Black-Scholes. Ser-Huang Poon. September 29, 2008

Black-Scholes Ser-Huang Poon September 29, 2008 A European style call (put) option is a right, but not an obligation, to purchase (sell) an asset at a strike price on option maturity date, T. An American style option is a European option that can be exercised prior to T. 1 Black-Scholes Model The Black-Scholes formula below is for pricing European call and put options: c = S 0 N (d 1 ) Ke rt N (d 2 ) (1) p = Ke rt N ( d 2 ) S 0 N ( d 1 ) d 1 = ln (S 0/K)+ r + 1 2 σ2 T σ (2) T d 2 = d 1 σ T Z 1 d1 N (d 1 ) = e 0.5z2 dz 2π where c (p) is the price of the European call (put), S 0 is the current price of the underlying assset, K is the strike or exercise price, r is the continuously compounded risk free interest rate, T is the time to option maturity. N (d 1 ) is the cumulativ probability distribution of a standard Normal distribution for area below d 1,andN ( d 1 )=1 N (d 1 ). As T 0, d 1 and d 2 N (d 1 ) and N (d 2 ) 1 N ( d 1 ) and N ( d 2 ) 0 1

which means c S 0 K, p 0 for S 0 >K (3) c 0, p K S 0 for S 0 <K. (4) As σ 0, again N (d 1 ) and N (d 2 ) 1, N ( d 1 ) and N ( d 2 ) 0. This will lead to c S 0 Ke rt,and (5) p Ke rt S 0. (6) The boundary conditions (3), (4), (5) and (6) are the boundary conditions for checking option prices before using them for empirical tests. These conditions are not specific to Black-Scholes. Option with market prices (transaction or quote) violating these boundary conditions should be discarded. 1.1 The Black-Scholes assumptions The key Black-Scholes assumptions are: The stock price follows a Geometric Brownian Motion with drift μ and volatility σ ds = μ+ σdw S (7) where W is a Wiener process Short selling of the stock is allowed with full use of proceeds There are no taxes and transactions cost It is possible to lend/borrow at a risk-free interest rate All securities are infinitely divisible The underlying stock does not pay any dividend (cf. this assumption can easily be extended to include dividend payments) There are no arbitrage opportunities (i.e. the market is in equilibrium) 2

Continuous trading (so that rebalancing of portfolio is done instantaneously) Empirical findings suggest that option pricing is not sensitive to the assumption of a constant interest rate. There are now good approximating solutions for pricing American style options that can be exercised early and options that encounter dividend payments before option maturity. The impact of stochastic volatility on option pricing is much more profound. Apart from the constant volatility and the related GBM assumptions, the violation of any of the remaining assumptions will result in the option price being traded within a band instead of at the theoretical price. 2 Black-Scholes and no-arbitrage pricing 2.1 The stock price dynamics The Black-Scholes model for pricing European equity options assumes stock price has the following dynamics and for the growth rate on stock ds = μs + σsdz, (8) ds = μ + σdz. (9) S From ito lemma, the logarithmic of stock price has the following dynamics d ln S = µμ 12 σ2 + σdz, (10) which means that stock price has a lognormal distribution or the logarithm of stock price has a normal distribution. In discrete time d ln S = µμ 12 σ2 + σdz ln S = µμ 12 σ2 t + σε t ln S T ln S 0 N µμ 12 σ2 T, σ T ln S T N ln S 0 + µμ 12 σ2 T, σ T. (11) 3

2.2 Black-Scholes PDE The derivation of the Black-Scholes partial differential equation is based on the fundamental fact that the option price and the stock price depend on the same underlying source of uncertainty. A portfolio can then be created consisting of the stock and the option which eliminates this source of uncertainty. Given that this portfolio is riskless and must therefore earn the risk-free rate of return. Here is how the logic works: S = μs t + σs z (12) f f f = μs + S t + 1 2 f 2 S 2 σ2 S 2 t + f σs z (13) S We set up a hedged portfolio, Π, consistingof f S number of shares and short one unit of the derivative security. The change in portfolio value is Π = f + f S S f f = μs + S t + 1 2 f 2 S 2 σ2 S 2 t f S σs z + f f μs t + S S σs z f = t + 1 2 f 2 S 2 σ2 S 2 t Note that uncertainty due to z is cancelled out and μ, the premium for risk (returns on S), is also cancelled out. No only that Π is has no uncertainty, it is also preference free and not depend on μ, a parameter controlled by investor s risk aversion. If the portfolio value is fully hedged, then no arbitrage implies that it must earn only risk free rate of return µ r f + f S S rπ t = Π rπ t = f + f S S t = f f μs + S t + 1 2 f 2 S 2 σ2 S 2 t f S σs z + f [μs t + σs z] S r ( f) t = rs f f f t μs t S S t t 1 2 f 2 S 2 σ2 S 2 t f f f σs z + μs t + S S S σs z 4

