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### Option Pricing: An Introduction

Ravi Mukkamala

Popular Models

- Black-Scholes options pricing
- Binomial model (Cox, Ross, and Rubinstein)
- Implied volatility tree model (Deman and Kani)
- Implied binomial tree (Rubinstein)

What are options?

- Options are contracts that give their holders right to buy or sell an underlying asset at a fixed price.
- Call option gives the holder the right to buy
- Put option gives the holder the right to sell
- Value of an option: what the contract is intrinsically worth
- Price of an option: what someone pays to have the contract

Specifications of an option

- Underlying instrument (e.g., stock S)
- Contract size (e.g., how may shares)
- Strike price (Price at which option can be exercised)
- Settlement date (Date when option contract is written)
- Type (call or put)
- Expiration date
- Style (American, European, Asian)
- Premium (Price paid to own the option contract)

(European option: Option to exercise at strike price at expiration only

American option: Option to exercise at strike price any time prior to the expiration

Asian option: Option to exercise at average price during the period prior to expiration)

Option Pricing Theory

- To determine fair value for options
- Provides vital information on hedging the option by allowing the writer of the option to calculate hedge parameters---provide a profile of the riskiness of the option

Long and Short positions

- An investor is long a security if he or she owns it. The value of this position is the current value of the security purchased.
- Selling a security short means borrowing the security and then selling the borrowed security in the open market. (Of course, the short-seller is required to buy back the security at a later date to return it from the borrower.)
- An investor is short a security if he or she has sold the security short. The value of this position is –S where S is the current value of the security.

Some basics

Converting standard deviation from one time period to another

- Commercial banks, for example, typically calculate a daily VAR, asking themselves how much they can lose in a day; pension funds, on the other hand, often calculate a monthly VAR; option pricing refers to annualized VAR
- The standard deviation of stock returns tends to increase with the square root of time (A classic idea in finance).
- Example: If the standard deviation of daily returns is 2.64% and there are 20 trading days in a month (T = 20), then the monthly standard deviation is represented by the following: 2.64% * √20.
- Similarly, Annualized std. Dev. Of the returns of the same would be 2.64% * √250

Basics (cont.)

- Present value (at time t) of money K to be received at time T (continuously compounded): Ke-r(T-t)
- Example: Suppose you were to receive $1000 after 6 months where interest rate is 5% per annum, then today it is worth, 1000e-0.05*(1/2) = $987.58.

Basics (Cont.)

ARBITRAGE

- Opportunities to make risk-free profits
- Example: The same asset is traded at two different prices at two exchanges: buy an asset in one and sell in the other.
- Assumption: Arbitrage is not possible

Using arbitrage argument

- Portfolio 1: At time t, buy a call option and sell a put option (strike price: K, risk-free interest r, Expiration at time T; spot price is S). Its value at time t is: C-P
- Portfolio 2: Long the stock (buy the stock) and sell a riskless bond maturing at time T to K. Its value at time t is: S-Ke-r(T-t)
- At time T: Case i: ST>K: Value of portfolio 1: (ST-K)-0; value of portfolio 2: (ST-K);
- Case ii: ST<K: Value of portfolio 1: 0-(K-ST) = ((ST-K); value of portfolio 2: (ST-K);
- Since terminal value of both portfolios is the same at time T, the initial value of both portfolios should also be the same;
- So C-P = S-Ke-r(T-t)
- This is called put-call parity for European options

Basics (cont.)

Rate of Return on an investment

- If value of an asset is St at time t, and it is St+Δt at time t+Δt, then annualized return is: r = (1/Δt)ln(St+Δt /St)
- How: Future value of St at time t+Δt is St*erΔt; St+Δt = St*erΔt; so r = (1/Δt)ln(St+Δt /St)

Basics (cont.)

- Assume: Continuous random variables---easier to deal with mathematically
- Normal distribution (Mean and standard distribution)

Risk-neutral World

- A risk neutral world is characterized as a place where the investors require no risk premium for their investments (i.e., the investors always demand only the risk free rate of interest as the average expected return on investment). In such an investment environment, the reason investors are neutral towards risk is because on an average there is no risk.
- Expected appreciation of every asset is given by the risk-free rate of growth.

