Mini Chapter One
Interest rate Options
Interest rate Options – are options on interest rates that at it’s very basic construct can be described in three types a) caps b) Floors c) Swaptions (option on swaps). Most other structures and strategies can be understood by breaking them down into these basic types. Before taking a plunge into rate options let’s first understand the key variations made to the original Black Scholes Merton or Black Scholes (equity options) pricing model to arrive at the Black Model (also known as Black 76 based on Fischer Black’s research on commodity futures pricing in 1976) used to price interest rate options.
Black Scholes Merton (BSM) vs Black 76:
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- BSM pricing model used spot value of the underlying as a key parameter for arriving at option values, while Black 76 replaced it with the future/forward value of the underlying. But this isn’t anything more than a mathematical substitution as , hence it shouldn’t be seen as a fundamental difference.
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- Both models assume the underlying to have a lognormal distribution i.e. for the underlying to attain only positive values (Black 76 wasn’t built for negative interest rates). A key difference between the models however was on their assumptions on the nature of the movement of the underlying – BSM assumed a Geometric Brownian motion with a constant drift and volatility through time (following generalised weiner) while Black 76 assumed a driftless stochastic movement with the underlying’s volatility being a function of its absolute level and time (akin to an Ito process).
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- By way of notation this can be expressed as: for an underlying with spot value S and future value F,
Terms on the right hand side of the equations are the drift and random process for the underlying. Notice that a constant r and σ under BSM imply that the value of a derivative of S would only depend on its current value and time t.
- Both models assume zero-arbitrage market conditions and use risk-neutral valuation (probability measure) i.e. risk preferences (degree of risk aversion on the underlying) do not play any role in pricing the option and all future expected values are discounted at the risk free rate. This implies that the value of the forward today (at time 0) is equal to the expected future value (spot earns the risk-free rate) discounted at the risk free rate.
- BSM however assumes a constant risk free rate at all times which is inappropriate for pricing interest rate options where the underlying would have a volatility of its own. Since the Black model prices options on the future values of the underlying any divergence between the expected future value and the realised forward is offset by margin calls in the run-up to the maturity of the futures contract (this is why the movement of the underlying is drift-less). Hence even if we considered a non-risk neutral world (where return on investment was higher or lower than the risk free rate) lending variability to the underlying rate of return r, this would make no difference to the present value/pay-off of the derivative investment if it’s future cashflows are discounted at the same rate of return, assuming of course that pay-off is only a function of r for the holding period t.
- Intuitively then, a stock that pays no dividend in the BSM world (a key assumption) can be seen as paying a dividend yield equal to the risk free rate in the Black 76 world.
- Pay off on an interest rate option is a non-linear function of interest rates. While vols get quoted in terms of the (basis point) interest rate vol, it’s conversion to a premium amount in cents needs to take into account the duration and convexity of the underlying swap across same notionals (more on this later).
Black model Equations
Where Df (T’) is the Discount factor for the time period T’ when the option pay-off is realised while T is the option expiry period
( note that
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Further,
Relative to the d1, d2 descriptions in the Black Scholes Merton model that has the ‘r’ component, in the Black notations above its already embedded in F0.
