Brief illustration of a fixed-for-fixed currency swap (e.g., dollars for euros). Please note: in a plain vanilla interest rate swap, we referred to the NOTIONAL because it is not exchanged (in that case, the notional is required only to compute the interest). However, in a currency swap the PRINCIPAL is exchanged.

At inception, the value of the swap is zero or nearly zero. Subsequently, the value of the swap will differ from zero. Under this approach, we simply treat the swap as two bonds: a fixed-coupon bond and a floating-coupon bond. The value of the swap is difference between the two.

This is a classic building block for Monte Carlos simulation: Brownian motion to model a stock price. The periodic return (note the return is expressed in continuous compounding) is a function of two components: 1. constant drift, and 2. random shock; i.e., volatility multiplied by a randomized critical z value

Relative to plain vanilla options, barrier options have an additional feature: c(S,K, H, volatility, T, r) where H is the barrier. The barrier either knocks-in the option (into existence) or knocks-out (out of existence) the option. Due to this “optionality on the option” the barrier is cheaper (lower premium) than its plain vanilla counterpart. Barrier is either: knock-in (up-and-in, down-and-in) or knock-out (up-and-out, down-and-out)

Put call parity derives from the idea we can have two portfolios (one with an option, the other with a put) that have identical payoffs regardless of what happens to the stock. This gives a way to link the value of a call option with a put option.

This is a brief review of the option Greeks. They are sensitivities: what is the change in option price with respect to [stock price | volatility | rate | term]. Delta: change in option price with respect to stock pric. Gamma: change in delta with respect to stock price. Vega: with respect to volatility. Rho: with respect to rate. Theta: with respect to term

This illustrates how an interest rate swap can transform a floating-rate obligation into a fixed-rate obligation and vice-versa

Delta is one of the option Greeks. It gives the sensitivity of the call option value to changes in stock price. In this example, a delta of 0.61 implies we can hedge a long position in 61 shares by writing (i.e., taking a short position in 100 call options. But note delta is a linear approximation; the hedge requires frequent rebalancing.

This is Black-Scholes for a European-style call option. You can download the XLS at my site @ www.bionicturtle.com