and finally we get the well known Black-Scholes PDE 2.3 Solving the PDE rf = rs f S + f t + 1 2 2 f S 2 σ2 S 2 (14) There are many solutions to (14) correspond to different derivatives, f, with underlying asset S. In another words, without further constraints, the PDE in (14) does not have a unique solution. The particular security being valued is determined by its boundary conditions of the differential equation. In the case of an European call, the value at expiry c (S, T )=max(s E,0) serves as the final condition for the Black-Scholes PDE. Here, we show how BS formula can be derived using the risk neutral valuation relationship. We need the following facts: (i) From (11), ln S N µ ln S 0 + μ 1 2 σ2,σ. Under risk neutral valuation relationship, μ = r and µ ln S N ln S 0 + r 1 2 σ2,σ. (ii) If y is a normally distributed variable Z µ e y μy a f (y) dy = N + σ y e μ y + 1 2 σ2 y. a (iii) From the definition of cumulative normal distribution Z µ µ a μy μy a f (y) dy =1 N = N a σ y σ y σ y Now we are ready to solve the BS formular. of a call is First, the terminal value c T = E [max (S K, 0)] = = Z K Z ln K (S K) f (S) ds e ln S f (ln S) d ln S K Z ln K f (ln S) d ln S. 5

Substituting facts (ii) and (iii) and using information from (i) to set μ y = lns 0 + r 1 2 σ2 σ y = σ a = lnk we get Ã! ln S0 + r + 1 2 σ2 ln K Ã! ln S0 + r 1 2 σ2 ln K c T = S 0 e r N ln K KN ln K = S 0 e r N (d 1 ) KN (d 2 ) (15) where d 1 = ln S 0 K + r 1 2 σ2 σ d 2 = d 1 σ. The present value of the call option is derived by applying e r to both sides. The put option price can be derived using put-call parity or use the same argument above. The σ in the above formular is volatility over the option maturity. If we use σ as the annualised volatility then we replace σ with σ T in the formula. There are important insights from (15), all valid only in a risk neutral world: (i) N (d 2 ) is the probability that the option will be exercise. (ii) Alternatively, N (d 2 ) is the probability that call finishes in the money. (iii) XN (d 2 ) is the expected payment. (iv) S 0 e rt N (d 1 ) is the expected value E [S T X] +,wheree [ ] + is expectation computed for positive values only. (v) In another words, S 0 e rt N (d 1 ) is the risk neutral expectation of S T, E Q [S T ] with S T >X. 6

3 Binomial method In a highly simplified example, we assume a stock price can only move up by one node or move down by one node over a three-month period as shown below. The option is a call option for the right to purchase the share at $21 at the end of the period (i.e. three month s time). stock price =20 option price = c @ @@ stock price =22 option price =1 stock price =18 option price =0 Construct a portfolio consist of amount of shares and short one call option. If we want to make sure the value of this portfolio is the same whether it is upstate or down state, then $22 $1 = $18 +$0 = 0.25 stock price =20 portfolio value =4.5e 0.12 3/12 =4.367 @ @@ stock price =22 portfolio value =22 0.25 1=4.5 stock price =18 portfolio value =18 0.25 = 4.5 Given that the portfolio s value is $4.367, this means that $20 0.25 f = $4.367 f = $0.633. This is the value of the option under no arbitrage. From the above simple example, we can do the following generalisation; 7

