i i i i Cutting edge: Option pricing
Model-free valuation of barrier options
Peter Austing and Yuan Li provide an analytic formula for valuing continuous barrier options. The model exactly fits the implied volatility smile in a manifestly arbitrage-free way and produces prices consistent with an underlying stochastic volatility dynamic
he standard approach to valuing continuous barrier options, devel- becomes more important. As a result, analytic approaches to valuation are oped in the literature by Jex (1999) and Lipton (2002), is to use a as interesting as ever. T local stochastic volatility (LSV) model. Such models begin with An analytic approach introduced by Lipton & McGhee (2002) and an underlying stochastic volatility (SV) process, for example, stochastic Wystup (2003), usually known as the Vanna-Volga method, works by alpha, beta, rho (SABR), Heston or a number of other choices. The SV is thoughtfully adding the smile cost of second-order Greeks to the theoret- first calibrated to give vanilla option prices that closely match the market- ical (flat smile) value. It was the workhorse for foreign exchange trading implied volatility smile at one or more expiries. Then, the volatility of the desks for a number of years, and particularlyMedia appreciated for the intuitive SV (the volatility of volatility) is reduced by a proportion known as the hedges it provided to traders. However, as the smile adjustment is built mixing fraction, so the smiles generated are flatter than the market. Finally, from a small number of vanillas, there are inevitable pricing anomalies a local volatility (LV)correction factor is introduced and calibrated to bring (see Bossens et al 2010; Moni 2011). vanillas back to the correct market prices. Our approach will be to write down a joint probability distribution for the
The mixing fraction measures how far the model is between SV (mixing terminal spot ST and the maximum value of spot in the interval t 2 Œ0; T , zero) and LV (mixing one).An added bonus is that, in the mixing zero case, which we denote MT . One starting point would be to use a simple copula, the LV correction can be used to fill in any mismatch between the pure SV with marginals for ST and MT determined from the volatility smile and smile and the true market smile. In practice, the mixing is tuned to match one-touch prices, respectively. This is problematic, because the copula
the price of a one-touch or double no-touch option, and it is often around would need to maintain the crucial relation MT > ST to avoid arbitrage. one half. Furthermore, as we know empirically that one-touch prices in SV are The underlying SV process chosen varies considerably, yet market par- determined from the volatility smile, a solution built from a full marginal ticipants agree closely on barrier option prices. Thus, although the true for MT is likely to be over-determined. underlying model is not known, this is not relevant as long as everyone Instead, we will write down a formula that determines both one-touch believes in local stochastic volatility. Such a remarkable fact makes barrier prices and more general barrier option prices directly from the volatility options safe to trade in large volumes, and they are often treated almost as smile. flow products. Incisive Consider the reason for the approximate model invariance. It has been argued by Austing (2014) that this can be explained by the reflection prin- Constructing the distribution ciple. If we consider a pure normal SV model with constant rates, and We wish to construct a joint probability distribution for S and M with without spot-volatility correlation, then the reflection principle holds; thus, T T M > S 1 valuation depends only on the terminal spot distribution, which is deter- the property that T T , with probability . As usual, we would like mined by the implied volatility smile. This remains approximately true our prices to reduce to Black-Scholes when the volatility smile is flat; this in a lognormal model, and two SV models that generate similar terminal motivates us to begin with a simple Brownian motion, Wt . We define the smile convexities will generate similar barrier option prices. Adding in a maximum to be AT D supt2Œ0;T Wt , and then the reflection principle local volatility correction smoothly deforms the price towards the Dupire tells us that: (1994) valuation, and so the impact is similar across the underlying SV 8 ˆ0; y < 0 models. < Such an argument is heuristic, and we shall not dwell upon it here. P.WT
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i i i i i i Cutting edge: Option pricing
gives us: The scaling parameter is irrelevant for constructing a copula and can be set to 1. Nevertheless, we shall retain it for reasons that will become 1 x D N .u1/ (2) apparent. 1 1 To simplify notation, we have denoted the joint distribution P.X < y D N . 2 .u2 C 1// (3) T x;AT
