# Pricing

Because the values of option contracts depend on a number of different variables in addition to the value of the underlying asset, they are complex to value. There are many pricing models in use, although all essentially incorporate the concepts of rational pricing (i.e. risk neutrality) , moneyness, option time value and put–call parity.

The valuation itself combines (1) a model of the behavior ("process") of the underlying price with (2) a mathematical method which returns the premium as a function of the assumed behavior.

## Put-call Parity

In financial mathematics, put–call parity defines a relationship between the price of a European call option and European put option, both with the identical strike price and expiry, namely that a portfolio of a long call option and a short put option is equivalent to (and hence has the same value as) a single forward contract at this strike price and expiry. This is because if the price at expiry is above the strike price, the call will be exercised, while if it is below, the put will be exercised, and thus in either case one unit of the asset will be purchased for the strike price, exactly as in a forward contract.

The validity of this relationship requires that certain assumptions be satisfied; these are specified and the relationship is derived below. In practice transaction costs and financing costs (leverage) mean this relationship will not exactly hold, but in liquid markets the relationship is close to exact.

### Assumptions

Put–call parity is a static replication, and thus requires minimal assumptions, namely the existence of a forward contract. In the absence of traded forward contracts, the forward contract can be replaced (indeed, itself replicated) by the ability to buy the underlying asset and finance this by borrowing for fixed term (e.g., borrowing bonds), or conversely to borrow and sell (short) the underlying asset and loan the received money for term, in both cases yielding a [wikipedia:[self-financing portfolio | self-financing portfolio]].

These assumptions do not require any transactions between the initial date and expiry, and are thus significantly weaker than those of the Black-Scholes model, which requires dynamic replication and continual transaction in the underlying.

Replication assumes one can enter into derivative transactions, which requires leverage (and capital costs to back this), and buying and selling entails transaction costs, notably the bid-ask spread. The relationship thus only holds exactly in an ideal frictionless market with unlimited liquidity. However, real world markets may be sufficiently liquid that the relationship is close to exact, most significantly FX markets in major currencies or major stock indices, in the absence of market turbulence.

### Statement

Put–call parity can be stated in a number of equivalent ways, most tersely as:

[$] C - P = D(F - K) [$]

where $C$ is the (current) value of a call, $P$ is the (current) value of a put, $D$ is the discount factor, $F$ is the forward price of the asset, and $K$ is the strike price. Note that the spot price is given by $D\cdot F = S$ (spot price is present value, forward price is future value, discount factor relates these). The left side corresponds to a portfolio of long a call and short a put, while the right side corresponds to a forward contract. The assets $C$ and $P$ on the left side are given in current values, while the assets $F$ and $K$ are given in future values (forward price of asset, and strike price paid at expiry), which the discount factor $D$ converts to present values.

Using spot price $S$ instead of forward price $F$ yields:

[$] C - P = S - D\cdot K [$]

Rearranging the terms yields a different interpretation:

[$]C + D \cdot K = P + S[$]

In this case the left-hand side is a fiduciary call, which is long a call and enough cash (or bonds) to pay the strike price if the call is exercised, while the right-hand side is a protective put, which is long a put and the asset, so the asset can be sold for the strike price if the spot is below strike at expiry. Both sides have payoff $\max(S(T), K)$ at expiry (i.e., at least the strike price, or the value of the asset if more), which gives another way of proving or interpreting put–call parity.

In more detail, this original equation can be stated as:

[$] C(t) - P(t) = S(t)- K \cdot B(t,T)[$]

where $C(t)$ is the value of the call at time $t$; $P(t)$ is the value of the put of the same expiration date; $S(t)$ is the spot price of the underlying asset; $K$ is the strike price; and $B(t,T)$ is the present value of a zero-coupon bond that matures to \$1 at time $T.$ This is the present value factor for K.

Note that the right-hand side of the equation is also the price of buying a forward contract on the stock with delivery price $K$. Thus one way to read the equation is that a portfolio that is long a call and short a put is the same as being long a forward. In particular, if the underlying is not tradeable but there exists forwards on it, we can replace the right-hand-side expression by the price of a forward.

