Derivative Pricing and Trading Strategy

This project focuses on the implementation of Black Scholes Model to price equity index options, specifically options on the S&P 500 index. The implementation aims to achieve 4 main objectives.

1. Selection of inputs for the BS Model, pricing at-the-money options and implied volatility calculation

Black-Scholes model was originally developed to price options on non-dividend paying stock. However, the model has been adapted to price option on stocks that pay dividend and other complex options. For pricing options, the model requires five inputs namely, the time to expiration, option’s strike price, risk-free rate, current underlying spot price and its volatility. The model assumes stock prices follow a lognormal distribution because asset prices cannot be negative.

Call (C) and Put (P) option price formulae are as follows:

where,

C(S, t) = European Call option price

P(S, t) = European Put option price

S = Current underlying price (spot price)

K = Strike price of the option

r = Risk-free interest rate

q = Dividend yield

sigma = volatility of the underlying (assumed to be constant)

t = Time to maturity (in years)

N = Normal Cumulative Distribution Function

Generally for an ATM option, an increase in the underlying security price would have a significant impact on the fair value of the option. This will drive the price higher for call options and vice versa in the case of put options. Since a risk-free rate reflects the time value of the exercise price, a higher risk-free rate increases the fair value of the call option. However, as the risk-free rate increases, the value of a put option decreases.

1. 1. Observations:

Using the inputs mentioned above, the price of the option was determined using the Black-Scholes model with dividend yield. The price obtained for at-the-money (ATM) call option was $22.92. The obtained ATM call price was then compared with the mid-point of its bid-ask spread of $22.05 and found to be relatively close.

Similar exercise was conducted for pricing an ATM put option. The price obtained for ATM put option was $19.78. The mid-point of the bid-ask spread was calculated to be $16.55. The slightly higher difference in the two prices could be attributed to limitations of the BS model since it only attempts to provide the fair value of the options under certain assumptions and these assumptions do not always hold true. The volume or demand and supply for option contracts can affect the actual bid-ask spread.

1. 2. Implied Volatility Calculation:

Implied volatility is calculated by equating the BSM option price equation to the mid-point of its bid-ask spread and then solving the equations to derive the volatility value. Using the Newton Raphson Method, one of the fastest root-finding approximation methods, we compute the implied volatility for which the BSM prices match exactly with the mid-points of both the ATM call & put bid-ask spreads.

Newton Raphson Method:

where,

f(x_n) is a function that is the theoretical (BSM) option price – the actual option price,

f`(x_n) is the Vega, or the option price sensitivity to implied volatility

The implied volatility calculated for the ATM call option from the above method was 16.40%. The same method was used to calculate the implied volatility of the ATM put option, which gave a value of 13.60%.

2. Option Greeks and sensitivities

Options are often used as risk management tools for hedging the portfolios. The option Greeks can be used to manage portfolio risk containing options, futures and stocks. The Greeks are referred to the quantities of sensitivities of option price due to changes in the factors that determine the value of the option. These factors include the stock price, volatility, interest rate and time to expiration.

2. 1. Delta:

It measures the changes in the option price to changes in the price of underlying. It is referred to as the first derivative of the option value with respect to the price of the underlying. When option risk is being monitored the delta is used as hedge ratio to have a delta neutral portfolio.

2. 2. Gamma:

It measure the changes in delta to changes in the price of the underlying. Gamma is the second derivative of the option value with respect to the price of the underlying. It is maximum when the option is at-the-money and has a high vega.

The graph shows the change in option prices for a given change in SPX spot prices. We can infer that the gap between BSM price and TS approximation is lower at the strike and the gap widens as we approximate away from the strike.

As per the above plot, the values obtained from both BSM and the Taylor series expansion were close at strike price when the option is at-the-money. When option is out-the-money or in-the-money, the change in stock does not result in equal change in call price considering the price interval of -30% to +30%, hence there is a gap between BSM price and the price derived from Taylor series. Another observation is that the second order TS approximations are closer to the actual BSM prices in comparison to the first order TS approximations, and the gap reduces further for higher order TS approximations.