S 0 u f u S 0 f @ @@ @ @ S 0 d f d The amount is calculated using S 0 u f u = S 0 d f d = f u f d S 0 u S 0 d. (16) Since the terminal value of the risk less portfolio is the same in the upstate and in the downstate, we could use any one of the values (say upstate) to establish the following relationship S 0 f = (S 0 u f u ) e rt f = S 0 (S 0 u f u ) e rt. (17) Substitute the value of from (16) into (17), we get f = S 0 f µ u f d S 0 u S 0 d S 0 u f u f d S 0 u S 0 d f u = f µ u f d u d u f u f d u d f u e rt µ e rt (f u f d ) = u (f u f d ) + uf u df u e rt u d u d u d µ e rt f u e rt f d = + uf d df u e rt u d u d µ e rt d = u d f u + u ert u d f d e rt. By letting p = ert d u d,weget e rt f = e rt [pf u +(1 p) f d ] (18) and 1 p = u d ert + d u d 8 = u ert u d.

As we can see from (18) that although p is not the real probability distribution of stock price, it has all the characteristics of a probability measure (viz. sum to one and nonnegative). Moreover, when the expectation is calculated based on p, the expected terminal payoff is discounted using the risk free interest. Hence, p is called the risk neutral probability measure. We can verify that the underlysing asset S also produce risk free rate of returns under this risk neutral measure. µ e S 0 e μt rt µ d u e rt = S 0 u + S 0 d u d u d e μt = uert ud + ud de rt u d (u d) ert = = e rt u d μ = r The actual return of the stock is no longer needed and neither is the actual distribution of the terminal stock price. (This is a rather amzing discovery in the study of derivative securities!!!) 3.1 Matching volatility with u and d We have already seen in previous section and equation (18) that the risk neutral probability measure is set such that the expected growth rate is the risk free rate, r. µ e rt d f = u d f u + u ert u d f d e rt = [pf u +(1 p) f d ] e rt p = ert d u d This immediately leads to the question how do we set the values of u and d? The key is that u and d are jointly determined such that the volatility of the binomial process equal to σ which is given or can be estimated from stock prices or prices of the asset underlying the option contract. Given that there are two unknown and there is only one constant σ, thereareafew different ways to specify u and d. The good or better ways are those that guarantee the nodes recombined after an upstate followed by a downstate, and vice versa. In Cox, Ross and Rubinstein (1979), u and d are defined as 9

follow: u = e σ δt,andd = e σ δt. It is easy to verify that the nodes recombines since ud = du =1. So after each up move and down move (and vice versa), the stock price will return to S 0. To verify that the volatility of stock returns is approximately σ δt under the risk neutral measure, we note that The expected stock returns is Var = E x 2 [E (x)] 2 ln u = σ δt, and ln d = σ δt E (x) = p ln S 0u +(1 p)ln S 0d S 0 S 0 = pσ δt (1 p) σ δt = (2p 1) σ δt, and E x 2 = µ p ln S 2 µ 0u +(1 p) ln S 2 0d S 0 S 0 = pσ 2 δt +(1 p) σ 2 δt = σ 2 δt. Hence Var = σ 2 δt (2p 1) 2 σ 2 δt = σ 2 δt 1 4p 2 +4p 1 = σ 2 δt 4p (1 p). It has been shown else where that as δt 0, p 0.5 and Var σ 2 δt. 10

3.2 A two-step binomial tree and American style options p 1 S 0 u f u p 2 S 0 u 2 f uu S 0 f @ @@ 1 p 2 @ @@ p 2 @ S 0 ud f ud 1 p 1 @ @ S 0 d f d @ @@ 1 p 2 @ @ S 0 d 2 f dd The binomial tree is often constructed such that the branches recombine. If the volatilities in the period 1 and period 2 are different, then, in order to make the binomial tree recombined, p 1 6= p 2. (This is a more advanced topic in option pricing.) Here, we take the simple case where volatility is constant, and p 1 = p 2 = p. Hence, to price a European option, we simply take the expected terminal value under the risk neutral measure and discount it with a risk free interest rate, as follows: h i f = e r 2δt p 2 f uu +2p (1 p) f ud +(1 p) 2 f dd. (19) Note that the hedge ratio for state 2 will be different depending on whether state 1 is an upstate or a downstate 0 = 1,u = 1,d = f u f d S 0 u S 0 d f uu f ud S 0 u 2 S 0 ud f ud f dd S 0 ud S 0 d 2 This also means that for such a model to work in practice, one has to be able to continuously and costlessly rebalance the composition of portfolio of stock and option. This is a very important assumption and should not be overlooked. 11

