Some Things I Have Learned About Over The Years

In this lecture. . .

• Why does volatility matter?

• How can you make money from volatility?

• What would happen if you with the wrong volatility?

• What would a good volatility model look like?

c Paul Wilmott www.wilmott.com 1 In the companion lecture (CQF Institute Christ- mas Lecture, Dec. 14th, London). . .

• What is sensitivity to volatility?

• Where do prices come from?

• What information is contained in market prices?

• How important is it to get volatility right?

• Do you need to dynamically hedge?

c Paul Wilmott www.wilmott.com 2 Introduction

There are thousands of papers on volatility. Some are about volatility modelling, some on the implementation of the models. The popularity of the models is driven by two main features:

• The model must not admit (or rather, it must not give the appearance of arbitrage opportu- nities)

• Ideally there will be an element of tractability, per- haps in terms of closed-form solutions

I don’t approve of either of these! And this disapproval guides much of my research...

c Paul Wilmott www.wilmott.com 3 Why does volatility matter?

It matters because of two things:

1. Randomness

2. Nonlinearity

Jensen’s Inequality!

c Paul Wilmott www.wilmott.com 4 Jensen’s Inequality

c Paul Wilmott www.wilmott.com 5 If f(x) is a convex function and x is a random variable then E[f(x)] ≥ f(E[x]).

If the variance of x is small then you can get an ap- proximation for the difference between the two sides as:

1 ∂2f var( ) x 2 . 2 ∂x x=E[x]

c Paul Wilmott www.wilmott.com 6 How can you make money from volatility?

Suppose that you believe an is mispriced. . . how can you profit from this?

Remember that if you are delta hedging then you are only exposed to volatility and not market direction.

So you can interpret a ‘mispriced’ option as one for which your estimate of volatility differs from the .

c Paul Wilmott www.wilmott.com 7 You have a forecast of volatility, and so does the market (implied).

Black–Scholes tells you all about how to hedge when there is just one volatility, now there are two!

So which delta do you choose? Delta based on actual or implied volatility?

c Paul Wilmott www.wilmott.com

8 Hedging with actual volatility or implied volatility?

Scenario: Implied volatility for an option is 20%, but we believe that actual volatility is 30%

Question: How can we make money if our forecast is correct?

c Paul Wilmott www.wilmott.com

9 Answer: Buy the option and delta hedge.

But what delta do we use?

∆ = N(d1) where 1 Z x s2 N(x) = √ e− 2 ds 2π −∞ and  1 2 ln(S/E) + r + 2σ (T − t) d1 = √ . σ T − t

c Paul Wilmott www.wilmott.com 10 We can all agree on S, E, T − t and r (almost), but not on σ.

So should we use σ = 0.2 or 0.3?

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11 σa = actual volatility, 30% and

σi = implied volatility, 20%.

c Paul Wilmott www.wilmott.com

12 Case 1: Hedge with actual volatility, σa

By hedging with actual volatility we are replicating a position in a correctly priced option.

The payoffs for our long option and our short replicated option will exactly cancel.

The profit we make will be exactly the difference in the Black–Scholes prices of an option with 30% volatility and one with 20% volatility.

(Assuming that the Black–Scholes assumptions hold.)

c Paul Wilmott www.wilmott.com 13 If V (S, t; σ) is the Black–Scholes formula then the guar- anteed profit is

V (S, t; σa) − V (S, t; σi).

But how is this guaranteed profit realized?

c Paul Wilmott www.wilmott.com

14 The model

dS = µS dt + σaS dX.

Set up a portfolio by buying the option for V i and hedge with ∆a of the stock.

Today:

Option V i Stock −∆a S Cash −V i + ∆a S

Superscript ‘a’ means actual and ‘i’ means implied, these can be applied to deltas and option values.

c Paul Wilmott www.wilmott.com 15 Tomorrow:

Option V i + dV i Stock −∆a S − ∆a dS Cash (−V i + ∆a S)(1 + r dt)

c Paul Wilmott www.wilmott.com

16 Therefore we have made, mark to market,

dV i − ∆a dS − r(V i − ∆a S) dt. After a bit of Itˆothis becomes

1  2 2 2 i i a 2 σa − σi S Γ dt + (∆ − ∆ ) ((µ − r)S dt + σaS dX) .

Conclusion: The final profit is guaranteed (the dif- ference between the theoretical option values with the two volatilities) but how that is achieved is random.

