Practical Guide to Quantitative Finance Interviews Part 9

Finance the value of an at the money call or put is almost a linear function of volatility. If the investor purchases an at-the-money straddiz, both the call and the put options have the price c= rho approx0.4 sigma . S * sqrt(tau) . where a, is the implied volatility. If the realized volatiäty both options are undervalued. When the market prices converge to the prices with the realized volatility, both the call and the put will become more valuable. sigma_{c} > sigma_{c} Although initially a straddle with an at-the-money call and an at-the-money put (X = S) Bas a delta close to 0, as the stock price moves away from the strike price, the delta longer close to 0 and the investor is exposed to stock price movements. So a straddle is not a pure het on stock volatility. For a pure bet on volatility, it is better to use volatility swaps or variance swaps." For example, a variance swap pays epsilon\ sigma i ^ dagger -K i*omega \ i where N is the notional value, sigma * ^ z is the realized variance and K es the strike for the variance Binary options What is the prize of a binary (cash-or-nothing digital) European call option on a no dividend paying stock if the stock price follows a geometric Brownian motion? How would you liedge a cash-or-nothing call option and what's the limitation of your hedging strategy? Solution: A cash-or-nothing call option with strike price & pays $1 if the asset price is above the strike price at the maturity date, otherwise it pays nothing. The price of the option is c_{xi} = e ^ (- epsilon) * N(d_{xi}) if the underlying asset is a non-dividend paying stock. As ve have discussed in the derivation of the Black-Scholes formula, N(d_{z}) the probability that a vanilla call option finishes in the money under the risk-measure. So its discounted value is epsilon^ = M(d_{epsilon}) . Tacoretically, a cash-or-nothing call option can be hedged using the standard deli hedging strategy. Since Delta = mathcal F C mu CS =e^ -i tau N^ prime (d 1 )- 1 S sigma sqrt r , a long position in a cash-or notsing call opsion can be hedged by shorting e ^ (- r) * N(d_{z}) * 1/(S*sigma * sqrt(r)) shares (and a risk-free money market position). Such a hedge works difference between Sand is large and 7 is not close to . But when the option is approuching manurity f(x -> 0) "For detailed cussion about volatility ywups, please refer to the piper "More Thise You Ever Wand to Know about Volatility Swaps" by Kresimir Detteterfi, et al. The paper shows then a valor swap be approximated by a portfoliu of straddles with proper weights inversely proportional 160 A Practical Guide To Quantitative Fire Interviews and the stock price 5 is close to K, A is extremely volatile and small changes in the stock price cause very large changes in A. In these cases, it is practically impossible to hedge a cash-or-nothing call option by delta hedging. We can also approximate a digital option using a bull spread with two calls. If call options are available for all strike prices and there are no transaction costs, we can long 1/2 call options with strike price - and short 1/2 call options with strike price K + xi - The payoff of the bull spread is the same as the digital call option if S_{r} <= K - epsilon (both have payoff 0) or \mathcal{S}_{j} >= \mathcal{K} = x_{j} (both have payoff $1). When K - epsilon < S_{r} < K + E their payoffs are different. Nevertheless, if we set kappa -> 0 such a strategy will exactly replicate the digital call. So it provides another way of hedging a digital call option. This hedging strategy suffers its own drawhack, In practice, not all strike prices are traded in the market. Even if all strike prices were traded in the market, the number of options needed for hedging, 1/2 * epsilon_{0} will be large in order to keep e small. Exchange options How would you price an exchange call option that pays max(S,, -5,. 