CUNY 607

Options volatility

We will now examine implied volatility in the market. We do this by acquiring data on ticker symbol SPY, which is an exchange-traded fund (ETF) that replicates the S&P 500 index. Using the underlying stock and options price data, we calculate and plot the volatility curve or “smile.”

Load libraries

library(plyr)
library(dplyr)
library(tidyr)
library(tibble)
library(ggplot2)
library(rvest)
require(quantmod)
require(derivmkts)
library(knitr)

Define variables

We set the ticker symbol to SPY and the interest rate to 2.25 percent.

interest_rate <- 0.0225
symbol <- "SPY"

Get last closing price

We use the quantmod package to look up historical stock prices and save the most recent closing price.

price_df <- getSymbols(symbol, auto.assign=F, from=Sys.Date()-3) %>% as.data.frame()
names(price_df) <- names(price_df) %>% gsub(paste0(symbol, "\\."), "", .)
price_df <- price_df %>% 
  rownames_to_column(var="Date") %>% 
  mutate(Date=as.Date(Date)) %>% 
  arrange(desc(Date)) %>% slice(1)
stock_price <- price_df$Close
price_df %>% head() %>% kable()

Date	Open	High	Low	Close	Volume	Adjusted
2019-05-08	287.53	289.43	286.87	287.53	91412500	287.53

Get dividend info

We look up the empirical dividend stream and convert it into a yield percent (to plug into the options pricing model.)

div_df <- getDividends(symbol, from = Sys.Date()-365, to=Sys.Date()) %>% as.data.frame()
names(div_df) <- names(div_df) %>% gsub(paste0(symbol, "\\."), "", .)
div_df <- div_df %>% rownames_to_column(var="Date") %>% mutate(Date=as.Date(Date))
div_yield <- suppressWarnings( tryCatch( sum(div_df$div)/stock_price, error=function(e) 0))
div_yield

## [1] 0.01821375

Get options chain

We use the quantmod package to retrieve options prices from Yahoo Finance. The implied volatility values are often missing or incorrect, so we will calculate them ourselves using the Black-Scholes formula from the derivmkts package. We do that by looping through the list of options and using the necessary information as inputs - stock price, strike, option type and price, time to expiration, dividend yield, and interest rate.

### Get option prices
option_chain_list <- getOptionChain(Symbols = symbol, Exp = NULL)
option_chain_df <- data.frame()

for (i in 1:length(option_chain_list)){
  if (!is.null(option_chain_list[[i]]$calls) & !is.null(option_chain_list[[i]]$puts)){

    exp <- names(option_chain_list)[[i]] %>% as.Date(., format="%b.%d.%Y")
    t <- (as.numeric(exp-Sys.Date()))/365
    call_df <- option_chain_list[[i]]$calls %>% mutate(callput="call", time=t, exp=exp)
    put_df <- option_chain_list[[i]]$puts %>% mutate(callput="put", time=t, exp=exp)

    ### Calculate Implied Vol
    bidIV <- vector()
    askIV <- vector()
    for (k in 1:nrow(call_df)){
      bidIV[k] <- tryCatch(bscallimpvol(s=stock_price, k=call_df$Strike[k], r=interest_rate, tt=call_df$time[k],
                                        d=div_yield, price=call_df$Bid[k]), error=function(e) NA)
      askIV[k] <- tryCatch(bscallimpvol(s=stock_price, k=call_df$Strike[k], r=interest_rate, tt=call_df$time[k],
                                        d=div_yield, price=call_df$Ask[k]), error=function(e) NA)
    }
    call_df <- call_df %>% mutate(bidIV=suppressWarnings(as.numeric(bidIV)),
                                  askIV=suppressWarnings(as.numeric(askIV)), midIV=(bidIV+askIV)/2)
    bidIV <- vector()
    askIV <- vector()
    for (k in 1:nrow(put_df)){
      bidIV[k] <- tryCatch(bsputimpvol(s=stock_price, k=put_df$Strike[k], r=interest_rate, tt=put_df$time[k],
                                       d=div_yield, price=put_df$Bid[k]), error=function(e) NA)
      askIV[k] <- tryCatch(bsputimpvol(s=stock_price, k=put_df$Strike[k], r=interest_rate, tt=put_df$time[k],
                                       d=div_yield, price=put_df$Ask[k]), error=function(e) NA)
    }
    put_df <- put_df %>% mutate(bidIV=suppressWarnings(as.numeric(bidIV)),
                                askIV=suppressWarnings(as.numeric(askIV)), midIV=(bidIV+askIV)/2)

    option_chain_df <- rbind.fill(option_chain_df, call_df, put_df)
  }
}

option_chain_df <- option_chain_df %>% 
  filter(!is.na(midIV)) %>% mutate(underlying=symbol,
                                   stock_price=stock_price,
                                   price_date = price_df$Date)
option_chain_df %>% head() %>% kable()

Strike	Last	Chg	Bid	Ask	Vol	OI	callput	time	exp	bidIV	askIV	midIV	underlying	stock_price	price_date
285.0	3.23	-1.2500000	3.16	3.20	33570	3226	call	0.0027397	2019-05-10	0.2644935	0.2726526	0.2685731	SPY	287.53	2019-05-08
286.0	2.41	-1.3599999	2.50	2.52	12691	941	call	0.0027397	2019-05-10	0.2704158	0.2739998	0.2722078	SPY	287.53	2019-05-08
287.5	1.56	-1.1600001	1.66	1.69	8908	4023	call	0.0027397	2019-05-10	0.2737221	0.2787196	0.2762208	SPY	287.53	2019-05-08
288.0	1.46	-0.9200001	1.42	1.45	50298	8764	call	0.0027397	2019-05-10	0.2734018	0.2784263	0.2759140	SPY	287.53	2019-05-08
295.0	0.04	-0.0400000	0.03	0.04	23826	33508	call	0.0027397	2019-05-10	0.2427960	0.2542577	0.2485269	SPY	287.53	2019-05-08
297.5	0.01	-0.0100000	0.00	0.01	49	4713	call	0.0027397	2019-05-10	0.0010000	0.2673100	0.1341550	SPY	287.53	2019-05-08

Plot volatility curve

We use ggplot to show the volatility smile and term structure for all options expirations between three and nine months out.

option_chain_df %>% 
  filter(exp>Sys.Date()+90, 
         exp<Sys.Date()+270, 
         Strike>=stock_price & callput=="call" | Strike<stock_price & callput=="put", 
         OI>1000, 
         Bid>=0.25) %>% 
  mutate(Expiration=as.character(exp), midIV=midIV*100) %>% 
  ggplot(aes(x=Strike, y=midIV, colour=Expiration)) + geom_line() + labs(title=paste0(symbol, " Options Implied Volatility by Expiration and Strike"), x="Strike Price", y="Mid-Market Implied Volatility")

We see that implied volatility is higher at lower strike prices than higher strike prices. This phenomenon is what options traders refer to as “volatility skew.”

Theoretically the Black-Scholes model considers volatility to be a single constant number. It assumes that stock price movement percentages are normally distributed.

If the options market agreed with that assumption, every strike price would have the same implied volatility. We can see from the plot that this is clearly not the case. Because the market believes the true underlying distribution of stock price movement has a “fatter” left tail than normal, lower strikes trade at a higher price, and therefore higher implied volatility, than Black-Scholes would calculate.