On a mark-to-market basis you could lose before you gain.

c Paul Wilmott www.wilmott.com 17 P&L for a delta-hedged option on a mark-to-market basis, hedged using actual volatility. c Paul Wilmott www.wilmott.com 18 When S changes, so will V . But these changes do not cancel each other out.

The fluctuation in the portfolio mark-to-market is ran- dom. It may even go negative.

Although the path of the profit is random, the final profit is simply the difference between the option valued using actual volatility and that using implied volatility.

This is a known quantity.

c Paul Wilmott www.wilmott.com 19 An analogy, a bond: Guaranteed outcome, but may lose on a mark-to-market basis in the meantime. May be difficult to exit.

c Paul Wilmott www.wilmott.com

20 Case 2: Hedge with implied volatility, σi

By hedging with implied volatility we are balancing the random fluctuations in the mark-to-market option value with the fluctuations in the stock price.

The evolution of the portfolio value is ‘deterministic.’

c Paul Wilmott www.wilmott.com

21 Let’s see how this works.

Buy the option today, hedge using the implied delta, and put any cash in the bank earning r.

The mark-to-market profit from today to tomorrow is

dV i − ∆i dS − r(V i − ∆iS) dt

i 1 2 2 i i i Θ dt + 2σa S Γ dt − r(V − ∆ S) dt

1  2 2 2 i = 2 σa − σi S Γ dt.

c Paul Wilmott www.wilmott.com 22 Add up the present value of all of these profits to get a total profit of

Z T 1  2 2 −r(t−t0) 2 i 2 σa − σi e S Γ dt. t0

This is always positive, but path dependent.

c Paul Wilmott www.wilmott.com

23 P&L for a delta-hedged option on a mark-to-market basis, hedged using implied volatility. c Paul Wilmott www.wilmott.com

24 An analogy, money in the bank: Can access money at any time, easy to exit. Always increasing in value. End result uncertain.

c Paul Wilmott www.wilmott.com

25 What would happen if you hedge with the wrong volatility?

We will briefly examine hedging using volatilities other than actual or implied.

c Paul Wilmott www.wilmott.com

26 Actual volatility = Implied volatility

For the first example let’s look at hedging a long po- sition in a correctly priced option, so that σ =σ ˜. We will hedge using different volatilities, σh. The figure shows the expected profit and standard deviation of profit when hedging with various volatilities. The chart also shows minimum and maximum profit. Parameters are E = 100, S = 100, µ = 0, σ = 0.2, r = 0.1, D = 0, T = 1, andσ ˜ = 0.2.

c Paul Wilmott www.wilmott.com 27 c Paul Wilmott www.wilmott.com 28 Actual volatility > Implied volatility

Next we see the expected profit and standard deviation of profit when hedging with various volatilities when actual volatility is greater than implied. The chart again also shows minimum and maximum profit. Parameters are E = 100, S = 100, µ = 0, σ = 0.4, r = 0.1, D = 0, T = 1, andσ ˜ = 0.2. Note that it is possible to lose money if you hedge at below implied, but hedging with a higher volatility you will not be able to lose until hedging with a volatility of approximately 75%. The expected profit is again insensitive to hedging volatility.

c Paul Wilmott www.wilmott.com 29 c Paul Wilmott www.wilmott.com 30 Actual volatility < Implied volatility

Next is shown properties of the profit when hedging with various volatilities when actual volatility is less than implied. We are now selling the option and delta hedg- ing it. Parameters are E = 100, S = 100, µ = 0, σ = 0.4, r = 0.1, D = 0, T = 1, andσ ˜ = 0.2. Now it is possible to lose money if you hedge at above implied, but hedging with a lower volatility you will not be able to lose until hedging with a volatility of approximately 10%. The expected profit is again insensitive to hedg- ing volatility. The downside is now more dramatic than the upside.

c Paul Wilmott www.wilmott.com 31 c Paul Wilmott www.wilmott.com 32 To a large extent how you hedge matters less than doing some hedging.

Already this is hinting that using the right model, say a calibrated model, does not matter as much as you’d expect.

