s(d-1)-\mu_m)^2=0.2*(2-(-2))^2+0.8*(-3-(-2))^2=3.2+0.8=4$

Now calculate the covariance of market and call option $cov_{m,c}$

firstly calculate the average return of the option $\mu_c$:

$\mu_c=p_u(su-K)^++p_d(sd-K)^+-c=0.2(12-9)+0.8*0-c=0.6-c$

and then

$cov_{m,c}=p_u(s(u-1)-\mu_m)((su-K)^+-c-\mu_c)+p_d(s(d-1)-\mu_m)((sd-K)^+-c-\mu_c) =0.2(2-(-2))((3-c)-(0.6-c))+0.8(-3-(-2))(0-c-(0.6-c))=0.2*4*2.4+0.8*(-1)*(-0.6)=2.4$

we get $\beta=\frac{cov_{m,c}}{\sigma_m^2}=0.6$

now use CAPM formula we get:

$\mu_c=rc+\beta(\mu_m-rs)$ which is:

$0.6-c=0.1c+0.6(-2-0.1*10)$

solve for $c$ and we get

$c=\frac{12}{5.5}$

So we get the same result using CAPM.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.

Pricing a Binomial Call Option with CAPM

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