# Derivatives Pricing

## Ágnes Vidovics-Dancs

November 2013

(For more resources related to this topic, see here.)

Derivatives are financial instruments which derive their value from (or are dependent on) the value of another product, called the underlying. The three basic types of derivatives are forward and futures contracts, swaps, and options. In this article we will focus on this latter class and show how basic option pricing models and some related problems can be handled in R. We will start with overviewing how to use the continuous Black-Scholes model and the binomial Cox-Ross-Rubinstein model in R, and then we will proceed with discussing the connection between these models. Furthermore, with the help of calculating and plotting of the Greeks, we will show how to analyze the most important types of market risks that options involve. Finally, we will discuss what implied volatility means and will illustrate this phenomenon by plotting the volatility smile with the help of real market data.

The most important characteristics of options compared to futures or swaps is that you cannot be sure whether the transaction (buying or selling the underlying) will take place or not. This feature makes option pricing more complex and requires all models to make assumptions regarding the future price movements of the underlying product. The two models we are covering here differ in these assumptions: the Black-Scholes model works with a continuous process while the Cox-Ross-Rubinstein model works with a discrete stochastic process. However, the remaining assumptions are very similar and we will see that the results are close too.

# The Black-Scholes model

The assumptions of the Black-Scholes model (Black and Sholes, 1973, see also Merton, 1973) are as follows:

• The price of the underlying asset (S) follows geometric Brownian motion:

Here μ (drift) and σ (volatility) are constant parameters and W is a standard Wiener process.

• The market is arbitrage-free.
• The underlying is a stock paying no dividends.
• Buying and (short) selling the underlying asset is possible in any (even fractional) amount.
• There are no transaction costs.
• The short-term interest rate (r) is known and constant over time.

The main result of the model is that under these assumptions, the price of a European call option (c) has a closed form:

Here X is the strike price, T-tis the time to maturity of the option, and N denotes the cumulative distribution function of the standard normal distribution. The equation giving the price of the option is usually referred to as the Black-Scholes formula. It is easy to see from put-call parity that the price of a European put option (p) with the same parameters is given by:

Now consider a call and put option on a Google stock in June 2013 with a maturity of September 2013 (that is, with 3 months of time to maturity).Let us assume that the current price of the underlying stock is USD 900, the strike price is USD 950, the volatility of Google is 22%, and the risk-free rate is 2%. We will calculate the value of the call option with the GBSOption function from the fOptions package. Beyond the parameters already discussed, we also have to set the cost of carry (b); in the original Black-Scholes model, (with underlying paying no dividends) it equals the risk-free rate.

```> library(fOptions)
> GBSOption(TypeFlag = "c", S = 900, X =950, Time = 1/4, r = 0.02,
+ sigma = 0.22, b = 0.02)
Title:
Black Scholes Option Valuation
Call:
GBSOption(TypeFlag = "c", S = 900, X = 950, Time = 1/4, r = 0.02,
b = 0.02, sigma = 0.22)
Parameters:
Value:
TypeFlag c
S 900
X 950
Time 0.25
r 0.02
b 0.02
sigma 0.22
Option Price:
21.79275
Description:
Tue Jun 25 12:54:41 2013```

This prolonged output returns the passed parameters with the result just below the Option Price label. Setting the TypeFlag to p would compute the price of the put option and now we are only interested in the results (found in the price slot—see the str of the object for more details) without the textual output:

```> GBSOption(TypeFlag = "p", S = 900, X =950, Time = 1/4, r = 0.02, sigma
= 0.22, b = 0.02)@price
[1] 67.05461```

We also have the choice to compute the preceding values with a more user-friendly calculator provided by the GUIDE package. Running the blackscholes() function would trigger a modal window with a form where we can enter the same parameters. Please note that the function uses the dividend yield instead of cost of carry, which is zero in this case.

