Department of Mathematics. Mathematics of Financial Derivatives

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1 Department of Mathematics MA408 Mathematics of Financial Derivatives Thursday 15th January, pm 4pm Duration: 2 hours Attempt THREE questions MA408 Page 1 of 5

2 1. (a) Suppose 0 < E 1 < E 3 and E 2 = 1(E E 3 ). A butterfly spread can be created by holding a European call option with exercise price E 1, and holding a European call option with exercise price E 3, and writing two European call options with exercise price E 2. (The call options are on the same asset and have the same expiry date.) (i) Give a formula for the payoff at expiry of the butterfly spread, in terms of the quantities above and the asset price at expiry, and sketch the corresponding payoff diagram. (5 marks) (ii) What type of final asset price does the holder of this butterfly spread wish to see? (2 marks) (iii) Show that a butterfly spread with exactly the same payoff diagram as that in (i) can be obtained using only a combination of European put options. Justify your answer. (5 marks) (iv) Use put-call parity to show that the cost of setting up the butterfly spread in (i) is identical to the cost of setting up the butterfly spread in (iii). [You may quote the put-call parity relation without proof.] (4 marks) (b) Set up a portfolio involving cash-or-nothing call options that recreates the payoff diagram shown below, justifying your answer. Give, with justification, conditions on E 4 and A under which this portfolio has greater time-zero value than the the butterfly spread in part (a). Payoff Diagram A Payoff 0 E1 E4 S(T) (4 marks) MA408 Page 2 of 5

3 2. Following the notation used in the lecture course, let C(S, t) denote the value of a European call option for asset price S and time t. The value C(S, t) satisfies the Black-Scholes partial differential equation (PDE) C t σ2 S 2 2 C C + rs S2 S rc = 0, (1) for S 0 and 0 t T, and satisfies the final-time condition C(S, T ) = max(s E, 0). Here, r is the interest rate, E is the exercise price and T is the expiry date. (a) What conditions must C(S, t) satisfy for S arbitrarily small and for S arbitrarily large? Explain your answer. (2 marks) (b) Explain and justify, in words, the no-arbitrage assumption. (3 marks) (c) Explain why an arbitrage opportunity exists if the inequality S + P (S, t) C(S, t) < Ee r(t t) holds. (Here, P (S, t) denotes the value of a European put option for asset price S and time t.) (4 marks) (d) What final-time condition on P (S, T ) must be imposed, and what conditions must P (S, t) satisfy for S = 0 and for large S? Explain your answer. (3 marks) (e) Show that C(S, t) = KS, where K is constant, solves the Black-Scholes PDE, and give an accompanying financial explanation. (3 marks) (f) Consider the following quote from Eugene Fama, who was Myron Scholes thesis adviser. If the population of price changes is strictly normal, on the average for any stock... an observation more than five standard deviations from the mean should be observed about once every 7,000 years. In fact such observations seem to occur about once every three to four years. Given that for X N(µ, σ 2 ), P( X µ > 5σ) = , deduce roughly how many observations per year Fama is implicitly assuming to be made. (5 marks) MA408 Page 3 of 5

4 3. Let r denote the interest rate, E the exercise price and T the expiry date, as in Question 2. The Black-Scholes formula for the value of a European call option has the form C(S, t) = SN(d 1 ) Ee r(t t) N(d 2 ), where N( ) is the cumulative Normal(0, 1) distribution function, so and N(x) = 1 2π x e 1 2 y2 dy, d 1 = log(s/e) + (r σ2 )(T t) σ, T t d 2 = log(s/e) + (r 1 2 σ2 )(T t) σ = d 1 σ T t. T t (a) Prove that SN (d 1 ) e r(t t) EN (d 2 ) = 0. (4 marks) (b) Using (a), show that C = S T tn (d σ 1 ). Give a financial explanation for the fact that C is never negative. σ (4 marks) (c) Regarding C as a function of σ alone, show that and lim C(σ) = S σ lim σ 0 + C(σ) = max(0, S Ee r(t t) ). Then, appealing to part (b) or otherwise, justify the inequality max(0, S Ee r(t t) ) C(σ) S. (5 marks) (d) Given that 2 C/ σ 2 may be written in the form 2 C σ 2 = T t 4σ 3 ( σ4 σ 4 ) C σ, where C/ σ has a unique maximum at σ = σ, show that, with an appropriate starting value, Newton s Method provides a convergent iteration scheme for computing an implied volatility. (7 marks) MA408 Page 4 of 5

