Option prices and costly short-selling

TLDR: In this paper, a dynamic analysis of option prices with costly short-selling and option market makers is presented, showing that stock short selling costs and bans have significant effects on option prices.

About: This article is published in Journal of Financial Economics.The article was published on 2019-10-01 and is currently open access. It has received 24 citations till now. The article focuses on the topics: Asian option & Binary option.

Figures

Figure 1: Effects of shorting fee. These panels plot the (at-the-money) option bid and ask prices for varying levels of the shorting fee φ. The call option prices are in Panel A and put option prices in Panel B. The horizontal dotted black lines correspond to the benchmark Black-Scholes prices. All other parameter values are as in Table 1 of Section 5 with α = 20.95%.

Figure 2: Effects of partial lending. These panels plot the (at-the-money) option bid and ask prices for varying levels of the partial lending α. The call option prices are in Panel A and put option prices in Panel B. The horizontal dotted black lines correspond to the benchmark Black-Scholes prices. All other parameter values are as in Table 1 of Section 5 with φ = 6.96%.

Table 2: Quantitative effects of costly short-selling on option prices. This table reports the effects of costly short-selling for a call option (Panel A) and a put option (Panel B) on a typical stock in the lowest (D1) and the highest (D10) shorting fee decile in Drechsler and Drechsler (2016) for three different option moneyness levels. CMidt and PMidt denote the mid-point prices of the call and put, e.g., CMidt = 0.5(CAskt + CBidt ) and PMidt = 0.5(PAskt + PBidt ). Implied volatilities in the last columns are obtained by employing the standard approach of inverting the Black-Scholes formula using the mid-point option prices as inputs. All parameter values are as in Table 1.

Table 1: Parameter values. This table reports the parameter values used in our quantitative analysis. The determination of these values is presented in the text.

Table 3: Quantitative effects of costly short-selling on Palm options. This table reports the effects of extreme short-selling on the Palm options and implied stock prices on March 17, 2000 for three different option maturity dates, May 20, August 19 and November 18, 2000. The values in the Evidence rows are from Lamont and Thaler (2003) (Table 6, p. 256). The parameter values used in our model and the Black-Scholes model are as discussed in text: φ = 35%, α = 41.18%, r = 6.21% (May), r = 6.41% (Aug, Nov), St = K = 55.25, σ = 104.6%, and λCs = λCb = λCb = λPb = 1.66.

Table 4: Quantitative effects of costly short-selling on call prices in the stochastic volatility economy. This table reports the effects of costly short-selling on call option prices in the stochastic volatility economy for a typical stock in the lowest (D1) and the highest (D10) shorting fee decile in Drechsler and Drechsler (2016) for three different option moneyness levels when the skewness is ρ = −0.70 (Panel A) and ρ = −0.35 (Panel B). CMidt denote the mid-point call prices, CMidt = 0.5(CAskt +CBidt ). Implied volatilities in the last columns are obtained by employing the standard approach of inverting the Black-Scholes formula using the mid-point option prices as inputs. The values of the parameters that are common to the baseline setting are as in Table 1, the values of the additional parameters of the stochastic volatility economy are based on Duffie, Pan, and Singleton (2000) as discussed in text: κ∗ν = 6.21, σν = 0.61, ν̄∗ = 0.16, νt = 0.0859.

Frequently asked questions

Q1. What are the contributions in "Option prices and costly short-selling" ?

The authors obtain simple, closed-form, unique option bid and ask prices that represent option marketmakers ’ expected hedging costs, and are weighted-averages of well-known benchmark prices ( Black-Scholes, Heston ). 

Q2. What are the future works mentioned in the paper "Option prices and costly short-selling" ?

The authors leave these considerations and the empirical tests of their new predictions for future research. 

Q3. What is the effect of the partial lending on the call and put bid-ask spread?