If the bond interest rate, $r$, is assumed to be constant then

[$] B(t,T) = e^{-r(T-t)} [$]

Note: $r$ refers to the force of interest, which is approximately equal to the effective annual rate for small interest rates. However, one should take care with the approximation, especially with larger rates and larger time periods. To find $r$ exactly, use $r = \ln (1+i)$, where $i$ is the effective annual interest rate.

When valuing European options written on stocks with known dividends that will be paid out during the life of the option, the formula becomes:

[$] C(t) - P(t) + D(t) = S(t) - K \cdot B(t,T) [$]

where $D(t)$ represents the total value of the dividends from one stock share to be paid out over the remaining life of the options, discounted to present value. We can rewrite the equation as:

[$] C(t) - P(t) = S(t) - K \cdot B(t,T)\ - D(t)[$]

and note that the right-hand side is the price of a forward contract on the stock with delivery price $K$, as before.

## Derivation

We will suppose that the put and call options are on traded stocks, but the underlying can be any other tradeable asset. The ability to buy and sell the underlying is crucial to the "no arbitrage" argument below.

First, note that under the assumption that there are no arbitrage opportunities (the prices are arbitrage-free), two portfolios that always have the same payoff at time T must have the same value at any prior time. To prove this suppose that, at some time $T$ before $T$, one portfolio were cheaper than the other. Then one could purchase (go long) the cheaper portfolio and sell (go short) the more expensive. At time $T$, our overall portfolio would, for any value of the share price, have zero value (all the assets and liabilities have canceled out). The profit we made at time $T$ is thus a riskless profit, but this violates our assumption of no arbitrage.

We will derive the put-call parity relation by creating two portfolios with the same payoffs (static replication) and invoking the above principle (rational pricing).

Consider a call option and a put option with the same strike $K$ for expiry at the same date $T$ on some stock $S$, which pays no dividend. We assume the existence of a bond that pays 1 dollar at maturity time $T$. The bond price may be random (like the stock) but must equal 1 at maturity.

Let the price of $S$ be S(t) at time t. Now assemble a portfolio by buying a call option $C$ and selling a put option $P$ of the same maturity $T$ and strike $K$. The payoff for this portfolio is $S(T)-K$. Now assemble a second portfolio by buying one share and borrowing $K$ bonds. Note the payoff of the latter portfolio is also $S(T)-K$ at time $T$, since our share bought for $S(T)$ will be worth $S(T)$ and the borrowed bonds will be worth $K$.

By our preliminary observation that identical payoffs imply that both portfolios must have the same price at a general time $t$, the following relationship exists between the value of the various instruments:

[$] C(t) - P(t) = S(t)- K \cdot B(t,T) \, [$]

Thus given no arbitrage opportunities, the above relationship, which is known as put-call parity, holds, and for any three prices of the call, put, bond and stock one can compute the implied price of the fourth.

In the case of dividends, the modified formula can be derived in similar manner to above, but with the modification that one portfolio consists of going long a call, going short a put, and $D(T)$ bonds that each pay 1 dollar at maturity $T$ (the bonds will be worth $D(t)$ at time $T$); the other portfolio is the same as before - long one share of stock, short $K$ bonds that each pay 1 dollar at $T$. The difference is that at time $T$, the stock is not only worth $S(T)$ but has paid out $D(t)$ in dividends.

## Binomial Option Pricing Model

The binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" (lattice based) model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting.

The binomial model was first proposed by William Sharpe in the 1978 edition of Investments (ISBN 013504605X),[1] and formalized by Cox, Ross and Rubinstein in 1979[2] and by Rendleman and Bartter in that same year.[3]

## Method

The binomial pricing model traces the evolution of the option's key underlying variables in discrete-time. This is done by means of a binomial lattice (Tree), for a number of time steps between the valuation and expiration dates. Each node in the lattice represents a possible price of the underlying at a given point in time.

Valuation is performed iteratively, starting at each of the final nodes (those that may be reached at the time of expiration), and then working backwards through the tree towards the first node (valuation date). The value computed at each stage is the value of the option at that point in time.

Option valuation using this method is, as described, a three-step process:

1. Price tree generation,
2. Calculation of option value at each final node,
3. Sequential calculation of the option value at each preceding node.

#### Step 1: Create the binomial price tree

The tree of prices is produced by working forward from valuation date to expiration.

