Option Pricing

Options and Futures

Zhiyu Fu

Roadmap

• Single-period Binomial Tree
  • Replicating portfolio
  • Risk-neutral probability
• Multi-period Binomial Model
• Black-Scholes Formula (just FYI)

Binomial Trees

• Real world: Stock prices can have infinitely many future paths
• Simplified world: Only two possible outcomes (states of the world): price goes Up or Down

Binomial Tree Representations

• A stock that can go up/down
• A bond that gives 1% return regardless of the state
• A call option with strike $106

Replicating Portfolio

Replicating Portfolio

• Replicating the payoff of a call option using the stock and bond

• The price of the call option has to be the same as the price of the replicating portfolio

• Let \Delta be the number of stocks, and B be the number of bonds, we have

  \left.\begin{align*}107\Delta+B=1\\ 97\Delta+B=0 \end{align*} \right\} \implies100\times\Delta+\frac{1}{1+1\%}\times B=?

 

\Large +

 

\Large =

Solving the Example

• Solve the system of equations of \Delta and B: \left.\begin{align*}107\Delta+B=1\\ 97\Delta+B=0 \end{align*} \right\} \implies\left\{ \begin{align*}\Delta &= 0.1\\ B &= -9.7 \end{align*} \right.

  • Interpretation: In this binomial world, the call option is equivalent to
    • holding 0.1 shares of stock, and
    • shorting (borrowing) 0.097 bonds

• The price of the call option is then given by:

  100\times\Delta+\frac{1}{1+1\%}\times B=100\times0.1+\frac{1}{1+1\%}\times(-9.7) = \frac{0.4}{1.01} \approx 0.396

Replicating a Generic Derivative

\small \left.\begin{align*}\Delta\times S_{u}+B=X_{u}\\ \Delta\times S_{d}+B=X_{d} \end{align*} \right\} \implies\left\{ \begin{align*}\Delta & =\frac{X_{u}-X_{d}}{S_{u}-S_{d}}\\ B & =X_{u}-\Delta S_{u}=X_{d}-\Delta S_{d} \end{align*} \right. \small X=S\times\Delta+\frac{1}{1+r}\times B

Exercise: Price a Future

• A standardized future contract with contract price F have the payoff of S_T-F
• To replicate the future, we have: \Delta = 1, \quad B = -F
• The value of the future contract is then given by: 1 \times S_0 - \frac{F}{1+r}
• A future contract has zero value at t=0, therefore: F = S_0 \times (1+r)

Risk-neutral Probability

Risk-neutral Probability

• Did we ever use the probability p? No.

• In fact, the physical probability p is not the relevant probability

• We can compute the implied “probability” such that the expected return of the stock equals the risk-free rate 1\%: \begin{aligned} 100 \times (1+1\%) &= 107 \times q + 97 \times (1-q) \\ q &= 40\% \end{aligned}

• This q is called the risk-neutral probability

Risk-neutral probability

The probability under which the expected return of the stock equals the risk-free rate r

Risk-neutral Probability vs. Physical Probability

• Suppose in the early example, the physical (actual) probability of the stock going up is p=50\%.
• The expected return of the stock is then higher than the risk-free rate r: r^S = \frac{107 * 0.5 + 97 * 0.5}{100} - 1 = 2\% > r = 1\%
• For an investor who doesn’t care which state (up or down) she is in, she would happily borrow from the bond to invest in the stock as much as possible.
• But then why the stock is only valued at 100 but not \frac{107 * 0.5 + 97 * 0.5}{1+1\%} = 101 so it offers the same risk-free return?
• Because investors value the same $1 dollar of future payoff differently in two states of the world. How different? By a valuation factor:

\underset{\text{Risk-neutral prob.}}{\underbrace{\left(\begin{array}{c} 40\% \uparrow\\ 60\% \downarrow \end{array}\right)}}=\underset{\text{Physical prob.}}{\underbrace{\left(\begin{array}{c} 50\% \uparrow\\ 50\% \downarrow \end{array}\right)}}\times\underset{\text{Value of \$1}}{\underbrace{\left(\begin{array}{c} \$0.8 \uparrow\\ \$1.2 \downarrow \end{array}\right)}}