2. 3. Vega:

It measures the sensitivity of option price to changes in volatility of the underlying asset. While hedging, it is important to monitor Vega since both call and put options are affected with increase in volatility. Some of the option strategies benefit with increase in the volatility while others could incur huge losses in changes in volatility. Vega is positive for long positions in options and negative for short exposures. Volatility risk can be neutralised by aiming for Vega which is neither positive nor negative. The Vega is maximum when the option is at-the-money due to the observed higher volatility.

As volatility of the underlying security increases, both call & put options tend to rise because, higher volatility increases liquidity and possibility of options finishing in-the-money, thus making them more valuable.

2. 4. Theta:

It measures the changes in option price due to the passage of time. Option value is characterised by time value and intrinsic value. With passage of time, the option loses its time value as it approaches its expiration date. Theta is always negative for long call and put positions and always positive for short positions. Theta is generally not used for hedging options. The objective of option traders is to ensure that profits due to vega and gamma exposures in long positions outwit the losses due to theta decay.

*As time to expiry decreases, the value of the option also decreases due to fall in the time value of the option (time decay).*

2. 5. Rho:

It measures the changes in the option price to changes in risk free interest. In the hedging process, rho is the least utilised among the Greeks as the option value is less sensitive to risk free rate compared to other factors mentioned above.

Call option prices have a positive relationship with the risk-free interest rates because as the interest rates rise, buying calls (as opposed to buying the underlying security) becomes more profitable and that drives the option prices higher. In the case of put options, however, the opportunity cost of buying put options (as opposed to short selling the underlying) is greater in high interest rate environments than in low interest rate environments, and this drives put prices lower with any rise in interest rate.

3. Volatility Forecasting and comparison with IV

The GARCH(1, 1) model was used to forecast the volatility of the price of the underlying and compared with the various observed volatility measures as described below.

Annualised volatility values (%) as on the option trade date

The realised volatility was lower than the forecasted volatility. Implied volatility of options was calculated to be higher which could be due to the fact the market takes into consideration the dynamic hedging transaction costs and gap risk. As can be inferred from the above table, the forecasted SPX volatility (14.50 %) is lesser than the VIX value (18.53 %) on the trade date. This indicates that the general market has priced in higher volatility levels than as per anticipated, and thus the option appears to be overvalued as on the trade date. It's further supported by the higher implied volatility values observed for both ATM call and put options, i.e. the option prices have factored in higher volatility values, thus shooting up their premiums which leaves room for some price correction over time to expiry.

4. Options-based trading strategy

Given the scenario, a profitable trading strategy would be to construct an option spread portfolio to trade the volatility using a Short Straddle strategy where we short both, ATM call and ATM put options with the same strike and time to maturity. The rationale for this strategy is that the Implied Volatility is expected to reduce significantly by option expiry, allowing most, if not all, of the premium received on the short put and short call positions to be retained. Volatility trading refers to the trading strategies that provide an exposure to implied and realised volatilities of underlying asset.

Payoff function plot for the Short Straddle strategy

Maximum Profit Potential:

Overall portfolio profit potential is limited to the total premiums ($ 42.7) received less any transaction costs (transactions not factored in the above plot). The maximum profit is earned if the short straddle is held to expiration and the index price closes exactly at the strike price ($ 1120.00), i.e. both the options expire worthless.

Maximum Loss Potential:

As seen in the above plot, potential loss is unlimited on the upside because the index price can rise indefinitely. There's a substantial downside risk too since the index prices can crash significantly from current levels to reach an absolute zero (highly unlikely for an index).

There are 2 potential breakeven points:

Strike price plus total premium => 1120.0 + 42.7 = $ 1162.7

Strike price minus total premium => 1120.0 - 42.7 = $ 1077.3

So, if the SPX index price on expiry date is within these bounds, then the portfolio of a short put and a short call will have a positive return and money-ness property of an ITM option payoff. Based on the volatility projections and the trading strategy devised, the portfolio was able to generate a profit of $ 8.30 on expiry date with the above mentioned portfolio composition.

The GitHub link to this project is here. You are free to use it for all your needs.