At each step, it is assumed that the underlying instrument will move up or down by a specific factor ($u$ or $d$) per step of the tree (where, by definition, $u \ge 1$ and $0 \lt d \le 1$). So, if $S$ is the current price, then in the next period the price will either be $S_{up} = S \cdot u$ or $S_{down} = S \cdot d$.

The up and down factors are calculated using the underlying volatility, $\sigma$, and the time duration of a step, $t$, measured in years (using the day count convention of the underlying instrument). From the condition that the variance of the log of the price is $\sigma^2 t$, we have:

[$]u = e^{\sigma\sqrt \Delta t}, \, d = e^{-\sigma\sqrt \Delta t} = \frac{1}{u}.[$]

Above is the original Cox, Ross, & Rubinstein (CRR) method; there are various other techniques for generating the lattice, such as "the equal probabilities" tree, see.[4]

The CRR method ensures that the tree is recombinant, i.e. if the underlying asset moves up and then down (u,d), the price will be the same as if it had moved down and then up (d,u)—here the two paths merge or recombine. This property reduces the number of tree nodes, and thus accelerates the computation of the option price.

This property also allows the value of the underlying asset at each node to be calculated directly via formula, and does not require that the tree be built first. The node-value will be:

[$]S_n = S_0 \times u ^{N_u - N_d},[$]

Where $N_u$ is the number of up ticks and $N_d$ is the number of down ticks.

#### Step 2: Find option value at each final node

At each final node of the tree—i.e. at expiration of the option—the option value is simply its intrinsic, or exercise, value:

[$] \begin{array} \max(S_n -K, 0) \quad \textrm{ for call options} \\ \max(K-S_n,0) \quad \textrm{for put options} \end{array} [$]

where $K$ is the strike price and $S_n$ is the spot price of the underlying asset at the $n$th period.

#### Step 3: Find option value at earlier nodes

Once the above step is complete, the option value is then found for each node, starting at the penultimate time step, and working back to the first node of the tree (the valuation date) where the calculated result is the value of the option.

The "binomial value" is found at each node, using the risk neutrality assumption; see Risk neutral valuation. If exercise is permitted at the node, then the model takes the greater of binomial and exercise value at the node.

Under the risk neutrality assumption, today's fair price of a derivative is equal to the expected value of its future payoff discounted by the risk free rate. Therefore, expected value is calculated using the option values from the later two nodes (Option up and Option down) weighted by their respective probabilities—"probability" $p$ of an up move in the underlying, and "probability" $(1−p)$ of a down move. The expected value is then discounted at $q$, the risk free rate corresponding to the life of the option.

The following formula to compute the expectation value is applied at each node:

[$]\text { Binomial Value }=[p \times \text { Option up }+(1-p) \times \text { Option down] } \times \exp (-r \times \Delta t)[$]

, or

[$]C_{t-\Delta t,i} = e^{-r \Delta t}(pC_{t,i} + (1-p)C_{t,i+1}) \,[$]

where $C_{t,i} \,$ is the option's value for the $i^{th} \,$ node at time $t$;

[$]p = \frac{e^{(r-q) \Delta t} - d}{u - d}[$]

is chosen such that the related binomial distribution simulates the geometric Brownian motion of the underlying stock with parameters $q$ and $\sigma$; and $q$ is the dividend yield of the underlying corresponding to the life of the option.

Note that for $p$ to be in the interval (0,1) the following condition on $\Delta t$ has to be satisfied $\Delta t \lt \frac{\sigma^2}{(r-q)^2}.$

## Black-Scholes Model

The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price given the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The equation and model are named after economists Fischer Black and Myron Scholes; Robert C. Merton, who first wrote an academic paper on the subject, is sometimes also credited.

The key idea behind the model is to hedge the option by buying and selling the underlying asset in just the right way and, as a consequence, to eliminate risk. This type of hedging is called "continuously revised delta hedging" and is the basis of more complicated hedging strategies such as those engaged in by investment banks and hedge funds.

The model is widely used, although often with some adjustments, by options market participants.[5]:751 The model's assumptions have been relaxed and generalized in many directions, leading to a plethora of models that are currently used in derivative pricing and risk management. It is the insights of the model, as exemplified in the Black–Scholes formula, that are frequently used by market participants, as distinguished from the actual prices.