5 4. (a) Define what is meant by an American call option, using the terms holder, writer, exercise price and expiry date, and explain why an American call option has the same value as a European call option. [You may quote the put-call parity relation without proof.] (4 marks) (b) Let S(t), C am (t) and P am (t) denote the values of an asset, an American call option and an American put option at time t, respectively, with exercise price E. Justify the American option put-call inequalities S(t) E C am (t) P am (t) S(t) Ee r(t t). (6 marks) (c) Recall that the Binomial Method models the asset price only at discrete time points t = iδt, based on the assumption that between successive time levels the asset price either moves up by a factor u or down by a factor d. We may define a random variable Y i such that Y i = 1 if the asset price goes up from time t = iδt to t = (i + 1)δt and Y i = 0 if the asset price goes down. Explain why the asset price S(nδt) at time t = nδt may be written S(nδt) = S(0)u P n i=1 Y i d n P n i=1 Y i, and, very briefly, say how the Central Limit Theorem can be used to tune this asset model to the Black-Scholes version. (4 marks) (d) In about 250 words, describe the concepts of implied volatility and historical volatility. Your description should state one advantage of implied volatility over historical volatility, and vice versa. (6 marks) END OF PAPER (Prof. D. J. Higham & Prof. X. Mao) MA408 Page 5 of 5

6 QUESTION 1) a) (i )Holding the two call options contributes max(s E 1, 0)+max(S E 3, 0) and writing the call option contributes 2 max(s E 2, 0). Hence, value at expiry date is max(s E 1, 0) + max(s E 3, 0) 2 max(s E 2, 0). (1) Payoff diagram is shown in Figure 1 below. Note for future reference that the payoff curve has maximum value of M := 1 2 (E 3 E 1 ) at S = E 2. Payoff Diagram: Butterfly Spread M Payoff 0 E_1 E_2 E3 S(T) Figure 1: Payoff diagram for 1 (a) (i). (ii) The spread holder is hoping that the final asset price, S(T ), is as close as possible to E 2, and certainly between E 1 and E 3. (iii) Consider holding a European put option with exercise price E 1, and holdng a European put option with exercise price E 3, and writing two European put options with exercise price E 2 = 1 2 (E 1 + E 3 ). The value at expiry is max(e 1 S, 0) + max(e 3 S, 0) 2 max( 1 2 (E 1 + E 3 ) S, 0) To show that this matches the payoff in equation (1), we note that it is piecewise linear with corners at S = E 1, S = E 3 and S = 1 2 (E 1 +E 3 ). At S = E 1 the payoff is 0 + E 3 E 1 2( 1 2 (E 1 + E 3 ) E 1 ) = 0. 1

7 At S = 1 2 (E 1 + E 3 ) the payoff is 0 + E (E 1 + E 3 ) + 0 = 1 2 (E 3 E 1 ). At S = E 3 the payoff is = 0. Hence, we have matched the payoff (1). (iv) The difference between the time zero values of the two portfolios is [C(E 1, 0) P (E 1, 0)] + [C(E 3, 0) P (E 3, 0)] 2 [C(E 2, 0) P (E 2, 0)]. (2) Now put-call parity says (with t = 0): In (2) we get C(E 1, 0) P (E 1, 0) = S 0 E 1 e rt, C(E 2, 0) P (E 2, 0) = S 0 E 2 e rt, C(E 3, 0) P (E 3, 0) = S 0 E 3 e rt. S 0 E 1 e rt + S 0 E 3 e rt 2(S 0 E 2 e rt ) = e rt (2E 2 (E 1 + E 3 )) = 0, as required. (b) Consider a portfolio consisting of: Hold a cash-or-nothing call option with exercise price E1. Write a cash-or-nothing call option with exercise price E4. Each one pays off the constant value A when it expires in the money. Then, along the x-axis, For x < E1 payoff is zero. For x = E1 payoff is A/2 + 0 = A/2. For E1 < x < E4 payoff is A + 0 = A. For x = E4 payoff is A A/2 = A/2. For x > E4 payoff is A A = 0. Hence we have the required payoff. If we set A M := 1 2 (E 3 E 1 ) and E 4 E 3, then this payoff curve always lies above (including intervals where it is strictly above) that of the butterfly spread. So, in this case, the portfolio always pays off at least as much as the butterfly, and sometimes pays more; hence it has a greater time-zero value. 2

