Question

Given the following prices of puts and calls and the risk free return of 10% until the maturity of those options, find

(i) the price of underlying asset price and

(ii) any trading which can generate arbitrage opportunity and verify that it actually generates an arbitrage opportunity

Strike Price	Call	Put
950	120.0	51.8
1000	93.8	74.2
1020	84.5	84.5
1050	71.8	101.2
1107	51.9	137.2

Answer 1

Call Option Intrinsic Value = Underlying Stock's Current Price – Call Strike Price

Put Option Intrinsic Value = Put Strike Price – Underlying Stock's Current Price

The most simple formula for put/call parity is Call – Put = Stock – strike price

(I) The price of underlying asset price=120-51.8= stock-950 =

68.2+ 950= stock price= 1018.2

ii) The price of underlying asset price=( 93.8-74.2)+1000=1019.6 stock price

iii)The price of underlying asset =( 84.5-84.5)+1020=1020( stock price)

iv) The price of underlying asset = (71.8-101.2)+1050=1020.6 ( stock price)

v)The price of underlying asset= (51.9-137.2)+1107=1021.7( stock price).

(ii)

An important principle in options pricing is called a put-call parity. It says that the value of a call option, at one strike price, implies a certain fair value for the corresponding put, and vice versa. The argument, for this pricing relationship, relies on the arbitrageopportunity that results if there is divergencebetween the value of calls and puts with the same strike price and expiration date. Arbitrageurs would step in to make profitable, risk-free trades until the departure from put-call parity is eliminated. Knowing how these trades work can give you a better feel for how put options, call options and the underlying stocks are all interrelated.

we will look at how we can seek arbitrage opportunities by using the put-call parity equation. As we know, the put-call parity equation is represented as follows:

c + PV(K) = p + s

If the prices of put and call options available in the market do not follow the above relationship then we have an arbitrage opportunity that can be used to make a risk-free profit. In the above equation the left side of the equation represents a fiduciary call and the right side of the equation is called a protective put. Depending on the asymmetry we can take our positions to earn a risk-free profit. We buy the underpriced side and sell the overpriced side. Let’s take an example to understand this.

Let’s say that we have we have the following information for a call and a put option on XYZ stock.

Exercise price: $100

Call option price: $7

Put option price: $5 o

Risk-free rate: 8%

Current market price of XYZ: $98

Time to maturity: 0.5 years

Let’s plug these values in the put-call parity equation:

7 + 100/(1.08)^0.5 = 5 + 99

103.225 = 104

As we can see, the right hand side is greater than the left hand side by (104 – 103.225) = 0.775

To make use of this arbitrage opportunity, we will buy the fiduciary call and sell the protective put.

1. Sell the protective put: We sell a put option and receive the $5 premium. We also short sell the ABC stock and receive $99. The total cash inflow is $104.
2. Buy fiduciary call: We payout a total of $103.225 for the fiduciary call option. That is we pay $7 as premium for the call option and invest 96.225 in a bond for 6 months at 8%.
3. Net cash inflow: Our net cash inflow is (104 – 103.225) = $0.775