### Fundamental hypotheses

The Black–Scholes model assumes that the market consists of at least one risky asset, usually called the stock, and one riskless asset, usually called the money market, cash, or bond.

Now assumptions are made on the assets (which explain their names):

Asset Related
• Riskless rate: The rate of return on the riskless asset is constant and thus called the risk-free interest rate.
• Random walk: The instantaneous log return of stock price is an infinitesimal random walk with drift; more precisely, the stock price follows a geometric Brownian motion, and it is assumed that its drift and volatility are constant (if they are time-varying, a suitably modified Black–Scholes formula can be deduced quite simply, as long as the volatility is not random).
• The stock does not pay a dividend.

Market Related
• No arbitrage opportunity (i.e., there is no way to make a riskless profit).
• Ability to borrow and lend any amount, even fractional, of cash at the riskless rate.
• Ability to buy and sell any amount, even fractional, of the stock (this includes short selling).
• The above transactions do not incur any fees or costs (i.e., frictionless market).

With these assumptions holding, suppose there is a derivative security also trading in this market. It is specified that this security will have a certain payoff at a specified date in the future, depending on the values taken by the stock up to that date. It is a surprising fact that the derivative's price can be determined at the current time, while accounting for the fact that the path the stock price will take in the future is unknown. For the special case of a European call or put option, Black and Scholes showed that "it is possible to create a hedged position, consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock".[6] Their dynamic hedging strategy led to a partial differential equation which governed the price of the option. Its solution is given by the Black–Scholes formula.

### Notation

The notation used throughout this page will be defined as follows, grouped by subject:

General and Market Related

Asset Related
• $\mu$ is the drift rate of $S$, annualized
• $\sigma$ is the standard deviation of the stock's returns. This is the square root of the quadratic variation of the stock's log price process, a measure of its volatility.

Option Related
• $V(S, t)$ is the price of the option as a function of the underlying asset $S$ at time $t$
• $C(S, t)$ is the price of a European call option
• $P(S, t)$ is the price of a European put option
• $T$ is the time of option expiration.
• $\tau$ is the time until maturity: $\tau = T - t$
• $K$ is the strike price of the option, also known as the exercise price

$N(x)$ will denote the standard normal cumulative distribution function:

[$]N(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-z^2/2}\, dz.[$]

$N'(x)$ will denote the standard normal probability density function:

[$]N'(x) = \frac{dN(x)}{dx} = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}. [$]

### Black–Scholes equation

The Black–Scholes equation is a parabolic partial differential equation, which describes the price of the option over time. The equation is:

[$]\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0[$]

The key financial insight behind the equation is that one can perfectly hedge the option by buying and selling the underlying asset and the bank account asset (cash) in just the right way and consequently "eliminate risk". This hedge, in turn, implies that there is only one right price for the option, as returned by the Black–Scholes formula.

### Black–Scholes formula

The Black–Scholes formula calculates the price of European put and call options. This price is consistent with the Black–Scholes equation as above; this follows since the formula can be obtained by solving the equation for the corresponding terminal and boundary conditions:

[]\begin{align*} & C(0, t) = 0\text{ for all }t \\ & C(S, t) \rightarrow S - K \text{ as }S \rightarrow \infty \\ & C(S, T) = \max\{S - K, 0\} \end{align*}[]

The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is:

[]\begin{align*} C(S_t, t) &= N(d_1)S_t - N(d_2)Ke^{-r(T - t)} \\ d_1 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\ d_2 &= d_1 - \sigma\sqrt{T - t} \\ \end{align*}[]

The price of a corresponding put option based on put–call parity with discount factor $e^{-r(T-t)}$ is:

[]\begin{align*} P(S_t, t) &= Ke^{-r(T - t)} - S_t + C(S_t, t) \\ &= N(-d_2) Ke^{-r(T - t)} - N(-d_1) S_t \end{align*}\,[]

### Alternative formulation

Introducing some auxiliary variables allows the formula to be simplified and reformulated in a form that is often more convenient (this is a special case of the Black '76 formula):