Computing Cumulative Normal Distribution Function

- N(x) is the Cdf of x. We refer to a standard normal distribution with mean = 0, and std. Dev. = 1
- To convert a normally distributed random variable X to a std. Normal dist. function Y, we let Y = (X-mean)/std.dev.
- Approx formulae for N(x):

N(x) = 1-0.5*(1+c1x+c2x2+c3x3+ c4x4)-4 if x>= 0

1-N(-x) otherwise

c1=0.196854; c2 = 0.115194; c3 = 0.000344; c4=0.019527

Geometric Brownian Motion Model

- Since the theory applies directly to stock returns, and hence as a percentage change, it is referred to as geometric (as opposed to arithmetic)
- “Displacement of particle subject to random collisions over a long period of time is normally distributed with mean and std. Dev. depending only on the amount of time that has passed.
- Basic assumption of the model:
- The return on a stock price between now and some very short time later (∆t) is normally distributed. The mean of the distribution is μ ∆t, and the standard deviation is σ√∆t.
- μ instantaneous expected return (annualized); σ is instantaneous std. dev. (annualized)
- Stocks are traded continuously; prices move continuously;
- Conclusions of the model: “If S is a stock price with spot price St at current time t, and S follows a geometric Brownian motion with instantaneous mean μ and instantaneous std. dev. σ, then the return on S between now (t) and a future time T is normally distributed with mean (μ – σ2/2)(T-t) and std. dev. σ √(T-t)

Example

- Suppose stock prices of S on 4 successive days is: 50, 51, 48, 47.
- The respective % returns are: 1/50*100, -3/51*100, and -1/48*100; or 2%, -5.88%, and

-2.08%.

- Average daily returns: -1.99%; average annualized instantaneous mean returns μ = 365*(-1.99)= -7.2635 or -726.35%
- Std. Dev. 3.94%; annualized instantaneous std. dev σ = 0.0394* √365 = 0.752 or 75.20%
- Using GBM, Mean return over a year = (-7.2635-0.752*0.752/2) = -7.5462 or -754.62%

Geometric Brownian Motion Model (cont.)

- When returns of a stock follow normal distribution, the future stock price has log-normal distribution.
- In other words, log(future stock price) follows a normal distribution.
- For this reason, Geometric Brownian Motion is also referred to as log-normal process

Brownian Motion and Call Options

- A call is in-money if ST≥ K
- Return on S from time t to T is given by: ln(ST/St)
- This is normally distributed with mean (T-t)(r-σ2/2) and std. dev. of σ√(T-t)
- So (ln(ST/St)- (T-t)(r-σ2/2))/[σ√(T-t)] has standard normal distribution with mean 0 and std. dev of 1.
- From this, we can easily derive that

Prob(ST≤ K) = N([ln(St/K)+(T-t)(r-σ2/2))/σ√(T-t)])

Evidence against Stocks having Geometric Brownian Motion

- Large movements in stock prices are more likely than a normally distributed stock price model prediction
- Stock returns do not scale as they are supposed to as per the model---monthly and quarterly volatilities are higher than annual volatility; on the other hand, daily volatilities are lower than annual volatilities

Black-Scholes Options Pricing

- Two components: An option and a strategy to hedge the option
- Basics used: Any riskless portfolio must have a return close to short-term interest rates on risk-less securities
- If there is a formula that tells how the value of a call option depends on the price of the underlying stock, the volatility of the stock, the exercise price and maturity, and the interest rate; then such a formula can be used to form a hedged position that is close to risk-less. This gives the option formula.

Self-financing, Replicating Hedging Strategies

- Example hedge: “A casino has a game---when three coins are flipped it all turn out to be H (head), then the gambler is paid $1.” Fair-price of this bet is 1/8*$1 or $0.125. What is a gambler wants to play for $100,000 by betting $12,500? Is the casino worried of not having $100,00 to pay out?
- It can self-finance the game by having a hedge: It contacts a bet broker and says that for every “H” of a coin flip it will bet $0.50 to receive $1.00. This is the hedge.
- How does it work? Gambler pays the casino the initial $12,500 bet. It pays this to the bet maker.
- If the 1st coin flip is “H”, then bet maker pays $25,000 to the casino. If it is “T”, the gambler lost his $12,500 and the casino just lost his money. Thus, no-gain/no-loss for casino.
- If 1st was “H”, then casino bets $25,000 on the next flip. If it is also “H”, then casino receives $50,000. Else, gambler lost his money. Casino has no-gain/no-loss
- If 2nd was also “H”, then casino bets $50,000 on the 3rd flip. If it is also “H”, then casino receives $100,000 from bet maker, which is in turn paid to the gambler. No-gain/no-loss for the casino.
- Thus, the hedging strategy is also self-financing.
- The strategy was replicating: In every scenario, against the risk, the casino was covered.