# The Cox-Ross-Rubinstein model

The Cox-Ross-Rubinstein(CRR) model (Cox, Ross and Rubinstein, 1979) assumes that the price of the underlying asset follows a discrete binomial process. The price might go up or down in each period and hence changes according to a binomial tree illustrated in the following plot, where u and dare fixed multipliers measuring the price changes when it goes up and down. The important feature of the CRR model is that u=1/d and the tree is recombining; that is, the price after two periods will be the same if it first goes up and then goes down or vice versa, as shown in the following figure:

To build a binomial tree, first we have to decide how many steps we are modeling (n); that is, how many steps the time to maturity of the option will be divided into. Alternatively, we can determine the length of one time step ∆t,(measured in years) on the tree:

If we know the volatility (σ) of the underlying, the parameters u and dare determined according to the following formulas:

And consequently:

When pricing an option in a binomial model, we need to determine the tree of the underlying until the maturity of the option. Then, having all the possible prices at maturity, we can calculate the corresponding possible option values, simply given by the following formulas:

To determine the option price with the binomial model, in each node we have to calculate the expected value of the next two possible option values and then discount it. The problem is that it is not trivial what expected return to use for discounting. The trick is that we are calculating the expected value with a hypothetic probability, which enables us to discount with the risk-free rate. This probability is called risk neutral probability (pn) and can be determined as follows:

The interpretation of the risk-neutral probability is quite plausible: if the probability that the underlying price goes up from any of the nodes was pn, then the expected return of the underlying would be the risk-free rate. Consequently, an expected value calculated with pn can be discounted by rand the price of the option in any node of the tree is determined as:

In the preceding formula, g is the price of an option in general (it may be call or put as well) in a given node, gu and gd are the values of this derivative in the two possible nodes one period later.

For demonstrating the CRR model in R, we will use the same parameters as in the case of the Black-Scholes formula. Hence, S=900, X=950, σ=22%, r=2%, b=2%, T-t=0.25. We also have to set n, the number of time steps on the binomial tree. For illustrative purposes, we will work with a 3-period model:

```> CRRBinomialTreeOption(TypeFlag = "ce", S = 900, X = 950,
+ Time = 1/4, r = 0.02, b = 0.02, sigma = 0.22, n = 3)@price
[1] 20.33618
> CRRBinomialTreeOption(TypeFlag = "pe", S = 900, X = 950,
+ Time = 1/4, r = 0.02, b = 0.02, sigma = 0.22, n = 3)@price
[1] 65.59803```

It is worth observing that the option prices obtained from the binomial model are close to (but not exactly the same as) the Black-Scholes prices calculated earlier. Apart from the final result, that is, the current price of the option, we might be interested in the whole option tree as well:

```> CRRTree <- BinomialTreeOption(TypeFlag = "ce", S = 900, X = 950,
+ Time = 1/4, r = 0.02, b = 0.02, sigma = 0.22, n = 3)
> BinomialTreePlot(CRRTree, dy = 1, xlab = "Time steps",
+ ylab = "Number of up steps", xlim = c(0,4))
> title(main = "Call Option Tree")```

Here we first computed a matrix by BinomialTreeOption with the given parameters and saved the result in CRRTree that was passed to the plot function with specified labels for both the x and y axis with the limits of the x axis set from 0 to 4, as shown in the following figure. The y-axis (number of up steps) shows how many times the underlying price has gone up in total. Down steps are defined as negative up steps.

The European put option can be shown similarly by changing the TypeFlag to pe in the previous code:

# Connection between the two models

After applying the two basic option pricing models, we give some theoretical background to them. We do not aim to give a detailed mathematical derivation, but we intend to emphasize (and then illustrate in R) the similarities of the two approaches. The financial idea behind the continuous and the binomial option pricing is the same: if we manage to hedge the option perfectly by holding the appropriate quantity of the underlying asset, it means we created a risk-free portfolio. Since the market is supposed to be arbitrage-free, the yield of a risk-free portfolio must equal the risk-free rate. One important observation is that the correct hedging ratio is holding underlying asset per option. Hence, the ratio is the partial derivative (or its discrete correspondent in the binomial model) of the option value with respect to the underlying price. This partial derivative is called the delta of the option. Another interesting connection between the two models is that the delta-hedging strategy and the related arbitrage-free argument yields the same pricing principle: the value of the derivative is the risk-neutral expected value of its future possible values, discounted by the risk-free rate. This principle is easily tractable on the binomial tree where we calculated the discounted expected values node by node; however, the continuous model has the same logic as well, even if the expected value is mathematically more complicated to compute. This is the reason why we gave only the final result of this argument, which was the Black-Scholes formula.