[]\begin{align*} C(F, \tau) &= D \left[ N(d_+) F - N(d_-) K \right] \\ d_+ &= \frac{1}{\sigma\sqrt{\tau}}\left[\ln\left(\frac{F}{K}\right) + \frac{1}{2}\sigma^2\tau\right] \\ d_- &= d_+ - \sigma\sqrt{\tau} \end{align*}[]

where $D = e^{-r\tau}$ is the discount factor, $F = e^{r\tau} S = \frac{S}{D}$ is the forward price of the underlying asset, and $S = DF$. Given put–call parity, which is expressed in these terms as

[$]C - P = D(F - K) = S - D K[$]

, the price of a put option is

[$]P(F, \tau) = D \left[ N(-d_-) K - N(-d_+) F \right].[$]

## The Options Greeks

"The Greeks" measure the sensitivity of the value of a derivative product or a financial portfolio to changes in parameter values while holding the other parameters fixed. They are partial derivatives of the price with respect to the parameter values. One Greek, "gamma" (as well as others not listed here) is a partial derivative of another Greek, "delta" in this case.

The Greeks are important not only in the mathematical theory of finance, but also for those actively trading. Financial institutions will typically set (risk) limit values for each of the Greeks that their traders must not exceed. [7]

Delta is the most important Greek since this usually confers the largest risk. Many traders will zero their delta at the end of the day if they are not speculating on the direction of the market and following a delta-neutral hedging approach as defined by Black–Scholes. When a trader seeks to establish an effective delta-hedge for a portfolio, the trader may also seek to neutralize the portfolio's gamma, as this will ensure that the hedge will be effective over a wider range of underlying price movements.

The Greeks for Black–Scholes are given in closed form below. They can be obtained by differentiation of the Black–Scholes formula.[8]

Call Put
Delta $\frac{\partial V}{\partial S}$ $N(d_1)\,$ $-N(-d_1) = N(d_1) - 1\,$
Gamma $\frac{\partial^{2} V}{\partial S^{2}}$ $\frac{N'(d_1)}{S\sigma\sqrt{T - t}}\,$
Vega $\frac{\partial V}{\partial \sigma}$ $S N'(d_1) \sqrt{T-t}\,$
Theta $\frac{\partial V}{\partial t}$ $-\frac{S N'(d_1) \sigma}{2 \sqrt{T - t}} - rKe^{-r(T - t)}N(d_2)\,$ $-\frac{S N'(d_1) \sigma}{2 \sqrt{T - t}} + rKe^{-r(T - t)}N(-d_2)\,$
Rho $\frac{\partial V}{\partial r}$ $K(T - t)e^{-r(T - t)}N( d_2)\,$ $-K(T - t)e^{-r(T - t)}N(-d_2)\,$

Note that from the formulae, it is clear that the gamma is the same value for calls and puts and so too is the vega the same value for calls and puts options. This can be seen directly from put–call parity, since the difference of a put and a call is a forward, which is linear in $S$ and independent of $\sigma$ (so a forward has zero gamma and zero vega). $N'$ is the standard normal probability density function.

In practice, some sensitivities are usually quoted in scaled-down terms, to match the scale of likely changes in the parameters. For example, rho is often reported divided by 10,000 (1 basis point rate change), vega by 100 (1 vol point change), and theta by 365 or 252 (1 day decay based on either calendar days or trading days per year).

## References

1. William F. Sharpe, Biographical, nobelprize.org
2. "Option pricing: A simplified approach" (1979). Journal of Financial Economics 7 (3). doi:10.1016/0304-405X(79)90015-1.
3. Richard J. Rendleman, Jr. and Brit J. Bartter. 1979. "Two-State Option Pricing". Journal of Finance 24: 1093-1110. doi:10.2307/2327237
4. Mark s. Joshi (2008). The Convergence of Binomial Trees for Pricing the American Put
5. Bodie, Zvi; Alex Kane; Alan J. Marcus (2008). Investments (7th ed.). New York: McGraw-Hill/Irwin. ISBN 978-0-07-326967-2.
6. Black, Fischer (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy 81 (3): 637–654. doi:10.1086/260062.
7. Martin Haugh (2016). Basic Concepts and Techniques of Risk Management, Columbia University
8. Although with significant algebra; see, for example, Hong-Yi Chen, Cheng-Few Lee and Weikang Shih (2010). Derivations and Applications of Greek Letters: Review and Integration, Handbook of Quantitative Finance and Risk Management, III:491–503.