Hedging in Black-Scholes Model

- Suppose you are writing (short) a call option
- If the price of the underlying stock goes above the strike price, you have to make a payout of ST-K
- So you can long on the underlying securities and (short) on some riskless bonds.
- How much of each to put into this hedging portfolio determines the option’s value (due to self financing property)

Dynamic Hedging

- Refers to trading strategy carried out by maintaining the portfolio, throughout the life of the option.
- (i) It replicates the payoff of the option (ii) It has a fixed and known total cost.
- For short call---the portfolio consist of long on stock and short on a bond.
- The value of the hedging portfolio is equal to the value of the option at any time.
- When market parameters change (e.g., stock price), the portfolio weights also change, and hence rebalancing
- Delta (Δ) of an option: Rate of change of the option’s value w.r.t. underlying stock price.

Black-Scholes Formula

- For European options,

current value of Ct = ΔtSt – e-r(T-t)Bt

Where (T-t) is the time to expiration and St is the current price of the underlying stock.

Bt is equal to K (strike price) times the probability of expiring in the money.

Recall that, Prob(ST≤ K) = N([ln(St/K)+(T-t)(r-σ2/2))/σ√(T-t)])

Δt = N([ln(St/K)+(T-t)(r+σ2/2))/σ√(T-t)])

Bt = K. N(([ln(St/K)+(T-t)(r-σ2/2))/σ√(T-t)])

http://www.anthony-vba.kefra.com/vba/vba6.htm

- ****************************************************************************'* Cumulative Standard Normal Distribution *'* (This function provides similar result as NORMSDIST( ) on Excel) *'****************************************************************************Function SNorm(z) c1 = 2.506628 c2 = 0.3193815 c3 = -0.3565638 c4 = 1.7814779 c5 = -1.821256 c6 = 1.3302744 If z > 0 Or z = 0 Then w = 1 Else: w = -1 End If y = 1 / (1 + 0.2316419 * w * z) SNorm = 0.5 + w * (0.5 - (Exp(-z * z / 2) / c1) * _ (y * (c2 + y * (c3 + y * (c4 + y * (c5 + y * c6))))))End Function'**********************************************************************'* Black-Scholes European Call Price Computation *'**********************************************************************Function Call_Eur(s, x, t, r, sd) Dim a As Single Dim b As Single Dim c As Single Dim d1 As Single Dim d2 As Single a = Log(s / x) b = (r + 0.5 * sd ^ 2) * t c = sd * (t ^ 0.5) d1 = (a + b) / c d2 = d1 - sd * (t ^ 0.5) Call_Eur = s * SNorm(d1) - x * Exp(-r * t) * SNorm(d2)End Function'*********************************************************************'* Black-Scholes European Put Price Computation *'*********************************************************************Function Put_Eur(s, x, t, r, sd) Dim a As Single Dim b As Single Dim c As Single Dim d1 As Single Dim d2 As Single a = Log(s / x) b = (r + 0.5 * sd ^ 2) * t c = sd * (t ^ 0.5) d1 = (a + b) / c d2 = d1 - sd * (t ^ 0.5) CallEur = s * SNorm(d1) - x * Exp(-r * t) * SNorm(d2) Put_Eur = x * Exp(-r * t) - s + CallEurEnd Function

http://www.maths.strath.ac.uk/~aas96106/rep01_2004.pdf

Although the basic European call option valuation problem has a simple analytical solution (under the Black-Scholes assumptions), there are many variations of the problem that, currently, require the use of numerical methods. In particular, a rich variety of exotic options are traded. These options differ from the European call in that the payoff depends not only on the final time asset price, but also on its behavior over all or part of the time interval [0; T). For example, the payoff may depend on the maximum, minimum or average asset price and may knock-in or knock-out depending upon whether the asset price crosses a pre-determined barrier. Also, the option may have an early exercise facility, giving its holder the freedom to exercise before the expiry date. The design and analysis of numerical methods for valuing exotic options is still a very active research topic.