8 QUESTION 2) a) If S(t) = 0 then the stochastic differential equation model gives S(u) = 0 for all u > t. Hence, the call option will definitely give zero payoff at expiry. So C(0, t) = 0. If S is very large, then the payoff is essentially the asset price. Ignoring the exercise price and the possible fluctuations in asset price between time t and T we have C(S, t) S for S large. b) The no-arbitrage assumption says that there must never be an opportunity for an investor to make a greater risk-free return than that arising from cash invested in a bank. The justification for the assumption is as follows. If we suppose that an opportunity exists to set up a risk-free portfolio that pays better than the bank s interest rate (and also suppose that there are traders in the market who have the ability to recognise the opportunity) then traders could borrow large amounts from the bank and invest in the portfolio. The bank would respond by raising interest rates until equilibrium was maintained. c) If Π = S + P C < Ee r(t t) at time t then we can buy the portfolio Π at time t and invest the money in the bank. At expiry our payoff is S(t) + max(e S(T ), 0) max(s(t ) E, 0), which simplies to E. Note that this is riskless. This gives us a greater profit than simply investing the money in the bank at time t (we need to invest the greater amount Ee r(t t) in the bank in order to get this return at time T.) By definition, this is arbitrage. (This is simply one half of the argument that shows put-call parity.) d) At the expiry time t = T the holder will sell if E > S giving a payoff E S, and will not sell if E S giving no payoff. Hence the overall payoff at time t = T is max(e S, 0). This gives value of P (S, T ) = max(e S, 0). We argued in part a) that if S(t) = 0 then the stochastic differential equation model gives S(u) = 0 for all u > t, whence the put option will definitely give a payoff of E at expiry. So, allowing for the interest rate, P (0, t) = Ee r(t t). For large S, the asset price is very unlikely to fall below E by the time t = T. Hence P (S, t) 0 for S large. e) For C(S, t) = KS, we have C t = 0, C S = K, 2 C S 2 = 0. Substituting into the Black-Scholes PDE gives C t σ2 S 2 2 C C + rs S2 S as required. rc = rks rks = 0, 3

9 Interpretation The Black-Scholes PDE must be satisfied by any option. In the particular case of a European call option with exercise price E = 0, the payoff is always S, so (by the no-arbitrage principle) at time t and asset price S, the value must be C(S, t) = S. (If the value were any diffferent, either buy/sell the option and sell/buy the asset to lock into a guaranteed no-risk profit.) Hence, C(S, t) = S must solve the Black-Scholes PDE. The same argument applies when the overall asset consists of a multiple of K times any individual asset. f) Let N denote the number of observations per year. Then, interpreting probability as frequency, we have Number of extremes Number of observations = N = , which gives N 249. [Hence, Fama is thinking of one observation per working day.] Q3) a) We have ( ) SN (d 1 ) log e r(t t) EN (d 2 ) = log(s/e) + log(n (d 1 )) + r(t t) log(n (d 2 )) = log(s/e) 1 2 d2 1 + r(t t) d2 2. Now Substituting this gives ( ) SN (d 1 ) log e r(t t) EN (d 2 ) d 2 2 = (d 1 σ T t) 2 = d 2 1 2d 1σ T t + σ 2 (T t). = log(s/e) + r(t t) d 1 σ T t σ2 (T t) = σ { } log(s/e) + (r T t σ2 )(T t) σ d 1 T t = 0 from the definition of d 1. Hence, SN (d 1 ) = e r(t t) EN (d 2 ). (b) Differentiating with respect to σ we have Using C σ = SN (d 1 ) d 1 σ Ee r(t t) N (d 2 ) d 2 σ. d 1 σ = d 2 σ + T t 4

