Options Pricing Models

Options Pricing Models

Option Value Bounds

• Upper Bound: The value of a call option cannot exceed the stock price (So).
• Lower Bound: The value of a call option cannot be less than the difference between the stock price and the exercise price (So - E) or 0, whichever is greater.
• The lower bound is applicable only when So > E.

Factors Determining Option Value

• Stock Price (So): Positive correlation; higher stock price generally increases call option value.
• Exercise Price (E): Negative correlation; higher exercise price decreases call option value.
• Expiration Date (t): Positive correlation; longer time to expiration increases option value.
• Stock Price Variability ($\sigma^2$): Positive correlation; higher volatility increases option value.
• Interest Rate (r): Positive correlation; higher interest rates increase call option value.
• The call option price $C<em>0$ is a function of these variables: $C</em>1 = f[S<em>0, E, \sigma^2, t, r</em>1]$

Binomial Model - Option Equivalent Method

• Single-Period Binomial Model: Assumes the stock price can take two possible values next year, uS or dS, where uS > dS.
• Assumptions:
  • B can be borrowed or lent at a risk-free rate (1 + r) = R.
  • d < R < u
  • E is the exercise price.
  • $C_u = MAX(uS - E, 0)$
  • $C_d = MAX(dS - E, 0)$
• Portfolio Construction: Construct a portfolio with A shares of stock and borrowing B rupees.
• Portfolio Payoffs:
  • Stock price rises: $A \cdot uS - RB = C_u$
  • Stock price falls: $A \cdot dS - RB = C_d$
• Spreads:
  • Spread of possible option prices: $C<em>u - C</em>d$
  • Spread of possible share prices: $S(u - d)$
• Calculating A and B:
  • $A = \frac{C<em>u - C</em>d}{S(u - d)}$
  • $B = \frac{dC<em>u - uC</em>d}{(u - d)R}$
• Call Option Value: Since the portfolio has the same payoff as the call option:
  • $C = AS - B$

Illustration of Binomial Model

• Given:
  • S = 200, u = 1.4, d = 0.9
  • E = 220, r = 0.10, R = 1.10
• Calculations:
  • $C_u = MAX(uS - E, 0) = MAX(1.4 \cdot 200 - 220, 0) = 60$
  • $C_d = MAX(dS - E, 0) = MAX(0.9 \cdot 200 - 220, 0) = 0$
  • $\Delta = \frac{C<em>u - C</em>d}{(u - d)S} = \frac{60 - 0}{0.5 \cdot 200} = 0.6$
  • $B = \frac{dC<em>u - uC</em>d}{(u - d)R} = \frac{0.9 \cdot 60 - 1.4 \cdot 0}{0.5 \cdot 1.10} = 98.18$
• Interpretation: 0.6 of a share + 98.18 borrowing.
• Repayt: 98.18 * 1.10 = 108
• Portfolio Value:
  • If stock price rises: 1.4 x 200 x 0.6 - 108 = 60 = $C_u$
  • If stock price falls: 0.9 x 200 x 0.6 - 108 = 0 = $C_d$
  • $C = AS - B = 0.6 \cdot 200 - 98.18 = 21.82$

Binomial Model - Risk-Neutral Method

• Concept: Establishes the call option price without specific knowledge of investors' risk attitudes.
• Steps:
  • Calculate the probability of a rise in a risk-neutral world.
  • Calculate the expected future value of the option.
  • Convert it into its present value using the risk-free rate.
  • P = Probability

Pioneer Stock Example (Risk-Neutral Valuation)

• Given:
  • Rise: 40% to 280
  • Fall: 10% to 180
  • Expected Return = 10%
• Calculations:
  • Probability of rise (p): [p x 40%] + [(1 - p) x -10%] = 10% => p = 0.4
  • Expected Future Value of the Option: (0.4 x 60) + (0.6 x 0) = 24
    (Where 60 and 0 are the call option values at the rise and fall, respectively)
  • Present Value of the Option: 24 / 1.10 = 21.82

Black-Scholes Model

• Formula:
  • $C<em>0 = S</em>0N(d<em>1) - Ee^{-rt}N(d</em>2)$
• Variables:
  • $N(d)$ = Value of the cumulative normal density function
  • $S_0$ = Current stock price
  • E = Exercise price
  • r = Continuously compounded risk-free annual interest rate
  • t = Time to expiration
  • $\sigma$ = Standard deviation of the continuously compounded annual rate of return on the stock
• Intermediate Calculations:
  • $d<em>1 = \frac{ln(\frac{S</em>0}{E}) + (r + \frac{\sigma^2}{2})t}{\sigma \sqrt{t}}$
  • $d<em>2 = d</em>1 - \sigma \sqrt{t}$

Black-Scholes Model - Illustration

• Given:
  • $S_0$ = 60, E = 56, t = 0.5, $\sigma$ = 0.30, r = 0.14
• Step 1: Calculate $d<em>1$ and $d</em>2$
  • $d_1 = \frac{ln(\frac{60}{56}) + (0.14 + \frac{0.30^2}{2})0.5}{0.30 \sqrt{0.5}} = 0.7614$
  • $d<em>2 = d</em>1 - \sigma \sqrt{t} = 0.7614 - 0.30 \sqrt{0.5} = 0.5493$
• Step 2: Find N($d<em>1$) and N($d</em>2$)
  • N($d_1$) = N(0.7614) = 0.7768
  • N($d_2$) = N(0.5493) = 0.7086
• Step 3: Calculate the Present Value of Exercise Price
  • $Ee^{-rt} = 56 \cdot e^{-0.14 \cdot 0.5} = 56 \cdot e^{-0.07} = 52.21$
• Step 4: Calculate Call Option Value
  • $C_0 = 60 \cdot 0.7768 - 52.21 \cdot 0.7086 = 46.61 - 37.00 = 9.61$

Assumptions of Black-Scholes Model

• The call option is a European option (can only be exercised at expiration).
• The stock price is continuous and lognormally distributed.
• There are no transaction costs or taxes.
• There are no restrictions on or penalties for short selling.
• The stock pays no dividends.
• The risk-free interest rate is known and constant.

Adjustment for Dividends - Short-Term Options

• Adjusted Stock Price:
  • $S' = S - \sum \frac{Div_t}{(1+r)^t}$
• Value of Call:
  • $C = S'N(d<em>1) - Ee^{-rt}N(d</em>2)$
• d1 Calculation:
  $d_1 = \frac{ln(\frac{S'}{E}) + (r + \frac{\sigma^2}{2})t}{\sigma \sqrt{t}}$

Adjustment for Dividends - Long-Term Options

• Formula:
  • $C = S \cdot e^{-yt}N(d<em>1) - E \cdot e^{-rt}N(d</em>2)$
  • $d_1 = \frac{ln(\frac{S}{E}) + (r - y+ \frac{\sigma^2}{2})t}{\sigma \sqrt{t}}$
  • $d<em>2 = d</em>1 - \sigma \sqrt{t}$
• Where:
  • y = dividend yield
• Adjustments:
  • Discounts the stock value to present at the dividend yield, reflecting the expected drop in value due to dividend payments.
  • Offsets the interest rate by the dividend yield to reflect the lower cost of carrying the stock.

Put-Call Parity - Revisited

• Just Before Expiration:
  • $C<em>0 = S</em>0 + P_0 - E$
• With Time Left:
  • $C<em>0 = S</em>0 + P_0 - Ee^{-rt}$
• Applications:
  • Used to establish the price of a put option.
  • Determine whether the put-call parity is working.