The surface in the presence of

Standard Bank quantitative analyst Roelof Sheppard shows how absolute and proportional payments give rise to arbitrage constraints on the implied volatility surface

Introduction In South African markets, European call and put options trade S + = S − − D liquidly at certain strikes and maturities on the TOPI40, an index t D t D (2) of the top-40 shares based on market capitalisation. In general, where – and + denote the times immediately before and after the it is assumed that dividends on indexes pay frequently, hence it tD tD is appropriate to make use of a dividend , and that implied dividend payment of size D. volatilities for non-market options can be interpolated from The just after the dividend payment is given by:

liquid options. In this article, we show that these assumptions r (T − t ) f + = St e − D can introduce arbitrage around dividend dates. We also provide t ,t D specific arbitrage formulae, which relate implied volatilities on either side of dividend dates. By substituting this into equation (1) we obtain the value of an maturing on +: It is reasonable to assume that different institutions will have tD

different dividend forecasts. Furthermore, these forecasts may be − r (T − t ) r (T − t ) V + (K) = e Black(St e − D,σ + ,K,T ) modelled as discrete dividends, either proportional to the t ,t D t D , K (3) price or an absolute cash amount. In this article we will show how absolute and proportional dividend payments give rise to The value of the option maturing immediately after the dividend different arbitrage constraints on the implied volatility surface. date is the same as the value of the option maturing just before Consequently, two institutions who agree on all the market option a dividend date with an adjusted strike. To see this, consider the prices may imply quite different volatilities and prices for non- following trades: standard options. Trade 1: Buy a maturing on – with strike . If this is tD K+D Arbitrage constraints on implied volatility in the money we buy the stock for K+D. Since we are then the The cash price and volatility of an option are related as usual via: stock when the dividend pays, we will receive D in cash, so we will have effectively paidK for the stock. − r (T − t ) Trade 2: Buy a call option maturing on + with strike . Equation (2) Vt ,T (K) = e Black( ft ,T ,σ T , K ,K,T ) (1) tD K shows that, when trade 1 is in the money so is trade 2 and both will where is the value of a European option at time with a give the holder the stock for a price of K. Similarly and trivially they Vt, T (K) t strike , maturity and implied volatility . This option is written both have the same zero value when they are out of the money. K T σT,K on an underlying with a forward price given by . The ft,T is denoted by r. Alternatively we can see this via: To obtain a relationship for the implied volatility over a dividend

date we consider a European call option with a dividend paid at the + + = + − Vt ,t (K) max(St K,0) maturity of the option. We will discuss discrete absolute dividends D D D = max((S − − D) − K,0) and discrete proportional dividends separately. t D

= max(S − − (K + D),0) Absolute dividends t D Making use of the fact that the spot price will decrease by the amount of = V − − (K + D) the dividend on the dividend date, we obtain the following formula: t D ,t D

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Combining the arbitrage argument above with the fact the value of an option maturing just before a dividend date is given by: 1 Comparison of implied volatilities over a dividend date

Implied volatilities before + + 60% dividend date Vt ,t (K D) D 55% Implied volatilities after − r (T − t ) r (T − t ) absolute dividend = e Black(St e ,σ − _ ,K + D,T ) 50% Implied volatilities after t D , K + D 45% proportional dividend we obtain the following implicit relationship for implied volatilities 40% over dividend dates: 35% 30% Implied volatilities r (T − t ) 25% Black(St e − D,σ + ,K,T ) t D , K (4) 40 50 70 r (T − t ) 60 80 90 100 110 120 130 140 150 = Black(St e ,σ − + ,K + D,T ) t D , K D Strikes Given market prices of options maturing just before a dividend date, equation (4) can be used to obtain arbitrage-free implied volatilities after dividend dates. A numerical root solver such as This equation implies the following arbitrage relationship: Newton’s method can be used to obtain the required implied σ + = σ − K (6) volatility s + . t D , K t D , − α tD,K (1 ) Proportional dividends Once again this equation can be used to obtain arbitrage-free If we assume now that the dividend is proportional to the stock price implied volatilities after dividend dates from market prices of options maturing just before a dividend date. just before the dividend, i.e., D=aS – equation (2) becomes: tD An example + = − − α (5) St St (1 ) Consider a case when the underlying spot price is 100, the agreed D D forward price is 105 and a dividend is paid in one year’s time at the The value of the option maturing just after the dividend is now maturity of the option. If we assume that the interest rate is 7%, it given by: can be shown that this market data implies an absolute dividend of − r (T − t ) r (T − t ) and a proportional dividend of a . V + (K) = e Black(S − (1− α)e ,σ + ,K,T ) D=2.25 =2.1% t ,t D t D t D , K Given a skew just before a dividend date, figure 1 shows which implied volatilities after a dividend date should be used to avoid In this case we have the following relationship: arbitrage. From figure 1, it is clear that absolute and proportional dividends will enforce different arbitrage constraints on the implied volatility V + + (K) = max(S + − K,0) t D ,t D t D surface.

= max(S − ()1− α − K,0) t D Conclusion These discontinuities in volatility over dividend dates are not seen = − α − − K (1 )max St ,0 on the market implied surface since options are not available ( D (1− α) ) immediately before and after the dividend dates. These arbitrage constraints are significant when: = (1− α)V − − K t D ,t D ()(1− α) Option traders transact over-the-counter options and interpolate to obtain volatilities. From this equation we obtain the following arbitrage relationship Models are calibrated to the whole implied volatility surface to for implied volatilities over a dividend date: price exotic options.

r (T − t ) Black(S − (1− α)e ,σ + ,K,T ) t D t D , K Roelof Sheppard Quantitative Analyst ⎛ r (T − t ) ⎞ = − α − σ − K (1 )Black St e , t , K , ,T T. +27 11 378 7152 ⎝ D D (1− α ) (1− α) ⎠ M. +27 71 603 7576

⎛ r (T − t ) ⎞ E. [email protected] = − − α σ − Black St (1 )e , t , K ,K,T ⎝ D D (1− α ) ⎠ www.standardbank.co.za

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