We can see from (19) that the intermediate nodes are not require for the pricing of European options. What are required are the range of possible values for the terminal payoff and the risk neutral probability density for each node. This is not the case for the American option and all the nodes in the intermediate stages are needed because of the possibility of early exercise. 4 Black-Scholes and Deterministic Volatility Here, we depart from Black-Scholes constant volatility assumption by allowing volatility to be a deterministic positive function of time and stock price, such that σ = σ (S, t): ds = μs + σ (S, t) SdW We denote by V (S, t) the option price and we form the portfolio We select = V S Π = V S =0 dπ = dv ds = ½ V t + 1 2 σ2 S 2 2 V S 2 and get dπ = ¾ + V ds ds S ½ V t + 1 ¾ 2 σ2 S 2 2 V S 2 As the portfolio change is deterministic, it should be equal to the risk-free return on the portfolio for no arbitrage, ½ rπ = r V V ¾ S S Therefore, we obtain dπ = rπ V t + 1 2 σ2 S 2 2 V S 2 = rv r V S S (20) V t + 1 2 σ2 S 2 2 V S 2 + r V S rv S = 0 (21) Here, the market is complete because the randomness of the volatility was introduced as a function of the existing randomness of the lognormal stock 12

price process. Hence, a unique risk neutral measure exists underwhich the stock price is a geometric Brownian motion with drift rate r and the same volatility σ (S, t). 4.1 Time-dependent volatility Here,wetakeaspecialcasebyassumingthatσ (S, t) =σ (t) is a function of time only (i.e. deterministic and time dependent). Now we introduce notations for three new variables S, V and t as follows 1 : From S = Se α(t),weget From t = β (t), weget S = Se α(t), V = Ve α(t), and t = β (t). S = Se α(t) ds ds = e α(t). Now we can evaluate 1 dβ (t) = = 1 dβ (t). V V V = e α(t) = e α(t) S S S eα(t) = V S 2 V S 2 = 2 V S S 2 S = 2 V S 2 eα(t). It is slightly more complex with V t.asv is a function of both S and t and S itself is a function of t, V t = V t + V S S t = V d e α(t) α(t) V + e t + V α(t) dα (t) α(t) V = Ve + e t α(t) dα (t) α(t) V = Ve + e t dα (t) Seα(t) S dβ (t) dβ (t) 1 The derivation here follows that in Wilmott (1998). + V S Se α(t) α(t) dα (t) e + V dα (t) Se α(t) S 13

Substitute them into (21) and get V dα (t) dβ (t) S 2 eα(t) + r V α(t) dα (t) α(t) V Ve + e t + 1 2 σ2 S 2 e 2α(t) 2 V + V t dβ (t) dβ (t) V t + 1 2 σ2 S 2 2 V S 2 + + V dα (t) Se α(t) S S Se α(t) rve α(t) = 0 + V dα (t) S + 1 S 2 σ2 S 2 2 V S 2 + r V S S rv = 0 µ µ r + r + V = 0 dα (t) S V S dα (t) (22) There are many possible solutions to (22) depending on the functional form of α (t) and β (t). Here, we want to show that one particular solution exists for the case α (t) =r (T t) and β (t) =σ 2 (T t). First, we choose α (t) = dα (t) Z T = r, ³ so that the last two terms, r + dα(t) zeroes. Next we choose β (t) = t Z T dβ (t) = σ 2, subtitute the result into (22), and get t r (τ) dτ = r (T t) S V S and ³ r + dα(t) σ 2 (τ) dτ = σ 2 (T t) σ 2 V t + 1 2 σ2 S 2 2 V S 2 = 0 V t V in (22) become = 1 2 S2 2 V S 2. (23) The important point about equation (23) is that it has coefficients which are independent of time and it does not involve σ and r. If we use V S; t to denote any solution of (23), then the corresponding solution of (22), in the original variables, is V = e α(t) V S; t ³ = e α(t) V Se α(t),β(t). (24) 14