for the cumulative probability distribution of the maximum MT , or equiv- and substituting these into (4). 1 alently: If we now look at the region 2 .u2 C 1/
Thus, we have constructed a joint probability distribution for ST , MT , with the property that MT > ST , with probability 1. At this stage, the Matching stochastic volatility prices marginal for the spot ST matches the input smile, while the marginal So far, we have constructed a joint probability distribution between the spot for the maximum MT is a consequence of the construction and does not at expiry, ST , and the maximum of the spot between start time and expiry necessarily match one-touch prices. We have not yet succeeded in meeting time, MT . We can use this distribution to value options that have a con- the other constraint, MT >S0. Instead, the marginal distribution (8) tells tinuous upper barrier and a European payout. The probability distribution 1 1 exactly matches the market prices of vanilla options, since the marginal us that MT >F1 . 2 /. To fix this, we will introduce two parameters, a drift and a volatility, into for ST is an input that can be derived from the market smile. However, the the Brownian motion from which we generated our copula. If we consider values of one-touch options are given by the other marginal distribution, a new process: which is an output from the model and not necessarily expected to match market prices. Our aim in what follows is to extend the model so that one- dXt D dt C dWt (9) touch prices become consistent with an underlying stochastic volatility and set AT D supt2Œ0;T Xt , then standard methods show that the joint model. distribution (1) becomes: Let us denote the distribution we have constructed so far by P ; .We 8 recall that it is given by: ˆ0; y < 0 ˆ ˆ 1 1 < x 2 x 2y P ; .K; B/ D P.P .F1.K//; P .F1.B/// (18) N e2y = N ; y>0;y>x X X P.x;y/ D ˆ ˆ where P.x;y/, P .x/ and F are as defined above. As we have noted, ˆ Copyrighty 2 y X 1 :N e2y = N ; y>0;y risk.net 81 i i i i i i i i Cutting edge: Option pricing 1 Numerical demonstration of the model (b) (a) 0.10 18 0.08 16 0.06 14 0.04 Local volatility 12 0.02 10 0 8 –0.02 6 Volatility (%) –0.04 4 2 Adjustment to TV –0.06 Model 0 –0.08 0 0.2 0.4 0.6 0.8 1.0 –0.10 Delta Theoretical value (TV) (d) (c) Media1.110 0–0.5 0.5–1.0 1.056 1.0 1.002 0.5 1.08 0.948 0 0.90 0.94 Terminal spot, S 0.98 1.06 Terminal spot, S 0.894 1.14 0.80 Max spot, M 1.22 0–50 50–100 100–150 0.840 0.98 1.01 1.04 1.07 1.10 1.13 Max spot, M (a) Implied volatility smile. (b) One-touch plot. (c) Joint cumulative distribution.Incisive (d) Joint probability density However, we can easily generate families of distributions by defining, In order to get realistic SV prices, we are going to extend our original for example: probability distribution to that generated by a mixture model. Reminiscent of Brigo & Mercurio (2000), we employ this approach not to use a mixture 1 1 Q 1; 1I 2; 2 .K; B/ D 2 P 1; 1 .K; B/ C 2 P 2; 2 .K; B/ (19) model directly, but rather to help us generate an appropriate and tractable model. We note at the outset that the following approach continues to work We have four parameters: , , and . As before, one parame- 1 1 2 2 if we extend to a continuum of volatilities. This corresponds to integrating ter is redundant through scaling, and one is fixed by the requirement over the terminal volatility distribution of an arbitrary (correlation-free) P.M 82 risk.net May 2017 i i i i i i i i Cutting edge: Option pricing 2 One-touch prices and densities for the model and two additional values of ˛ (b) 1.110 (a) 0.10 1.056 0.08 Local volatility 0.06 Model 0.04 α = 0 α 1.002 0.02 = 0.6 0 –0.02 –0.04 0.948 Adjustment to TV –0.06 Terminal spot, S –0.08 –0.10 0.894 Theoretical value (TV) 0–50Media 50–100 100–150 0.840 0.98 1.01 1.04 1.07 1.10 1.13 Max spot, M (c) (d) 1.110 1.110 1.056 1.056 1.002 1.002 0.948 0.948 Terminal spot, S Terminal spot, S 0–20 20–40 0.894 0.894 40–60 60–80 0–50 50–100 0.840 0.840 0.98 1.01 1.04 1.07 1.10 1.13 Incisive0.98 1.01 1.04 1.07 1.10 1.13 Max spot, M Max spot, M (a) One-touch plot. (b) Joint probability density. (c) Joint probability density: ˛ D 0. (d) Joint probability density: ˛ D 0.6 Having fixed ˛ and, as a result, 1 and 2, we choose: where y D log.K=FT / is the log moneyness with respect to the forward 1 FT . Our plots were generated with expiry T D 1 year, spot S0 D 1, 1 2 1 D ˇ 2 1 (23) risk-free rate 3% and dividend yield 5%. The volatility parameters were 1 2 2 D ˇ 2 2 (24) a D 0:005, b D 0:0005 and c D 0:0001, giving an at-the-money volatility of 10%, a 25-delta strangle of 0.54% and a 25-delta risk-reversal and solve numerically for ˇ to achieve P.MT risk.net 83 i i i i i i i i Cutting edge: Option pricing makes sense, as, for example, the line MT D S0 represents those paths maximum