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Contents Brevan Howard Group What is a Hedge Fund ? Journey to the Mathematics of Finance Career opportunities Contact us

The Brevan Howard Group Brevan Howard manages one of the largest and most successful hedge funds in the world. The group: manages more than $11 billion dollars employs over 250 personnel across offices in London, New York, Washington , Tel Aviv, Hong Kong, and Dublin was founded in 2002 by world renown traders The fund’s investor base is comprised of hundreds of institutional clients across the world, including pension funds, endowments, insurance companies and leading banks

What is a Hedge Fund ? A hedge fund is a private investment fund often characterized by complex investment strategies. Hedge funds typically seek to provide positive returns from declining markets as well as rising ones, often profiting from falling or volatile markets. There are a variety of hedge fund investment strategies, some examples of which include global macro trading (taking views and financial positions on countries’ economies through currency, government bonds, commodities, money markets, equity indices and derivatives on these asset classes); and relative value (using advanced quantitative models to take advantage of pricing or spread inefficiencies). In return for managing the investors' funds, the hedge fund management will typically receive a management fee and a performance or incentive fee.

Journey to the Mathematics of Finance Introduction Mathematical models are used in finance to compute theoretical price of financial products (derivatives), and to help traders to hedge their positions. The few next slides will take us on a short journey to the world of financial mathematics, in order to give you a flavour of the mathematics used in the industry. The theme of the journey is the simplest and most well known model Black – Scholes. Note: this presentation doesn’t pretend to be complete and/or rigorous.

Journey to the Mathematics of Finance European Option A simple financial option, a European call option, is a contract to buy a financial asset, say a stock, at a given future date T, the expiry date, for a given price K, the strike price. Options are among the simplest financial products. If you are the buyer of the option, then at the option expiry T the stock price S(T)≡ST either: exceeds the strike price K and by exercising the option and selling the stock you make an immediate profit of ST-K. is less than the strike, and the option is worthless.

Journey to the Mathematics of Finance European Option We say that the pay-off or value of the option at expiry is C(T) = Max(ST-K,0) K ST C Call option pay-off at expiry T

Journey to the Mathematics of Finance Arbitrage We call an arbitrage a strategy which guarantees (risk free) a rate of return better than the “bank” interest rate. Main Problem How much would you pay today for an option to buy a stock at price K in a future date T in an arbitrage-free world ?

Journey to the Mathematics of Finance In order to answer that question one needs to model the evolution of the stock price S(t). Black-Scholes lognormal model The Black-Scholes model assumes that the returns of the stock in one unit of time dt is normally distributed where N is the normal distribution of mean μ(t) and variance σ2dt. Formally, S(t) is assumed to follow a “standard” stochastic process:

Journey to the Mathematics of Finance Black-Scholes lognormal model The Black-Scholes model assumes as well, that if you deposit 1$ to a bank account, the value of your bank account B(t) (cash-bond) follows: where r is the interest rate.

Journey to the Mathematics of Finance There are two approaches to solve the problem based on the arbitrage-free assumption. PDE approach One can prove that the price of the option C(t,S(t)) satisfies the following Partial Differential Equation (PDE): With terminal boundary condition: C(T,ST) = max (ST-K,0) So calculating the price of the option boils down to solving a deterministic PDE (which can be analytically solved in this context).

Journey to the Mathematics of Finance Martingale approach The price of the option is itself a random variable, and more specifically a stochastic process. One can show that there is a probability measure under which the “discounted” price of the option C(t)/B(t) is a martingale and hence is value today (t = 0) satisfies: where E stands here for the mathematical expectation of a random variable. Given that B(t) is deterministic (not random), C(T) = Max(ST – K,0) and the fact that the distribution probability of the random variable ST can be derived from the Black-Schles model, the option price can be calculated.

Journey to the Mathematics of Finance Black-Scholes formula for a European call option The two approaches give the same formula (known as the Black-Scholes formula) : where S is today’s stock price, Φ is the inverse normal cumulative function and T is the time to expiry.

Journey to the Mathematics of Finance Conclusion The fact that the two approaches lead to the same result is not coincidence. This results from a deep and central theorem of the mathematical finance, known as the Feymann-Kac theorem, that relates expectation of stochastic process to solutions of (deterministic) partial differential equation. Pricing financial derivatives often leads to solving PDE’s or calculating expectation of stochastic process.

Career Opportunities at Brevan Howard Brevan Howard offers challenging and highly remunerating career opportunities for graduates, PhD and PostDoc students with strong scientific skills and education. Quantitative Analysts (“Quants”) work on: Pricing and hedging tools Sophisticated models are often used to valuate financial products and to “hedge”. Hedging – reducing the risk exposure – plays a central role in a hedge fund. Mathematical stochastic models are applied to price financial products and to predict the impact of various scenarios (increase in interest rate, increase in volatility, etc.) on the portfolios. Using those risk scenarios, portfolios can be readjusted to reduce exposure to losses.

Career Opportunities Systematic trading tools Of the most promising areas of trading. Mathematical models are developed and applied to actively trade in the markets, using algorithms that make automatic trading decisions based on for example signal processing/pattern recognition, statistical arbitrage, technical (eg trend following) and fundamental data analysis. Relative value tools Traders use quantitative tools to monitor and take relative value positions between various financial instruments.

Career Opportunities General Job requirements PhD or MS (mandatory) in Physics, Mathematics, Computer science or any related scientific discipline. Strong mathematical and problem solving skills (mandatory) Experience in C/C++ (mandatory) Experience in applied and numerical mathematics (preferred) : numerical solutions for PDEs, optimizations, etc. Experience with stochastic process and probabilities (preferred). Good English verbal and written communication skills (mandatory).

Contact information Sari Lorber Brevan Howard (Israel) Ltd. Sari.lorber@brevanhoward.com 03-576-8400