Now use V BS to mean any solution of the Black-Scholes equation for constant volatility σ c (and constant interest rate r c ). This solution can be written in the form V BS = e rc(t t) V BS ³Se rc(t t),σ 2 c (T t) (25) for some function V BS. By comparing (24) with (25), it follows that the solution of the time dependent parameter problem is the same as the solution of the constant parameter problem if Z T r c (T t) = r (τ) dτ, or r c = 1 Z T r (τ) dτ, and t T t t Z T σ 2 c (T t) = σ 2 (τ) dτ, or σ 2 c = 1 Z T σ 2 (τ) dτ. T t t That is, the Black-Scholes solution holds if r τ and σ τ are deterministic function of time and we can simply use the Black-Scholes equation based on r c and σ c. However, this comparison is valid only for European options, where there is no possibility of early exercise. t 5 Dividend and early exercise premium As option holders are not entitled to dividends, option price should be adjusted for known dividends to be distributed during the life of the option and the fact that option may have the right to exercise early to receive the dividend. 5.1 Known and finite dividends Assume that there is only one dividend at τ. Should the call option holder decide to exerise the option, she will receive S τ K at time τ and if she decides not to exercise the option, her option value will be worth c (S τ D τ,k,r,t,σ). The Black (1975) approximation involves making such comparisons for each dividend date. If the decision is not to exercise, then the option is priced now at c S t D τ e r(τ t),k,r,t,σ. If the decision is to exercise, then the option is priced according to c (S t,k,r,τ,σ). We note that if the decision isnottoexercise,theamericancalloptionwillhavethesamevalueasthe European call opton calculated by removing the discounted dividend from the stock price. 15

A more accurate formula that takes into account of the probability of early exercised is that by Roll (1977), Geske (1979), and Whaley (1981), and is presented in Hull (2002, appendix 11). These formulae work quite well (even the Black-approximation) for American calls. In the case of American put, a better solution is to implement the Barone-Adesi and Whaley (1987) formula below. 5.2 Dividend yield method When the dividend is in the form of yield it can be easily netted off from the risk free interest rate as in the case of currency option. To calculate the dividend yield of index option, the dividend yield, q, is the average annualised yield of dividends distributed during the life of the option. Ã q = 1 t ln S + P! n i=1 D ie r(t t i) S where D i and t i are the amount and the timing of the ith dividend on the index (t i should also be annualised in a similar fashion as t). The dividend yield rate computed here is thus from the actual dividends paid during the option s life which will therefore account for the monthly seasonality in dividend payments. 5.3 Barone-Adesi and Whaley quadratic approximation Define M = 2r σ 2 and N = 2(r q),then σ 2 for an American call option 2 ( c (S)+A S q2 2 S when S<S S X when S S (26) C (S) = The variable S is the critical price of the index above which the option should be exercised. It is estimated by solving the equation S X = c (S )+ 1 e qt N [d 1 (S )] ª S 2 Note that in Barone-Adesi and Whaley (1987), K(t) is 1 e rt,andisb is (r q). q 2 16

iteratively. The other variables are " r # q 2 = 1 1 N + (N 1) 2 + 4M 2 1 e rt A 2 = S q 2 1 e qt N [d 1 (S )] ª d 1 (S ) = ln (S /X)+(r q +0.5σ 2 )t σ t (27) To compute delta and vega for hedging purpose 3 : ( C = C S = e qt N(d 1 (S)) + A 2 q 2 SS (q2 1) S when S<S 1 when S S ( Λ C = C σ = S t N 0 (d 1 ) e qt when S<S 0 when S S (28) For an American put option, the valuation formula is ( p (S)+A S q1 1 P (S) = S when S>S X S when S S (29) The variable S is the critical index price below which the option should be exercised. It is estimated by solving the equation X S = p (S ) 1 e qt N [ d 1 (S )] ª S iteratively. The other variables are " r # q 1 = 1 1 N (N 1) 2 + 4M 2 1 e rt A 1 = S q 1 1 e qt N [ d 1 (S )] ª d 1 (S ) = ln (S /X)+(r q +0.5σ 2 )t σ t 3 Vega for the American options cannot be evaluated easily because C partly depends on S, which itself is a complex function of σ. The expression for vega in the case when S < S in equation (28) represents the vega for the European component only. This value has been compared with vega produced by numerical method and was found to be quite close to the latter. We used the numerically derived vega in the implementation of vega-neutral trading strategies. q 1 17