10 and the result in part (a), we have C σ = d 2 ( SN (d 1 ) Ee r(t t) N (d 2 ) ) + SN (d 1 ) T t σ = SN (d 1 ) T t. as required. The non-negativity of C/ σ can be understood by considering that an increase in volatility leads to a wider spread of asset prices. Assets moving deeper out of the money have no effect on the option price (the payoff remains zero) while assets moving deeper into the money lead to a greater payoff. Because of this asymmetry, increasing σ has a net positive effect. c) As σ we see that d 1 and d 2, so N(d 1 ) 1 and N(d 2 ) 0. It follows in the BS formula that as σ, C(σ) S. To look at the limit σ 0 + we separate out three cases. Case 1 S Ee r(t t) > 0. In this case log(s/e) + r(t t) > 0, so as σ 0 + we have d 1, N(d 1 ) 1, d 2 and N(d 2 ) 1. Hence, C S Ee r(t t). Case 2 S Ee r(t t) < 0. In this case log(s/e) + r(t t) < 0, so as σ 0 + we have d 1, N(d 1 ) 0, d 2 and N(d 2 ) 0. Hence, C 0. Case 3 S Ee r(t t) = 0. In this case log(s/e) + r(t t) = 0, so as σ 0 + we have d 1 0, N(d 1 ) 1, d and N(d 2 ) 1. Hence, C 2 1 (S 2 Ee r(t t) ) = 0. The three cases can be summarised in the formula lim C(σ) = max(0, S Ee r(t t) ). σ 0 + Since C(σ) is continuous with a positive first derivative (see part (b)), we conclude that C is monotonic increasing on [0, ), and it follows from the limits above that max(0, S Ee r(t t) ) C(σ) S. (d) We will write our nonlinear equation for the implied volatility σ in the form F (σ) = 0, where F (σ) := C(σ) C. Newton s Method takes the form where F (σ) = C/ σ. Since F (σ ) = 0, we have σ n+1 = σ n F (σ n) F (σ n ), (3) σ n+1 σ = σ n σ F (σ n) F (σ ) F (σ n ) = σ n σ (σ n σ )F (ξ n ) F (σ n ) 5

11 for some ξ n between σ n and σ (from the Mean Value Theorem). Hence, we may write σ n+1 σ = 1 F (ξ n ) σ n σ F (σ n ). (4) We know that F (σ) is positive and takes its maximum at the point σ. Hence, using the starting value σ 0 = σ we must have 0 < F (ξ 0 ) < F ( σ), so that 0 < σ 1 σ < 1. (5) σ 0 σ This means that the error in σ 1 is smaller but has the same sign as the error in σ 0. To proceed we suppose that σ < σ. Then (5) tells us that σ 0 < σ 1 < σ. Now, we know from the expression given in the question that F (σ) < 0 for all σ > σ, and we also know that ξ 1 in (4) lies between σ 1 and σ. Hence 0 < F (ξ 1 ) < F (σ 1 ) and (4) gives 0 < σ 2 σ σ 1 σ < 1. Continuing this argument gives 0 < σ n+1 σ σ n σ < 1, for all n 0. (6) So the error decreases monotonically as n increases. Completely analogously, it can be shown that (6) holds in the case where σ > σ. Overall, we conclude that with the choice σ 0 = σ the error will always decrease monotonically as n increases. It follows that the error must tend to zero. (The general convergence result for Newton s Method then shows that convergence must be quadratic). Hence, σ 0 = σ is a foolproof starting value for this particular nonlinear equation. [Much less detail would be OK here.] QUESTION 4) (a) An American call option gives its holder the right (but not the obligation) to purchase from the writer a prescribed asset for a prescribed exercise price at any time between the start time and a prescribed expiry date in the future. The put-call parity relation for Europeans, S + P C = Ee r(t t), may be rearranged to C S + Ee r(t t) = P. Now, since the put value must be non-negative (holding the option is no worse than having nothing) we conclude that C S + Ee r(t t) 0. 6