0) at maturity Atsume that S, and ate rxan-dividend paying steckt and both follow pometric Brownian motions with correlation p S_{2} Selacion: The solution to this problem uses change of numeraire. Nunseraire means a ubit of measurement. When we express the price of an asset, we utually use the local currency as the numeraire. But for modeling purposes, it is often casier to use a different awet as the numeraire. The only requirement for a mameraire is that it must always be positive The payoff of the exchange option depends on both S VS (price of S_{t} at maturity date 7) and S r2 price of S_{1} at 7), so it appears that we need two geometric Brownian motions: dS j = mu_{i}*S_{i} * dl + sigma i S i dW ij dS z = mu_{2}*S_{2} * dt + sigma 2 S 2 dW 1,2 Yet if we use \mathcal{S}_{1} as the namernire, we can convert the problem to just one geometrie When Brownian motion. The final payoff is max (S gamma, z -S gamma,z ,0)=S gamma,z max ( S r, 1 S r,1 -1,0) SK and 05/04 Finanos Delta_{y} and \mathcal{S}_{j} are geometrical Browian motions, a geometric Brownian mution as well. One intuitive explanation is that both in S_{1} and S_{7} follow normal distributions. so in f* ln(S) t -ln S t follows a normal distribution as well and/ follows a tognormal distribution. More rigorously, we can apply the Ito's lemma to f = S_{z}/S_{t} overline f = S_{1}/S_{1} ; vartheta^ prime = partial f partial S 1 c partial S 1 + partial f partial S 2 dS 1 + 1 2 partial^ 2 f partial S 2 ^ 2 (d*S_{1}) ^ 3 + 1 2 partial^ 3 f partial S 1 ^ 2 (dS 2 )^ 3 + partial^ 2 f partial S f partial S 2 dS 1 dS 2 = - mu * t_{1} * S_{2}/S_{z} * dt - sigma z S 2 S z dW zz + mu 2 S z S z dt+ sigma z S z S z dW z,z + sigma z ^ z S z S z di- rho sigma z sigma z S z S z dt = (mu_{2} - mu_{1} + sigma_{1} ^ 2 - mu*sigma_{1}*sigma_{1}) * f * dt = sigma_{z}*fd*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta*delta partial f partial S 1 = -S 2 S 1 ^ 2 + partial f partial S 2 = 1 S 1 , partial^ 2 f partial S 1 ^ 2 = 2S 1 S 3 ^ 2 = 2S 1 S 4 ^ 2 , partial^ 3 f partial S 2 ^ 2 =0, overline sigma ^ 2 f d*S_{2} * f/(d*S_{2}) = - 1/S_{1} take To make f = S_{2}/S_{1} a martingale, set mu_{2} - mu_{1} + sigma_{1} ^ 2 - rho*sigma_{1}*sigma_{2} = 0 and we have and S z S ey a martingale under the new measure. The value of the exchange option using S_{1} as the numeraite is with underlying asset price which is just the value of a call option strike price K = I_{i} interest rate r = 0 and volatility sqrt(sigma_{1} ^ 2 - 2rho*sigma_{1}*sigma_{2} + sigma_{1} ^ 2) So its value is (S/S)+0.50 where and d_{2} = d_{1} - sigma * sqrt(pi) The payoff of the exchange option expressed in local currency is S_{i}*C_{j} = S_{z}*N(d_{z}) - S_{z}*N(d_{z}) C z = S_{z}/S_{t} * N(d_{t}) -N(d t ) t S = S_{1}/S_{1} E[ S r, 2 S t,3 ]= S t S t , C r E[ max ( S r,2 S r,3 -1,0)]. A Practical Guide To Quamtarive Finance viz 6.4. Other Finance Questions Besides option pricing problems, a variety of other quantitative finance problems are tested in quantitative interviews as well. Many of these problems tend to be position specific. For example, if you are applying for a risk management job, prepare to answer questions about VaR; fer fixed-income jobs, get ready to answer questions about interest mte models. As I explained in Chapter 1, it always helps if you grasp the basic knowledge before the interview. In this section, we use several examples to show some typical interview problems. Portfolio optimization You are constructing a simple portfolio using two stocks A and B. Both have the same expected return of 12%. The standard deviation of A's retum is 20% and the standard deviation of B ^ * S retum is 30%, the correlation of their retums is 50%, How will you allocate your investment between these two stocks to minimize the risk of your portfolio? Solution: Portfolio optimization has always been a crucial topic for investment management fires. Harry Markowitz's mean-variance portfolio theory is by far the most well-known and well-studied portfolio optimization model. The essence of the mean variance portfolio theory assumes that investors prefer (1) higher expected returns for a given level of standard deviationyvariance and (2) lower standard deviations/variances for a given level of expected return. Portfolios that provide the mininum standard deviation for a given expected return are termed efficient portfolios. The expected return and the variance of a portfolio with N assets can be expressed as mu p =w 1 mu 1 +w 1 mu 1 +***+ w_{n}*mu_{n} = w ^ r * mu . where i, forall i = 1 ,***,N is the weight of the i-th asset in the portfolio, mu i ^ s forall i=1,***,N i Is the expected of the ith asset, sigma_{i} ^ 2 is the variance of ith asset's return, sigma_{i} = rho_{g}*sigma_{i}*sigma_{j} is the covariance of the returns of the i-th and the/-th assets and alpha_{s} : their w is an N * 1 column vector of w_{s}' <= mu an N * 1 column vector of mu_{i} ^ * Sigma is the covariance matrix of the returns of an NN matrix Since the optimal portfolio minimizes the variance of the retum for a given level of expected return, the efficient portfolio can be formulated as the following optimization problem 162 163