Assumptions of Black-Scholes Model

1) The stock pays no dividends during the option's life

2) European exercise terms are used

3) Markets are efficient

4) No commissions are charged

5) Interest rates remain constant and known

6) Returns are lognormally distributed

Binomial trees

- Binomial---spot price will either move up or move down at the end of each time period. Up ratio (Su/S0) and down ratio (Sd/S0);
- Two steps: Stock price modeling and option pricing
- Cox-Ross-Rubenstein trees---allow only constant volatility (e.g., Geometric Brownian motion)
- Flexible trees---allow variable volatility
- Binomial trees---discrete time model; Geometric Brownian---continuous time model;

Flexible trees

- No restrictions on up and down ratios and the transition probabilities of the tree
- But the tree should be recombining (up followed by a down is the same as a down followed by an up; i.e. u.d = d.u)

Binomial trees and stock returns

- One-period expected stock price = pSu + (1-p) Sd
- By setting up transition probability p suitably, we can have the desired average rate of return (in one period).
- In an “n” period standard binomial tree, if the one-period expected stock value is erΔt times the initial value, then the n-period expected value is enrΔt
- Expected rate of return is proportional to the number of periods (nrΔt)
- In a standard binomial tree, the local volatility is the same at every node σloc = (1/√t) √[(p)(1-p)] log(u2)
- This is the same assumption in geometric Brownian motion model

Option pricing with Binomial trees

- If we consider a one period European option struck at K, Cu=max(Su-S0,0); Cd=max(Sd-K,0); What is the value of C0?
- It may be easily derived (using arbitrage argument and hedging portfolio with Δ shares of S and risk-free bond with maturity value of B at time t1) that:

C0 = e-rΔt(pCu + (1-p)Cd) where Cu=ΔSu+ B and Cd=ΔSd+ B

Hedging Strategy

- We used a hedging portfolio with Δ shares of S and risk-free bond with maturity value of B at time t1 to value the option at t0.
- Suppose C is available for a price less than what is suggested, then we could short the portfolio, use the proceeds to buy C, and be guaranteed a profit.
- If C is available for a price higher than suggested, then we could short the option, use the proceeds to purchase the portfolio, and make a guaranteed profit.

Example

- A call option expiring in a year struck at $100 with a spot price of $100/share. In 1-year, the price of a share may rise to $120 or fall to $80. Risk free rate of interest is 5%.
- Risk-neutral probability p = (erΔtS0-Sd)/(Su- Sd)= 0.6282;
- Δ = (Cu-Cd)/(Su- Sd)= 0.5 shares of S
- B = (SuCd- SdCu)/(Su- Sd)= -$40. WE short a riskless bond maturing to $40 in one year.
- Total cash flow: 50-40e-0.05 = $11.95; This is the cost of the portfolio
- After 1-year, if the stock price goes to $120, the portfolio is worth $60-$40 =$20; If the stock price goes down to $80, then the portfolio is worth $40-$40=0. This is exactly the value of the call option at the end of 1-year.
- So value of the call option at t0 is $11.95

The value of an option is equal to its risk-neutral expected value, discounted to today’s value at the risk-free rate of interest.

- C0 = e-rΔt(pCu + (1-p)Cd)
- In the case of multiple intervals, this backward induction is used repeatedly, to compute C0.

Risk-neutral Worlds and Binomial Trees

- Create a risk-neutral world by choosing the transition probabilities to ensure that the expected value is equal to risk-free rate.

pSu + (1-p) Sd = erΔtS0

p = (erΔtS0 - Sd)/(Su- Sd)= Risk-neutral probability

Implied Volatility

- Many options are traded on and are therefore priced by, the market. This is the market price.
- When the price of an option is known, we can compute its Black-Scholes implied volatility
- If we take different options (on a given stock) with different K and (T-t), we find that the implied volatility are not necessarily consistent; this means that underlying stock does not follow geometric Brownian motion model.
- Black-Scholes formula:

C(St, K, T-t, σ, r) = N(d1)St – N(d2)Ke-r(T-t)

When all other variables are known, determine σ using a numerical approximation technique (e.g. Newton-Raphson method)

Volatility Smile

- Implied volatility structure has two components: volatility smile and term structure of volatility
- Volatility smile is the way in which implied volatility varies with strike price for options of a fixed expiration.

Implied Volatility Trees (IVT)

- Black-Scholes is based on constant volatility assumption; the volatility smile contradicts this;
- IVT model says: Black-Scholes model is wrong; market is right
- Use of IVT:
- To compute hedge parameters for standard parameters that we wish to hedge
- To value other derivatives whose prices are not readily available from the market

Implied Binomial Trees

- Start with a given stock price distribution for a particular future date and then build a binomial tree whose terminal distribution is equal to the given distribution. The tree is then used to value options.

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