Now we know that the two models have the same pricing principles and ideas (delta-hedging and risk-neutral valuation), but we also observed that their numerical results are not equal. The reason is that the stochastic processes assumed to describe the price movements of the underlying asset are not identical. Nevertheless, they are very similar; if we determine the value of u and d from the volatility parameter as we did it in The Cox-Ross-Rubinstein model section, the binomial process approximates the geometric Brownian motion. Consequently, the option price of the binomial model converges to that of the Black-Scholes model if we increase the number of time steps (or equivalently, decrease the length of the steps).

To illustrate this relationship, we will compute the option price in the binomial model with increasing numbers of time steps. In the following figure, we compare the results with the Black-Scholes price of the option:

The plot was generated by a loop running N from 1 to 200 to compute CRRBinomialTreeOption with fixed parameters:

```> prices <- sapply(1:200, function(n) {
+ CRRBinomialTreeOption(TypeFlag = "ce", S = 900, X = 950,
+ Time = 1/4, r = 0.02, b = 0.02, sigma = 0.22, n = n)@price
+ })```

Now the prices variable holds 200 computed values:

```> str(prices)
num [1:200] 26.9 24.9 20.3 23.9 20.4...```

Let us also compute the option with the generalized Black-Scholes option:

```> price <- GBSOption(TypeFlag = "c", S = 900, X = 950, Time = 1/4, r =
0.02, sigma = 0.22, b = 0.02)@price```

And show the prices in a joint plot with the GBS option rendered in red:

```> plot(1:200, prices, type='l', xlab = 'Number of steps',
+ ylab = 'Prices')
> abline(h = price, col ='red')
> legend("bottomright", legend = c('CRR-price', 'BS-price'),
+ col = c('black', 'red'), pch = 19)```

# Greeks

Understanding the risk-types that an option might involve is crucial for all market participants. The idea behind Greeks is to measure the different types of risks; they represent the sensitivity of the option to different factors. The Greeks of a plain vanilla option are: delta (∆, sensitivity to the underlying price), gamma (Γ, sensitivity of delta to the underlying price, delta of delta), theta(θ, sensitivity to time), rho(ρ, sensitivity to the risk-free rate), and vega (V, sensitivity to the volatility). In terms of mathematics, all Greeks are partial derivatives of the derivative price:

The Greeks can be computed easily for each option with the GBSGreeks function:

```> sapply(c('delta', 'gamma', 'vega', 'theta', 'rho'), function(greek)
+ GBSGreeks(Selection = greek, TypeFlag = "c", S = 900, X = 950,
+ Time = 1/4, r = 0.02, b = 0.02, sigma = 0.22)
+ )
delta gamma vega theta rho
0.347874404 0.003733069 166.308230868 -79.001505841 72.82355323```

It is often useful to analyze how a given Greek changes if some market parameters change. Such analysis might help us to understand risks better. For example, delta of a call option as a function of the underlying price is an increasing curve taking an S shape, ranging from 0 to 1. These characteristics are always valid, but if time passes and we are approaching the maturity of the option, the curve becomes steeper and steeper (see the next figure). The interpretation is as follows: if it is very probable that the call option will be exercised, then it is very similar to a long forward contract; hence, delta is close to 1. If the chance of exercising is very low, holding the call option is similar to holding nothing and delta is 0. As time passes, the interval of those underlying prices where the exercising is really uncertain (that is, neither very probable, nor very improbable) gets narrower; as a result, the curve of the delta becomes steeper. To illustrate this behavior, we will plot the delta of a call as a function of the underlying price, with three different maturities.

To compute the deltas, we run two loops: one with three different time values and S running from 500 to 1500:

```> deltas <- sapply(c(1/4, 1/20, 1/50), function(t)
+ sapply(500:1500, function(S)
+ GBSGreeks(Selection = 'delta', TypeFlag = "c",
+ S = S, X = 950, Time = t, r = 0.02, b = 0.02, sigma = 0.22)))```

