Option Valuation

BUSI 722: Data-Driven Finance II

Kerry Back, Rice University

Comparison pricing

• How do you decide if a house is fairly priced?
• An analogue to price / square foot for valuing companies is price-to-earnings.
• For valuing an option, we could use other options or we can just start with the underlying price.
• Option value = intrinsic value + adjustment for time and uncertainty.

Replication

• If we could create the option value at maturity by dynamically trading the underlying, then the value of the option should be the cost of the underlying portfolio.
• Call $\sim$ long underlying with leverage, so value of call = cost of underlying minus amount borrowed
• Put $\sim$ short underlying not fully collateralized, so value of put = implicit collateral

Replication in a single period two-state example

Simple Example

Suppose a stock priced at $100 will either go up by 10% or down by 10%.

Call Option

• Consider a call option with a strike of 105.
• It ends with a value of 5 if the stock goes to 110 and a value of 0 if the stock goes to 90.
• We want to find its value at the beginning.

Delta

• Define $\Delta$ as the difference in the option values divided by the difference in the stock values.
• This is $(5-0)/(110-90) = 1/4$.
• Here is the value of 1/4 share of the stock.

Debt

• Consider borrowing the PV of the bottom value from the previous figure.
• Suppose the interest rate is 5%. The PV of 22.50 is 21.43. Here is how the debt evolves.

Buy $\Delta$ shares on margin

The value of delta shares less the value of the debt is:

Conclusion

• In this simple example, we can get the call option payoff at maturity by investing 3.57, borrowing 21.43, and buying 1/4 share.
• The value of the call must be 3.57.

Risk-neutral probability

• If there were no risk premium, the call value would be the expected value discounted at the risk-free rate:

$$C = \frac{p \times \text{\$}5 + (1-p)\times \text{\$}0}{1.05}$$

• where $p=$ up probability. The stock price would also be the discounted expected value:

$$\text{\$}100 = \frac{p \times \text{\$}110 + (1-p) \times \text{\$}90}{1.05} \quad \Leftrightarrow \quad S = \frac{p(1+r_u)S + (1-p)(1+r_d)S}{1+r_f} $$

• Solve the stock equation for $p$:

$$p = \frac{r_f - r_d}{r_u - r_d} = \frac{0.05 - (-0.1)}{0.1 - (-0.1)} = \frac{.15}{.2} = 0.75$$

• Substitute into the call option equation. Obtain

$$C = \frac{0.75 \times \text{\$}5 + 0.25\times \text{\$}0}{1.05} = \text{\$}3.57$$

• Why does this work? Delta hedge argument didn't depend on risk preferences, so we can act as if investors don't require risk premia.

Risk-neutral probability in two-period example

Stock dynamics

• A $100 stock goes up by 5% or down by (1/1.05-1) = -4.76% in each of two periods.
• Interest rate is 3% per period

Risk-neutral probability

The risk-neutral probability of "up" is

$$ p = \frac{r_f - r_d}{r_u - r_d} = \frac{0.03 - (-0.0476)}{0.05 - (-0.0476)} = 0.795$$

Call option with strike = 105

The call evolves as

Discounting the expected call value (using prob of up = 0.795) at the risk-free rate yields

Exercise

Price a call with a strike of 95.

Calibration

• Estimate $\sigma=$ std dev of annual stock return
• Find $r_f =$ annualized continuously compounded interest rate = log(1+annual rate)
• $T=$ time to maturity of an option in years
• $N=$ number of periods in a binomial model

• $\Delta t = T/N=$ period length
• Set the up rate of return as $u = e^{\sigma\sqrt{\Delta t}}-1$ and set $d=1/(1+u)-1$ as in the two-period example
• Set the 1-period interest rate as $r=e^{r_f \Delta t}-1$
• The risk-neutral probability of "up" is

$$p = \frac{r-d}{u-d} = \frac{e^{r_f \Delta t} - e^{-\sigma \Delta t}}{e^{\sigma \Delta t} - e^{-\sigma \Delta t}}$$

Exercise

Price a six-month call option using a two-period model.

• $\sigma = 0.4$
• $r_f = 0.05$
• $T = 0.5$
• $N = 2$
• $S = 100$
• $K = 95$

Black-Scholes formulas

• As $N \rightarrow \infty$ the distribution of the stock price at date $T$ converges to lognormal, meaning that the log stock price has a normal distribution.
• The values of European options converge to the Black-Scholes formulas.
• More about American options (and dividends) coming.

Black-Scholes call formula

import numpy as np 
from scipy.stats import norm 

def BS_call(S, K, T, sigma, r, q=0):
    d1 = np.log(S/K) + (r-q+0.5*sigma**2)*T
    d1 /= sigma*np.sqrt(T)
    d2 = d1 - sigma*np.sqrt(T)
    N1 = norm.cdf(d1)
    N2 = norm.cdf(d2)
    return np.exp(-q*T)*S*N1 - np.exp(-r*T)*K*N2

Black-Scholes put formula

def BS_put(S, K, T, sigma, r, q=0):
    d1 = np.log(S/K) + (r-q+0.5*sigma**2)*T
    d1 /= sigma*np.sqrt(T)
    d2 = d1 - sigma*np.sqrt(T)
    N1 = norm.cdf(-d1)
    N2 = norm.cdf(-d2)
    return np.exp(-r*T)*K*N2 - np.exp(-q*T)*S*N1

Example

S, K, T, sigma, r = 100, 95, 0.5, 0.4, 0.05
print(f"Value of call option is ${BS_call(S, K, T, sigma, r):.2f}")
print(f"Value of put option is ${BS_put(S, K, T, sigma, r):.2f}")

Value of call option is $14.89
Value of put option is $7.55

Dividends and early exercise

Dividends

• To use binomial model, build a tree for "stock minus PV of future dividends," the future being until the option maturity.
• Try to set tree nodes near ex-dividend dates
• Everything else as before. As time passes, dividends get paid and "stock minus PV of future dividends" becomes "stock."

Early exercise

• It may be optimal to exercise an American put at any time (though just after ex-dividend date is better than just before).
• It may be optimal to exercise an American call just before a dividend is paid.
• To use binomial model, replace "discounted risk-neutral expectation of option value" with max of discounted risk-neutral expectation and intrinsic value.