That is, an increase in the shorting feeincreases, while an increase in the partial lending decreases the option bid-ask spreads more for relatively illiquid options with lower offsetting order arrival rates. 

Q4. What is the effect of the shorting fee and the partial lending on the call and put bid?

The effects of the shorting fee and the partial lending on the call and put bid-ask spreads are decreasing in the offsetting order arrival rates λCs, λCb, λPs, λPb.Property (i) reveals that the call and put bid and ask prices are decreasing in their offsetting sell order arrival rates λCs, λPs, while they are increasing in their offsetting buy order arrival rates λCb, λPb, respectively. 

Q5. What is the effect of partial lending on the call bid and ask price?

a higher partial lending reduces the call ask price more than the call bid price since the hedge portfolio for a call seller requires holding a share long, resulting in a lower call bid-ask spread. 

Q6. what is the effect of short-selling costs on option bid-ask spreads?

Most notably, the authors show that (i) the call and put bid-ask spreads are decreasing in the partial lending, (ii) the option marketmakers’ participation in the stock lending market is decreasing in the shorting fee for each call option sold, (iii) the effects of short-selling costs on option bid-ask spreads are more pronounced for relatively illiquid options, and (iv) the presence of stock short-selling costs generates bid-ask spreads for corporate bonds. 

Q7. How much of the total market capitalization was lent out to short-sellers?

Saffi and Sigurdsson (2010) report that the amount of global supply of lendable shares in December 2008 was $15 trillion (about 20% of the total market capitalization) and $3 trillion of this amount was lent out to short-sellers. 

Q8. Why does the corporate bond price have an implied shorting fee?

This is because the corporate bond has an option-like payoff with its underlying being the firm value, which also has an implied shorting fee (Lemma 3). 

Q9. How much of the deviation is able to be generated by the Palm option prices?

The option prices implied by their model are able to generate roughly half of this deviation as they imply 10.66% (bid) and 8.72% (ask) lower prices. 

Q10. What are the distribution functions of offsetting orders?

The respective distribution functions of offsetting orders are denoted by FCs, FCb, FPs, FPb, which along with the objective of marketmakers given above are sufficient to determine the option prices as follows. 

Q11. How do the authors obtain the partial lending parameter values for these deciles?

Applying these values to SIRIO/(1 + SIRIO), the authors obtain the partial lending parameter values for these deciles as 4.31% and 20.95%, respectively. 

Q12. Why is the stock price correlated with the firm value?

This is because the firm value and stock price are (perfectly) correlated, and the value of any payoff that depends on the firm value, such as a corporate bond, is obtained via a hedge portfolio in the stock. 

Q13. What is the basic framework for the study of short-selling?

To study the effects of costly short-selling on option prices, in this Section, the authors adopt the classic Black-Scholes framework as their baseline setting and incorporate costly short-selling in the underlying stock. 

Q14. How can the authors extend their model to a setting where the stock pays a constant dividend yield?

their model can be extended to a setting in which the stock pays a constant dividend yield δ, by adding it to both the shorting fee φ and the lending income αφ. 

Q15. What is the expected cost of hedging?

Lemma 2 indicates that in the economy with costly short-selling and marketmakers, option bid and ask prices are given by the marketmakers’ expected cost of hedging sell and buy orders, respectively. 

Q16. What are the risks and costs of short-selling the Palm stock?

As Lamont and Thaler (2003) also highlight, high levels of short-selling costs were only part of the story as there were several other extreme risks and costs for short-sellers of the Palm stock during that time (e.g., search costs, uncertainties about collateral levels, shorting fee and early recall of the shares by lenders). 

Q17. What is the reason why the result is intuitive?

This result is also intuitive as it simply says that option bid-ask spreads are decreasing in liquidity (e.g., investors’ buying/selling activity). 

Q18. What is the effect of shorting fees on the spreads of options?

the authors demonstrate that the effects of short-selling costs on option bidask spreads are more pronounced for relatively illiquid options with lower trading activity.