The Stochastic Discount Factor (SDF)

\underset{\text{Risk-neutral prob.}}{\underbrace{\left(\begin{array}{c} 40\% \uparrow\\ 60\% \downarrow \end{array}\right)}}=\underset{\text{Physical prob.}}{\underbrace{\left(\begin{array}{c} 50\% \uparrow\\ 50\% \downarrow \end{array}\right)}}\times\underset{\text{Value of \$1}}{\underbrace{\left(\begin{array}{c} \$0.8 \uparrow\\ \$1.2 \downarrow \end{array}\right)}}

Interpretation of the valuation factor:

• We value $1 dollar differently in different states of the world
• In particular, $1 is more valuable when you are poor (Down state, $0.8) than when you are rich (Up state, $1.2)
  • “Aid in need outweighs luxury in plenty. （锦上添花不如雪中送炭）”
    
• Because the valuation factor is used to discount the future payoff, and it is different across different stochastic states of the world, it is also often called the stochastic discount factor (SDF)

Risk-neutral Pricing

Fundamental Asset Pricing Equation

An asset price equals to the expected value of the discounted payoffs:

S=\sum_{s\in\{up,down\}}\underset{\text{Risk-neutral probability}_{s}}{\underbrace{\text{Probability}_{s}\times\text{Value of \$1 in state }s}}\times\frac{1}{1+r}\times\text{Payoff}_{s}

For example, the price of the stock is given as:

\$100=\frac{1}{1+1\%}\left(\$107\times\underset{40\%}{\underbrace{50\%\times0.8}}+\$97\times\underset{60\%}{\underbrace{50\%\times1.2}}\right)

• To price a stock, we need to know the probability p of each state of the world, and the value of $1 in each state.
  • Beyond the scope of this course!
• We do relative pricing: Given the stock price, how do we price other derivatives?

Use Risk-neutral Pricing to Price Derivatives

Given the stock price and the risk-free rate, we can back out the implied risk-neutral probabilities, and use it to price any other derivative!

1. Solve the risk-neutral probability of the Up state q from the following equation: S_0 = \frac{1}{1+r}\left[S_u\times q + S_d\times (1-q)\right] \implies q = \frac{(S_0 \times (1+r)) - S_d}{S_u - S_d} = \frac{r - d}{u - d} We define S_u = S_0 \times (1+u) and S_d = S_0 \times (1+d) to have the last equality.

2. For a derivative that pays X_u in the Up state and X_d in the Down state, use the fundamental asset pricing equation, the price of the derivative is given by: X = \frac{1}{1+r}\left[X_u\times q + X_d\times (1-q)\right]

Example: Price the Call Option

• Compute the risk-neutral probability q = \frac{r - d}{u - d} = \frac{1\% - (-3\%)}{7\% - (-3\%)} = 40\%
• The call price would be: C = \frac{1}{1+1\%}\left[1\times40\% + 0\times60\%\right] = 0.396

The Arrow-Debreu Securities and State Prices

• The risk-neutral probability q discounted by 1+r is the state prices:

State Price

The price of an asset that pays $1 in a given state and $0 in all other states. Such assets are called Arrow-Debreu securities.

• The price of the derivative is then given by: X=\frac{1}{1+r}\sum_{s\in\{up,down\}}\text{X}_{s}\times q_{s}
• Alternative interpretation: The price of an asset X equals a replicating portfolio with X_u units of the Up Arrow-Debreu security and X_d units of the Down Arrow-Debreu security.

The Connections between Two Approaches

• Same no-arbitrage principle: If two portfolios offer the same payoffs under any state of the world, they should have the same price
• The replicating portfolio approach: Using the stock and bond to directly replicate the payoff of the derivative
• The risk neutral-probability approach:

X=\frac{1}{1+r}\sum_{s\in\{up,down\}}\text{X}_{s}\times q_{s} where q_s is the risk-neutral probability of the state s.

• By computing the state prices, we ask: If we use Arrow-Debreu securities to replicate the payoff of the stock and the bond, what would be the price of the Arrow-Debreu securities so there is no arbitrage?
• Then we use Arrow-Debreu securities to replicate the payoff of the derivative X, and its price has to equal to the price of the Arrow-Debreu securities