To compute delta and vega for hedging purpose: ( P = P S = e qt N (d 1 (S)) + A 1 q 1 S Λ P = P σ = S (q1 1) S when S>S 1 when S S ( C σ = S t N 0 (d 1 ) e qt when S>S 0 when S S 6 Measurement errors and bias Early studies of option implied volatility suffered many estimation problems 4 such as the improper use of the Black-Scholes model for American style option, the omission of dividend payments, the option price and the underlying asset prices were not recorded at the same time, or stale prices were used. Since transactions may take place at bid or ask prices, transaction prices of option and the underlying assets are subject to bid-ask bounce making the implied volatility estimation unstable. Finally, in the case of S&P 100 OEX option, the privilege of a wildcard option is often omitted. 5 In more recent studies, much of these measurement errors have been taken into account. Many studies use futures and options futures because these markets are more active than the cash markets and hence the smaller risk of prices being stale. Conditions in the Black-Scholes model include no arbitrage, transaction cost is zero and continuous trading. The lack of such a trading environment will result in option being traded within a band around the theoretical price. This means that implied volatility estimates extracted from market option prices will also lie within a band. Figlewski (1997) shows that implied volatility estimates can differ by several percentage points due to bid-ask spread and discrete tick size alone. To smooth out errors caused by bidask bounce, Harvey and Whaley (1992) use a nonlinear regression of ATM option prices observed in a ten-minute interval before the market close on model prices. 4 Mayhew (1995) gives a detailed discussion on such complications involved in estimating implied volatility from option prices, and Hentschel (2001) provides a discussion of the confidence intervals for implied volatility estimates. 5 This wildcard option arises because the stock market closes later than the option market. Option trader is given the choice to decide, before the stock market closes, whether or not to trade on an option whose price is fixed at an earlier closing time. 18

Indication of non-ideal trading environment is usually reflected in poor trading volume. This means implied volatility of options written on different underlying assets will have different forecasting power. For most option contracts, ATM option has the largest trading volume. This supports the popularity of ATM implied volatility in forecasting. 6.1 Investor risk preference In the Black-Scholes s world, investor risk preference is irrelevant in pricing options. Given that some of the Black-Scholes assumptions have been shown to be invalid, there is now a model risk. Figlewski and Green (1999) simulate option writer s positions in the S&P 500, DM/$, US LIBOR and T-Bond markets using actual cash data over a 25 year period. The most striking result from the simulations is that delta hedged short maturity options, with no transaction costs and a perfect knowledge of realised volatility, finished with losses on average in all four markets. This is a clear evidence of Black-Scholes model risk. If option writers are aware of this model risk and mark up option prices accordingly, the Black-Scholes implied volatility will be greater than the true volatility. In some situations, investor risk preference may override the risk neutral valuation relationship. Figlewski (1997), for example, compares the purchase of an OTM option to buying a lottery ticket. Investors are willing to pay a price that is higher than the fair price because they like the potential payoff and the option premium is so low that mispricing becomes negligible. On the other hand, we also have fund managers who are willing to buy comparatively expensive put options for fear of the collapse of their portfolio value. Both types of behaviour could cause market price of option to be higher than the Black-Scholes price, translating into a higher Black-Scholes implied volatility. Arbitrage argument does not apply here because these are unique risk preference (or aversion) associated with some groups of individuals. Franke, Stapleton and Subrahmanyam (1998) provide a theoretical framework in which such option trading behaviour may be analysed. 19