spot, respectively. Thus, certain contracts, such as one-touches that spend a brief time around S0 before heading downwards. and digital payouts with a continuous barrier, can be valued analytically. The parameter ˛ is fixed to generate stochastic volatility-style prices by Vanilla payouts with continuous barriers can be valued fast with a single imposing (22). While the ability to obtain SV barrier prices analytically is numerical integration. a significant step forward, the true market price of a barrier option usually The model takes the market-implied volatility smile as an input and lies between the stochastic volatility price and the local volatility price. ensures the vanilla options are exactly repriced. This is reminiscent of local The most sophisticated market participants use local-stochastic volatility stochastic volatility, in which the local volatility correction adjusts for any models to achieve a mixing between SV and LV. This invites the question mismatch between the underlying SV smile and the true market smile. of whether we can tune ˛ in order to find a proxy for the LSV price. We have emphasised two main assumptions in our model. First, we rely Figure 2 shows what happens when we vary ˛. We note that one-touch on the reflection principle and, therefore, have implicitly approximated prices do not change much. In particular, even when ˛ is zero, so that our the drift in any underlying SV model by a constant. This means we are underlying mixture model has no stochasticity, the one-touch prices do assuming interest rates are not strongly varying over time. Similarly, since not come close to the local volatility prices. This shows that the parameter we assume a lognormal underlying spot process, volatility contributes ˛ does not act like a mixing between SV and LV. This should not be sur- to the drift and this means we are assuming individual volatility paths prising, as there is no reason to believe the original copula we constructed generated by the SV are not too stronglyMedia varying in time. earlier in this article would provide a proxy for LV prices. The second assumption is that we used an implicit underlying mixture However, the joint density function, also shown in figure 2, varies dra- model to generate the smile convexity. We used the fact that this underly- matically with ˛. This is an important point. It means we can construct ing model is known to generate SV-style barrier prices to argue that our families of models that price vanilla options correctly, and match one-touch model is sensible. However, as the mixture model does not have spot- prices fairly closely, yet in which other barrier contracts vary significantly. volatility correlation, it does not generate any skew. Therefore, the skew Thus, the choice of ˛ to satisfy (22) is a crucial part of our model. It is this fit is achieved entirely through the copula, without reference to any under- choice that allows us to rely on the argument of the underlying mixture lying dynamic model. The full impact of this remains open, and it would model to obtain meaningful barrier option prices. be interesting to extend to SV models with correlation using the methods An alternative way of generating prices close to LSV is to create a dis- of McGhee & Trabalzini (2014). tribution that mixes true local volatility (calculated by a numerical partial In addition to providing an arbitrage-free alternative to Vanna-Volga differential equation solution) with the SV proxy from our model. In prac- and a simple means of generating LSV-style prices by mixing with local tice, this is as simple as calculating the price in LV and mixing with the volatility, it is pleasing to see a simple analytic explanation for the classic price from our model. While this may leave some feeling uneasy due to the one-touch moustaches. Indeed, those who are energetic enough could use Piterbarg (2003) hangover, there is probably no need for such qualms. It the formula to generate continuous vanilla hedge portfolios to improve on is a legitimate means of creating a joint terminal probability distribution, the classic trader rules. and it does not suffer from the temporal arbitrage problem. Peter Austing is a quantitative researcher at Citadel in London. IncisiveYuan Li is an associate director within the FX quant team at ANZ in London. The authors are grateful to Anqi Zhou and Minying Conclusion Lin for indispensable discussions. The opinions expressed in We have developed an analytic model that generates barrier option prices this paper are those of the authors, and do not necessarily that approximate stochastic volatility models. The model provides the joint represent the views of their employers. cumulative probability distribution of ST and MT , the terminal spot and Emails: [email protected], [email protected]. 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