I shall return to this subject in Part 2 of this talk.

c Paul Wilmott www.wilmott.com

33 What would a good volatility model look like?

What people want from a volatility model:

• The academic wants a complex piece of mathemat- ics • The trader wants whatever will hide the most risk • The risk manager wants to use whatever models other people are using • The boss wants to be able to trust his employees’ models • The regulator wants to seem cleverer than he really is Above everything, they all want “Plausible Deniability.”

c Paul Wilmott www.wilmott.com 34 What doesn’t matter

• The distribution of returns doesn’t matter. If you are hedging often enough then all that matters is finite variance of returns

• Fat tails are misleading. Extreme events should not treated in a probabilistic way

What really matters

• If selling OTC derivatives: Hedging, both static and dynamic; Adding a decent profit

• If speculating: A good model for what you are spec- ulating on. (And no arbitrage is obviously wrong!)

c Paul Wilmott www.wilmott.com 35 What people should want from a volatility model

Being a responsible person you might say you want. . .

• A good approximation to reality

• Usable and transparent. It’s no good having a model that is only understood by the professors!

• Makes its faults known. Accept that there will be risks you don’t know, and model error

c Paul Wilmott www.wilmott.com 36 What models can do

Mathematical models can do a lot more than you think.

The vast majority of derivatives models use the same mathematics. The end result is in 99% of cases:

“The value of an option is the present value of the expected payoff under the risk-neutral random walk.”

And this then allows valuation by Monte Carlo simula- tion.

So what is the point of quant finance research, if every paper is the same?

c Paul Wilmott www.wilmott.com 37 Some good models

• Constant volatility. Why is no one taught the basics of volatility arbitrage?

• Uncertain Volatility Model: σ− < σ < σ+. Nonlin- earity

• without dynamic hedging of one option with another. Nonlinearity

• Anchoring model: dS = µS dt+σ(S/A)S dX where A R t −λ(t−τ) is weighted average of past prices, A = λ ∞ e S(τ) dτ

c Paul Wilmott www.wilmott.com 38 The Anchoring model

We need an antidote to the deterministic volatility model — Yeugh!

dS = µ S dt + σ(S) S dX

And the antidote is...the Anchoring model!...

c Paul Wilmott www.wilmott.com

39 We are going to work with a model that has a memory. Let us introduce a new variable A, representing averag- ing or, in the language of behavioral finance, anchoring

Z t A = λ e−λ(t−τ)S(τ) dτ. ∞

Note that this is just about the simplest memory func- tion that has tractable properties.

We have a volatility function σ(ξ), to be determined, where S ξ = . A

c Paul Wilmott www.wilmott.com 40 If you multiply S everywhere by a constant then the variable ξ is unchanged. Thus the model has a nice scaling property.

Our model is thus

dS = µS dt + σ(ξ)S dX.

Such a volatility function can be made consistent with the common observation that volatility increases when prices are low, but crucially this means low relative to some historical average, not simply low in an absolute sense since that would be contrary to our scaling re- quirement.

c Paul Wilmott www.wilmott.com 41 Qualitative Observation On Share Prices Generally

c Paul Wilmott www.wilmott.com 42 In the figure are two lines, one the logarithm of the Standard and Poor’s Index from 1950 until March 2012, detrended, and the second is the inverse of the index.

If we were dealing with simple, classical lognormal ran- dom walks then the choppy (i.e. Brownian Motion) nature of these two lines would be qualitatively simi- lar. But there is one aspect of the real financial time series that is different from its mirror image: The ar- tificial data, the mirror image, has peaks and troughs like a classical sea wave, the peaks being pointy and the troughs being rounded. With the real data this picture is upside down. The real data has pointed troughs and rounded crests. This is suggestive of higher volatility for lower share prices.

c Paul Wilmott www.wilmott.com 43 Observations On ξ

By examining the stochastic differential equation for ξ in isolation we can tell a great deal about the behavior of this model, and get some clues as to how to approach the determination of a functional form for σ and for determining a value for λ.

dξ = (µ + λ − λξ) ξ dt + σ(ξ)ξ dX.

c Paul Wilmott www.wilmott.com 44 After a bit of Fokker–Planck and some data analysis we find that a good fit for σ(ξ) is the sigmoidal function b − a a + . 1 + e−c(ln(ξ)−d)

The best-fit parameters are a = 0.60, b = 0.12, c = 10.82, d = −0.27 and λ = 0.90. The last of these shows a typical memory of the order of one year.

c Paul Wilmott www.wilmott.com 45 c Paul Wilmott www.wilmott.com 46 c Paul Wilmott www.wilmott.com 47 Summary

Please take away the following important ideas

• There are probably more math models than you know about

• People choose models for very narrow, personal, reasons

• If a model is popular it’s almost certainly bad

c Paul Wilmott www.wilmott.com 48