The resulting deltas holds 1001 rows (for the S values) and three columns (for the specified times) that we show in a joint plot:

```> plot(500:1500, deltas[, 1], ylab = 'Delta of call option',
+ xlab = "Price of the underlying (S)", type = 'l')
> lines(500:1500, deltas[, 2], col='blue')
> lines(500:1500, deltas[, 3], col='red')
> legend("bottomright", legend = c('t=1/4', 't=1/20', 't=1/50'),
+ col = c('black', 'blue', 'red'), pch = 19)```

The following figure shows the delta of the call options with three different values of time to maturity:

Determining or plotting the Greeks of complex option strategies is very similar. For example, calculating the delta of a straddle position (a portfolio of a call and a put option with the same parameters) means simply calculating deltas separately for the call and the put and then adding them. We will plot delta of a straddle as a function of the underlying price. We may observe that the shape is very similar to the delta of the previous call, but now the S-curve is ranging from -1 to 1:

```> straddles <- sapply(c('c', 'p'), function(type)
+ sapply(500:1500, function(S)
+ GBSGreeks(Selection = 'delta', TypeFlag = type, S = S,
X = 950, Time = 1/4, r = 0.02, b = 0.02, sigma = 0.22)))```

So we call a nested loop running S from 500 to 1500 for both the call and put options keeping the other parameters fixed, and save the resulting deltas in a matrix. With the next command, the sum of these rows (put and call options) is rendered:

```> plot(500:1500, rowSums(straddles), type='l',
+ xlab='Price of the underlying (S)', ylab = 'Delta of straddle')```

The resulting plot illustrates the delta of a straddle position as a function of the underlying's price as shown in the following figure:

# Implied volatility

The Black-Scholes model is often criticized because of some shortcomings. One important problem is that the model assumes constant volatility for the underlying asset, which does not hold in reality. Furthermore, since it is not observable directly, the volatility is the most complicated parameter of the model to calibrate. Due to this difficulty, the Black-Scholes formula is often used in an indirect way for estimating the volatility parameter; we observe the market price of an option, then in view of all the other parameters we can search for σ that results a BlackScholes price equal to the observed market price. This σ parameter is called the implied volatility of the option. As Riccardo Rebonato famously stated, implied volatility is "the wrong number to put in the wrong formula to get the right price" (Rebonato, 1999, p.78).

We will illustrate the calculation of implied volatility with the help of some Google options. The options are call options with the maturity of September 21, 2013 and strike prices ranging from USD 700 to USD 1150 (76 different options). We collected the ask prices of these options on June 25, 2013 from finance.google.com and put them in a CSV file. For the calculations, we need to know that the price of Google on the given day was USD 866.2. Since the time to maturity is 88 days, we will use 88/360 years for the Time parameter. The risk-free rate and the cost of carry are assumed to remain 2% further on.

`> goog <- read.csv('goog_calls.csv')`

And then run a loop for each line of the dataset to compute the volatility with the given parameters:

```> volatilites <- sapply(seq_along(goog\$Strike), function(i)
+ GBSVolatility(price = goog\$Ask.Price[i], TypeFlag = "c",
+ S = 866.2, X = goog\$Strike[i], Time = 88/360, r = 0.02, b = 0.02))```

The volatilities variable is a vector holding the computed values:

```> str(volatilites)
num [1:76] 0.258 0.253 0.269 0.267 0.257...```

That can be shown against the strike price:

```> plot(x = goog\$Strike, volatilites, type = 'p',
+ ylab = 'Implied volatiltiy', xlab = 'Strike price (X)')```

Hence, the following figure shows the implied volatilities for different strike prices:

It is worth noticing that the implied volatilities calculated for Google options vary according to the strike prices. This is contrary with the Black-Scholes model, which assumes constant volatility. The observed implied volatility pattern (lower volatilities for medium strike prices) is not unique and appears in financial markets quite frequently. Because of the specific form of the curve, the phenomenon is called the volatility smile.

# Summary

In this article, we have used R to price plain vanilla options with the Black-Scholes and Cox-Ross-Rubinstein models. Furthermore, we examined the basic Greeks and the implied volatility of these options. Besides getting to know some tools from the fOptions package, we have also created a few loops and custom functions programmatically for simulation purposes.

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