7 Appendix: Implementing Barone-Adesi and Whaley s efficient algorithm The determination of S and S in equations (26) and (29) above are not exactly straightforward. We have some success in solving S using NAG routing C05NCF. Barone-Adesi and Whaley (1987), however, have proposed an efficient method for determing S details of which can be found in Barone- Adesi and Whaley (1987, hereafter refer to as BAW) pp. 309 to 310. BAW claimed convergence of S and S canbeachievedwiththreeiterationsor less. American Calls The followings are step-by-step procedures for implementing BAW s efficient menthod for estimating S of the Americal Call. Step 1. Make initial guess of σ and denote this initial guess as σ j with j =1. Step 2. Make initial guess of S, S i (with i =1), as follow; denoting S at T =+ as S ( ) : h i S 1 = X +[S ( ) X] 1 e h 2, (30) where S ( ) = q 2 ( ) = 1 2 X 1 1 q 2 ( ), 1 N + q (N 1) 2 +4M ³ h 2 = (r q) t +2σ ½ t X S ( ) X ¾. (31) Note that the lower bound of S is X. So if S 1 <X, reset S 1 = X. However, the condition S <X rarely occurs. Step 3. Compute LHS and RHS of equation (30) as follow : LHS (S i ) = S i X, and (32) RHS (S i ) = c (S i )+ 1 e qt N [d 1 (S i )] ª S i /q 2. (33) Compute starting value of c (S i ) using the simple Black-Scholes equation (26) and d 1 (S i ) using equation (27). It will be useful to set up a function (or subroutine or variable) for d 1. 20

Step 4. Check tolerance level LHS (S i ) RHS (S i ) /X < 0.00001. (34) Step 5. If equation (34) is not satisfied; compute the slope of equation (33), b i, and the next guess of S, S i+1,asfollow h b i = e qt N [d 1 (S i )] (1 1/q 2 )+ 1 e qt n [d 1 (S i )] /σ i t /q 2, S i+1 = [X + RHS (S i ) b i S i ] / (1 b i ). where n (.) is the univariate normal density function. step 3. Repeat from Step 6. When equation (34) is satisfied, compute C (S) accordingtoequation (26). If C (S) is geater than the observed American call price, try smaller σ j+1, otherwise try a larger σ j+1. Repeat steps 1 through to 5untilC (S) is the same as the observed American call price. Step 6 could be handled by a NAG routine such as C05ADF for a quick solution. American Puts To approximate S for American puts, steps 2, 3 and 5 have to be modified. Step 1. Make initial guess of σ and denote this initial guess as σ j with j =1. Step 2. Make initial guess of S, S i (with i =1), as follow; denoting S at T =+ as S ( ) : S 1 = S ( )+[X S ( )] e h 1, (35) where S ( ) = q 1 ( ) = 1 2 h 1 = X 1 1 q 1 ( ), 1 N ³ (r q) t 2σ ½ t q (N 1) 2 +4M X X S ( ) ¾. (36) 21

Note that the upper bound of S is X. SoifS 1 >X,resetS 1 = X. Again, the condition S >Xrarely occurs. According to Barone- Adesi and Whaley (1987), footnote 9, the influence of (r q) must be bounded in the put exponent to ensure critical prices monotonically decreasing in t, for very large values of (r q) and t. A reasonable bound on (r q) is 0.6σ t, so the critical commodity price declines at least with a velocity e 1.4σ t. This check is required before computing h 1, in equation (36) above. Step 3. Compute LHS and RHS of equation (35) as follow : LHS (S i ) = X S i, and RHS (S i ) = p (S i ) 1 e qt N [ d 1 (S i )] ª S i /q 1. (37) Step 4. Check tolerance level as before LHS (S i ) RHS (S i ) /X < 0.00001. (38) Step 5. If equation (38) is not satisfied; compute the slope of equation (37), b i, and the next guess of S, S i+1,asfollow b i = e qt N [ d 1 (S i )] (1 1/q 1 ) h 1+e qt n [d 1 (S i )] /σ i t /q 1, S i+1 = [X RHS (S i )+b i S i ] / (1 + b i ). Repeat from step 3 above. Step 6. When equation (38) is satisfied, compute P (S) using equation (29). If P (S) is geater than observed American call price, try larger σ j+1, otherwise try a smaller σ j+1. Then repeat steps 1 through to 5 until P (S) is the same as the observed American put price. Similar to the case for the American call, step 6 could be handled by a NAG routine such as C05ADF for a quick solution. 22