Question

class: Derivative Securities

. Company X wishes to borrow US dollars at a fixed rate of interest. Company Y wishes to borrow Japanese Yen at a fixed rate of interest. The amount required by the two companies is the same at current exchange rate. The companies are subject to the following interest rates:

	Yen	Dollar
Company X	5%	8.5%
Company Y	6.3%	9%

Design a swap that will net a bank, acting as intermediary, 30 basis points per annum and will appear equally attractive to X and Y.

Q5. Briefly explain Delta, Theta, Gamma, and Vega.

Answer 1

X wants to borrow US Dollars (USD)

Y wants to borrow Yen (JPY)

Per annum Differentials of JPY Rates = 6.3 - 5 = 1.3%

Per annum Differentials of USD Rates = 9-8.5 = 0.5%

Per annunm Net gain for all the 3 parties in this swap (X, Y, Bank) = 1.3 -0.5 = 0.8 %

Bank requires 30 basis points i.e. 0.30%

So the remaining net gain for the 2 parties is = 0.8 - 0.3 = 0.5%

Given that the swap should appear equally attractive to X and Y. so the net gains should be divided equally between X and Y i.e. 0.25% each

So the swap should lead to X borrowing dollars at 8.5 -0.25 = 8.25%

So the swap should lead to Y borrowing yen at 6.3 - 0.25 = 6.05%

Swap arrangement can be viwed as below:

Delta:

Delta is a measure of option's price sensitivity to changes in stock price. Qunatatively, It measures the amount of option value changes per 1$ change in the underlying stock.

Calls delta is posiitve i.e. between 0 to 1. That means if the stock price goes up a the price for the call will go up.

Put delta is negative i.e. between 0 to -1. That means if the stock goes up the price of the option will go down.

As an option gets further in-the-money, the probability it will be in-the-money at expiration increases as well. So the option’s delta will increase.

Gamma:

Gamma is a measure of  option price sensitivity to changes in Delta. This is a second level derivative of delta. Gamma is the rate that delta will change based on a $1 change in the stock price. Delta and Gamma can be visualized as Speed and Acceleration.

Gamma is always posiitve.

Theta:

Theta is a measure of the time decay of an option as the option move towards to expiry. Theta is also referred to as time decay. Thus theta is the amount the price of calls and puts will decrease for a one-day change in the time to expiry.

Theta is always negative because the value of an option always goes down with each passing day

Vega:

Vega is a measure of sensitivity of the option price to changes in volatility. if the volatility in the market goes up then the value of the call and put options go up. Vega is the amount by which a call and put prices will change for a corresponding change in implied volatility.

Increase in volatility will increase the value of call and put, while a decrease in volatility will decrease the value of call and put.

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