12 This rearranges to and so, C S Ee r(t t), C S E. Now, S E is the amount that we would receive from exercising our American option at time t, and this inequality therefore shows that it is never optimal to exercise before expiry the European call value, obtained by holding on to the option until expiry, has at least as high a value. Since it is never optimal to exercise an American call option before the expiry date, an American call option must have the same value as a European call option. (b) At time t, consider two portfolios (1) American put plus asset (2) Amercian call plus an amount E of cash At any time t t there are two cases Case 1 : S(t ) E. In this case we could exercise in portfolio (1) to obtain an amount E S(t ) + S(t ) = E. Exercising in portfolio (2) would produce an amount Ee r(t t), which is at least as large. Case 2 : S(t ) > E. In this case we could exercise in portfolio (1) to obtain an amount 0 + S(t ) = S(t ). Exercising in portfolio (2) would produce an amount S(t ) E + Ee r(t t) which is at least as large. Hence, exercising portfolio (1) always gives payoff that is no better than that given by exercising portfolio (2). So the value of portfolio (2) at time t must be at least as large as the value of portfolio (1) at time t, i.e., P am (t) + S(t) C am (t) + E, which rearranges to S(t) E C am (t) P am (t). The other inequality follows immediately from the European put-call parity relation. For Europeans we have S + P C = Ee r(t t) ; that is, P = C S + Ee r(t t). Because an American put is worth at least as much as a European put, we have P am (t) P (t) = C(t) S(t)+Ee r(t t), and hence, since C(t) = C am (t), C am (t) P am (t) S(t) Ee r(t t). (OK to quote European put-call parity without proof.) (c) After n time increments the asset has undergone n i=1 Y i upward movements and n n i=1 Y i downward movements. Hence, the asset price S(nδt) at time t = nδt is given by S(nδt) = S(0)u P n i=1 Y i d n P n i=1 Y i. 7

13 We may rearrange this to S(nδt) S(0) = d n ( u d ) P n i=1 Y i. Taking logs gives log ( ) S(nδt) = n log d + log S(0) ( u n Y i. (7) d) i=1 Now, by the Central Limit Theorem, for large n, the sum n i=1 Y i behaves like a normal random variable. Hence, for large n, log(s(nδt)/s(0)) will be close to normal. Since we know the mean and variance of Y i, we can calculate the mean and variance of this limiting nornmal random variable. To match the Black Scholes asset price model, which has the same lognormal distribution, we would like the mean of log(s(nδt)/s(0)) to be (r 1 2 σ2 )nδt and the variance to be σ 2 nδt. This leads to two equations in the three unknowns, p, u and d. [Much less detail would be OK here.] (d) The Black-Scholes option value C depends on S, E, r, T t and σ 2. Of these five quantities, only σ cannot be observed directly. Hence, given an option value C from the market, we can solve the equation C(σ) = C to get a value σ for the volatility. In other words, the implied volatility σ is the value that needs to go into the Black-Scholes formula to reproduce the option value. We have seen that for European calls and puts any sensible (no-arbitrage) option value has a unique C and Newton s method with an appropriate starting value will always converge to it. The Black-Scholes formula is derived under an assumption about the behaviour of the asset. This asset model includes the volatility paremeter. The idea behind computing a historical volality estimate is to deal directly with the asset price data. For equally spaced time values t i := i t, the log returns U i := log S(t i) S(t i 1 ). (8) may be regarded as independent samples from a normally distributed random variable with mean (µ 1 2 σ2 ) t and variance σ 2 t. Hence a Monte Carlo approach will provide a sample mean and sample variance that can be matched to these expressions. So σ can be fitted to the asset data alone. Historical volality has the advantage of relying only on the asset price model assumptions and not on the extra assumptions that go into the derivation of 8

14 the Black-Scholes formula (no trading costs, no dividends, ability to trade in continuous time,... ). Also, historical volality comes with a confidence interval. The implied volatility has the advantage of being forward looking it is based on the market s view of how the asset price will evolve and, when available, the time frame over which the volatility is needed can be matched to that of a corresponding option product. [Note: there is no exact answer to this question: any sensible explanation that deals with key points is acceptable.] 9

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